Statistics – O. R. 891 Object Oriented Data Analysis J. S. Marron Dept. of Statistics and...

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Statistics ndash O R 891Object Oriented Data Analysis

J S Marron

Dept of Statistics and Operations Research

University of North Carolina

Directional Data

Eg ldquoaveragerdquo of =

Should

Use

Unit

Circle

Structure

Manifold Feature Spaces

xx

xx

Freacutechet Mean in General

Big Advantage

Works in Any Metric Space

Manifold Feature Spaces

n

ii

xxXdX

1

2minarg

Directional Data Examples of Freacutechet Meanbull Not always easily interpretable

ndash Think about distances along arcndash Not about ldquopoints in rdquondash Sum of squared distances

strongly feels the largestbull Not always unique

ndash But unique with probability one ndash Non-unique requires strong symmetryndash But possible to have many means

Manifold Feature Spaces

Directional Data Examples of Freacutechet Meanbull Also of interest is Freacutechet Variance

bull Works like Euclidean sample variancebull Note values in movie reflecting spread in databull Note theoretical version

bull Useful for Laws of Large Numbers etc

Manifold Feature Spaces

22 min xXdEXx

n

iixxXd

n 1

22 1

min

OODA in Image Analysis

First Generation Problems

bull Denoising

bull Segmentation

bull Registration

(all about single images

still interesting challenges)

OODA in Image Analysis

Second Generation Problems

bull Populations of Images

ndash Understanding Population Variation

ndash Discrimination (aka

Classification)

bull Complex Data Structures (amp Spaces)

bull HDLSS Statistics

Image Object Representation

Major Approaches for Image Data

Objects

bull Landmark Representations

bull Boundary Representations

bull Medial Representations

Boundary Representations

Main Drawback

Correspondence

bull For OODA (on vectors of parameters)

Need to ldquomatch up pointsrdquo

bull Easy to find triangular meshndash Lots of research on this driven by

gamers

bull Challenge to match mesh across objectsndash There are some interesting ideashellip

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

Directional Data

Eg ldquoaveragerdquo of =

Should

Use

Unit

Circle

Structure

Manifold Feature Spaces

xx

xx

Freacutechet Mean in General

Big Advantage

Works in Any Metric Space

Manifold Feature Spaces

n

ii

xxXdX

1

2minarg

Directional Data Examples of Freacutechet Meanbull Not always easily interpretable

ndash Think about distances along arcndash Not about ldquopoints in rdquondash Sum of squared distances

strongly feels the largestbull Not always unique

ndash But unique with probability one ndash Non-unique requires strong symmetryndash But possible to have many means

Manifold Feature Spaces

Directional Data Examples of Freacutechet Meanbull Also of interest is Freacutechet Variance

bull Works like Euclidean sample variancebull Note values in movie reflecting spread in databull Note theoretical version

bull Useful for Laws of Large Numbers etc

Manifold Feature Spaces

22 min xXdEXx

n

iixxXd

n 1

22 1

min

OODA in Image Analysis

First Generation Problems

bull Denoising

bull Segmentation

bull Registration

(all about single images

still interesting challenges)

OODA in Image Analysis

Second Generation Problems

bull Populations of Images

ndash Understanding Population Variation

ndash Discrimination (aka

Classification)

bull Complex Data Structures (amp Spaces)

bull HDLSS Statistics

Image Object Representation

Major Approaches for Image Data

Objects

bull Landmark Representations

bull Boundary Representations

bull Medial Representations

Boundary Representations

Main Drawback

Correspondence

bull For OODA (on vectors of parameters)

Need to ldquomatch up pointsrdquo

bull Easy to find triangular meshndash Lots of research on this driven by

gamers

bull Challenge to match mesh across objectsndash There are some interesting ideashellip

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

Freacutechet Mean in General

Big Advantage

Works in Any Metric Space

Manifold Feature Spaces

n

ii

xxXdX

1

2minarg

Directional Data Examples of Freacutechet Meanbull Not always easily interpretable

ndash Think about distances along arcndash Not about ldquopoints in rdquondash Sum of squared distances

strongly feels the largestbull Not always unique

ndash But unique with probability one ndash Non-unique requires strong symmetryndash But possible to have many means

Manifold Feature Spaces

Directional Data Examples of Freacutechet Meanbull Also of interest is Freacutechet Variance

bull Works like Euclidean sample variancebull Note values in movie reflecting spread in databull Note theoretical version

bull Useful for Laws of Large Numbers etc

Manifold Feature Spaces

22 min xXdEXx

n

iixxXd

n 1

22 1

min

OODA in Image Analysis

First Generation Problems

bull Denoising

bull Segmentation

bull Registration

(all about single images

still interesting challenges)

OODA in Image Analysis

Second Generation Problems

bull Populations of Images

ndash Understanding Population Variation

ndash Discrimination (aka

Classification)

bull Complex Data Structures (amp Spaces)

bull HDLSS Statistics

Image Object Representation

Major Approaches for Image Data

Objects

bull Landmark Representations

bull Boundary Representations

bull Medial Representations

Boundary Representations

Main Drawback

Correspondence

bull For OODA (on vectors of parameters)

Need to ldquomatch up pointsrdquo

bull Easy to find triangular meshndash Lots of research on this driven by

gamers

bull Challenge to match mesh across objectsndash There are some interesting ideashellip

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

Directional Data Examples of Freacutechet Meanbull Not always easily interpretable

ndash Think about distances along arcndash Not about ldquopoints in rdquondash Sum of squared distances

strongly feels the largestbull Not always unique

ndash But unique with probability one ndash Non-unique requires strong symmetryndash But possible to have many means

Manifold Feature Spaces

Directional Data Examples of Freacutechet Meanbull Also of interest is Freacutechet Variance

bull Works like Euclidean sample variancebull Note values in movie reflecting spread in databull Note theoretical version

bull Useful for Laws of Large Numbers etc

Manifold Feature Spaces

22 min xXdEXx

n

iixxXd

n 1

22 1

min

OODA in Image Analysis

First Generation Problems

bull Denoising

bull Segmentation

bull Registration

(all about single images

still interesting challenges)

OODA in Image Analysis

Second Generation Problems

bull Populations of Images

ndash Understanding Population Variation

ndash Discrimination (aka

Classification)

bull Complex Data Structures (amp Spaces)

bull HDLSS Statistics

Image Object Representation

Major Approaches for Image Data

Objects

bull Landmark Representations

bull Boundary Representations

bull Medial Representations

Boundary Representations

Main Drawback

Correspondence

bull For OODA (on vectors of parameters)

Need to ldquomatch up pointsrdquo

bull Easy to find triangular meshndash Lots of research on this driven by

gamers

bull Challenge to match mesh across objectsndash There are some interesting ideashellip

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

Directional Data Examples of Freacutechet Meanbull Also of interest is Freacutechet Variance

bull Works like Euclidean sample variancebull Note values in movie reflecting spread in databull Note theoretical version

bull Useful for Laws of Large Numbers etc

Manifold Feature Spaces

22 min xXdEXx

n

iixxXd

n 1

22 1

min

OODA in Image Analysis

First Generation Problems

bull Denoising

bull Segmentation

bull Registration

(all about single images

still interesting challenges)

OODA in Image Analysis

Second Generation Problems

bull Populations of Images

ndash Understanding Population Variation

ndash Discrimination (aka

Classification)

bull Complex Data Structures (amp Spaces)

bull HDLSS Statistics

Image Object Representation

Major Approaches for Image Data

Objects

bull Landmark Representations

bull Boundary Representations

bull Medial Representations

Boundary Representations

Main Drawback

Correspondence

bull For OODA (on vectors of parameters)

Need to ldquomatch up pointsrdquo

bull Easy to find triangular meshndash Lots of research on this driven by

gamers

bull Challenge to match mesh across objectsndash There are some interesting ideashellip

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

OODA in Image Analysis

First Generation Problems

bull Denoising

bull Segmentation

bull Registration

(all about single images

still interesting challenges)

OODA in Image Analysis

Second Generation Problems

bull Populations of Images

ndash Understanding Population Variation

ndash Discrimination (aka

Classification)

bull Complex Data Structures (amp Spaces)

bull HDLSS Statistics

Image Object Representation

Major Approaches for Image Data

Objects

bull Landmark Representations

bull Boundary Representations

bull Medial Representations

Boundary Representations

Main Drawback

Correspondence

bull For OODA (on vectors of parameters)

Need to ldquomatch up pointsrdquo

bull Easy to find triangular meshndash Lots of research on this driven by

gamers

bull Challenge to match mesh across objectsndash There are some interesting ideashellip

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

OODA in Image Analysis

Second Generation Problems

bull Populations of Images

ndash Understanding Population Variation

ndash Discrimination (aka

Classification)

bull Complex Data Structures (amp Spaces)

bull HDLSS Statistics

Image Object Representation

Major Approaches for Image Data

Objects

bull Landmark Representations

bull Boundary Representations

bull Medial Representations

Boundary Representations

Main Drawback

Correspondence

bull For OODA (on vectors of parameters)

Need to ldquomatch up pointsrdquo

bull Easy to find triangular meshndash Lots of research on this driven by

gamers

bull Challenge to match mesh across objectsndash There are some interesting ideashellip

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

Image Object Representation

Major Approaches for Image Data

Objects

bull Landmark Representations

bull Boundary Representations

bull Medial Representations

Boundary Representations

Main Drawback

Correspondence

bull For OODA (on vectors of parameters)

Need to ldquomatch up pointsrdquo

bull Easy to find triangular meshndash Lots of research on this driven by

gamers

bull Challenge to match mesh across objectsndash There are some interesting ideashellip

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

Boundary Representations

Main Drawback

Correspondence

bull For OODA (on vectors of parameters)

Need to ldquomatch up pointsrdquo

bull Easy to find triangular meshndash Lots of research on this driven by

gamers

bull Challenge to match mesh across objectsndash There are some interesting ideashellip

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

Medial RepresentationsMain Idea Represent Objects asbull Discretized skeletons (medial atoms)bull Plus spokes from center to edgebull Which imply a boundary

Very accessible early referencebull Yushkevich et al (2001)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

A Challenging Examplebull Male Pelvis

ndash Bladder ndash Prostate ndash Rectumndash How do they move over time (days)ndash Critical to Radiation Treatment (cancer)

bull Work with 3-d CTndash Very Challenging to Segmentndash Find boundary of each objectndash Represent each Object

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

Male Pelvis ndash Raw Data

One CT Slice

(in 3d image)

Tail Bone

Rectum

Bladder

Prostate

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

13

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

14

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

15

UNC Stat amp OR

PGA for m-reps Bladder-Prostate-Rectum

Bladder ndash Prostate ndash Rectum 1 person 17 days

PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

16

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

17

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

Counterexample

Data on sphere along equator

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

18

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

19

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

Counterexample

Data follows Tropic of Capricorn

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

20

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

21

UNC Stat amp OR

PCA Extensions for Data on Manifolds

bull Fletcher (Principal Geodesic Anal)bull Best fit of geodesic to data

bull Constrained to go through geodesic mean

bull Huckemann Hotz amp Munk (Geod PCA)bull Best fit of any geodesic to data

bull Jung Foskey amp Marron (Princ Arc Anal)bull Best fit of any circle to data

(motivated by conformal maps)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

22

UNC Stat amp OR

PCA Extensions for Data on Manifolds

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

23

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

24

UNC Stat amp OR

Principal Arc Analysis

Jung Foskey amp Marron (2011)bull Best fit of any circle to data

bull Can give better fit than geodesics

bull Observed for simulated m-rep example

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

25

UNC Stat amp OR

Challenge being addressed

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

26

UNC Stat amp OR

Composite Nested Spheres

Idea Use Principal Arc Analysis

Over Large Products of

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

27

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

(recall major monographs)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

28

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

29

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

30

UNC Stat amp OR

Landmark Based Shape Analysis

bull Kendall

bull Bookstein

bull Dryden amp

Mardia

Digit 3 Data

(digitized to 13 landmarks)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

31

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

32

UNC Stat amp OR

Landmark Based Shape Analysis

Key Step mod out

bull Translation

bull Scaling

bull Rotation

Result

Data Objects

points on Manifold ( ~ S2k-

4)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

33

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections

(Tangent Plane Analysis)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

34

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

35

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

36

UNC Stat amp OR

Landmark Based Shape Analysis

Currently popular approaches to PCA on Sk

Early PCA on projections Fletcher Geodesics through mean Huckemann et al Any Geodesic

New Approach

Principal Nested Sphere Analysis

Jung Dryden amp Marron (2012)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

37

UNC Stat amp OR

Principal Nested Spheres Analysis

Main Goal

Extend Principal Arc Analysis (S2 to Sk)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

38

UNC Stat amp OR

Principal Nested Spheres Analysis

Key Idea

Replace usual forwards view of PCA

With a backwards approach to PCA

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

39

UNC Stat amp OR

Terminology

Multiple linear regression

ikkiii xxxY 2211

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

40

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model

ikkiii xxxY 2211

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

41

UNC Stat amp OR

Terminology

Multiple linear regression

Stepwise approaches Forwards Start small iteratively

add variables to model Backwards Start with all

iteratively remove variables from model

ikkiii xxxY 2211

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

42

UNC Stat amp OR

Illustrsquon of PCA View Recall Raw Data

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

43

UNC Stat amp OR

Illustrsquon of PCA View PC1 Projections

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

44

UNC Stat amp OR

Illustrsquon of PCA View PC2 Projections

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

45

UNC Stat amp OR

Illustrsquon of PCA View Projections on PC12 plane

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

46

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

47

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

48

UNC Stat amp OR

Principal Nested Spheres Analysis

Replace usual forwards view of PCA

Data PC1 (1-d approx)

PC2 (1-d approx of Data-PC1)

PC1 U PC2 (2-d approx)

PC1 U hellip U PCr

(r-d approx)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

49

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

50

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

51

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

52

UNC Stat amp OR

Principal Nested Spheres Analysis

With a backwards approach to PCA

Data PC1 U hellip U PCr (r-d approx)

PC1 U hellip U PC(r-1)

PC1 U PC2 (2-d approx)

PC1 (1-d approx)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

53

UNC Stat amp OR

Principal Nested Spheres Analysis

Top Down Nested (small) spheres

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

54

UNC Stat amp OR

Digit 3 data Principal variations of shape

Princ geodesics by PNS Principal arcs by PNS

>
>
>
>

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

55

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Discussion

Jung et al (2010)

Pizer et al (2012)

An Interesting Question

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

56

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Anywhere this is already being done

An Interesting Question

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

57

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Hastie amp Stuetzle (1989)

(Foundation of Manifold Learning)

An Interesting Question

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

58

UNC Stat amp OR

Goal Find lower dimensional manifold that well approximates data

ISOmap

Tennenbaum (2000)

Local Linear Embedding

Roweis amp Saul (2000)

Manifold Learning

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

59

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Usual Smooth

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

60

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

61

UNC Stat amp OR

1st Principal Curve

Linear Regrsquon

Projrsquos Regrsquon

Usual Smooth

Princrsquol Curve

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

62

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Potential Application Principal Curves

Perceived Major Challenge

How to find 2nd Principal Curve

Backwards approach

An Interesting Question

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

63

UNC Stat amp OR

Key Component

Principal Surfaces

LeBlanc amp Tibshirani (1994)

An Interesting Question

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

64

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Nonnegative Matrix Factorization

= PCA in Positive Orthant

(early days)

An Interesting Question

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

65

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Nonnegative Matrix Factorization

Discussed in Guest Lecture

Tuesday November 13

Lingsong Zhang Thanks to statpurdueedu

An Interesting Question

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

66

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

Another Potential Application

Trees as Data

(early days)

An Interesting Question

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

67

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

An Interesting Question

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

68

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Geometry

Singularity

Theory

An Interesting Question

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

69

UNC Stat amp OR

How generally applicable is

Backwards approach to PCA

An Attractive Answer

James Damon UNC Mathematics

Key Idea Express Backwards PCA as

Nested Series of Constraints

An Interesting Question

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

70

UNC Stat amp OR

Define Nested Spaces via Constraints

Satisfying More Constraints

Smaller Subspaces

General View of Backwards PCA

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

71

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

(Singular Value Decomposition =

= Not Mean Centered PCA)

(notationally very clean)

General View of Backwards PCA

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

72

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Have Nested Subspaces

General View of Backwards PCA

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

73

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

-th SVD Subspace

Scores

Loading Vectors

General View of Backwards PCA

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

74

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

General View of Backwards PCA

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

75

UNC Stat amp OR

Define Nested Spaces via Constraints

Eg SVD

Now Define

Constraint Gives Nested Reduction of Dimrsquon

General View of Backwards PCA

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

76

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

Reduce Using Affine Constraints

General View of Backwards PCA

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

77

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

Use Affine Constraints (Planar Slices)

In Ambient Space

General View of Backwards PCA

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

78

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

General View of Backwards PCA

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

79

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

Spline Constraint Within Previous

Been Done Already

General View of Backwards PCA

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

80

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

Sub-Manifold Constraints

(Algebraic Geometry)

General View of Backwards PCA

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

81

UNC Stat amp OR

Define Nested Spaces via Constraints

bull Backwards PCA

bull Principal Nested Spheres

bull Principal Surfaces

bull Other Manifold Data Spaces

bull Tree Spaces

Suitable Constraints

General View of Backwards PCA

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

82

UNC Stat amp OR

Why does Backwards Work Better

General View of Backwards PCA

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

83

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

(Ie Add Constraints

Using Information in Data)

General View of Backwards PCA

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

84

UNC Stat amp OR

Why does Backwards Work Better

Natural to Sequentially Add Constraints

Hard to Start With Complete Set

And Sequentially Remove

General View of Backwards PCA

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

85

UNC Stat amp OR

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

86

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea Represent Shapes as Coordinates ldquoMod Outrdquo Translrsquon Rotatrsquon Scale

Variation on Landmark Based Shape

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

87

UNC Stat amp OR

Typical Viewpoint Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

88

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Interesting Alternative Study Variation in Transformation Treat Shape as Nuisance

Variation on Landmark Based Shape

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

89

UNC Stat amp OR

Context Study of Tectonic Plates

bull Movement of Earthrsquos Crust (over time)

bull Take Motions as Data Objects

Royer amp Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

90

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

91

UNC Stat amp OR

Non - Euclidean Data Spaces

What is ldquoStrongly Non-Euclideanrdquo Case

Trees as Data

Special Challenge

bull No Tangent Plane

bull Must Re-Invent

Data Analysis

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

92

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

93

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edges

Thanks to Burcu Aydin

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

94

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Thanks to Burcu Aydin

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

95

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory

bull Graph is set of nodes and edgesbull Tree has root and direction

Data Objects set of treesThanks to Burcu Aydin

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

96

UNC Stat amp OR

Strongly Non-Euclidean Spaces

General Graph

Thanks to Sean Skwerer

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

97

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Special Case Called ldquoTreerdquo

bull Directed

bull Acyclic

5

43

21

0

Graphical note

Sometimes ldquogrow

uprdquo

Others ldquogrow downrdquo

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

98

UNC Stat amp OR

Strongly Non-Euclidean Spaces

Motivating Example

bull From Dr Elizabeth Bullittbull Dept of Neurosurgery UNC

bull Blood Vessel Trees in Brains

bull Segmented from MRAs

bull Study population of trees

Forest of Trees

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

99

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRI view

Single Slice

From 3-d Image

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

100

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

101

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

102

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

103

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

104

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

105

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

MRA view

ldquoArdquo for

ldquoAngiographyrdquo

Finds blood

vessels

(show up as white)

Track through 3d

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

106

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

107

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

108

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

109

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

110

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

111

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

112

UNC Stat amp OR

Blood vessel tree data

Marronrsquos brain

From MRA

Reconstruct trees

in 3d

Rotate to view

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

113

UNC Stat amp OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation)

PCA in strongly non-Euclidean Space

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

114

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

115

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

116

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

117

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

118

UNC Stat amp OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

bull Predict Stroke Tendency (Collateral

Circulation)

bull Screen for Loci of Pathology

bull Explore how age affects connectivity

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

119

UNC Stat amp OR

Blood vessel tree data

Big Picture 3 Approaches

1Purely Combinatorial

2Euclidean Orthant

3Dyck Path

  • Statistics ndash O R 891 Object Oriented Data Analysis
  • Manifold Feature Spaces
  • Manifold Feature Spaces (2)
  • Manifold Feature Spaces (3)
  • Manifold Feature Spaces (4)
  • OODA in Image Analysis
  • OODA in Image Analysis (2)
  • Image Object Representation
  • Boundary Representations
  • Medial Representations
  • A Challenging Example
  • Male Pelvis ndash Raw Data
  • PGA for m-reps Bladder-Prostate-Rectum
  • PGA for m-reps Bladder-Prostate-Rectum (2)
  • PGA for m-reps Bladder-Prostate-Rectum (3)
  • PCA Extensions for Data on Manifolds
  • PCA Extensions for Data on Manifolds (2)
  • PCA Extensions for Data on Manifolds (3)
  • PCA Extensions for Data on Manifolds (4)
  • PCA Extensions for Data on Manifolds (5)
  • PCA Extensions for Data on Manifolds (6)
  • PCA Extensions for Data on Manifolds (7)
  • Principal Arc Analysis
  • Principal Arc Analysis (2)
  • Challenge being addressed
  • Composite Nested Spheres
  • Landmark Based Shape Analysis
  • Landmark Based Shape Analysis (2)
  • Landmark Based Shape Analysis (3)
  • Landmark Based Shape Analysis (4)
  • Landmark Based Shape Analysis (5)
  • Landmark Based Shape Analysis (6)
  • Landmark Based Shape Analysis (7)
  • Landmark Based Shape Analysis (8)
  • Landmark Based Shape Analysis (9)
  • Landmark Based Shape Analysis (10)
  • Principal Nested Spheres Analysis
  • Principal Nested Spheres Analysis (2)
  • Terminology
  • Terminology (2)
  • Terminology (3)
  • Illustrsquon of PCA View Recall Raw Data
  • Illustrsquon of PCA View PC1 Projections
  • Illustrsquon of PCA View PC2 Projections
  • Illustrsquon of PCA View Projections on PC12 plane
  • Principal Nested Spheres Analysis (3)
  • Principal Nested Spheres Analysis (4)
  • Principal Nested Spheres Analysis (5)
  • Principal Nested Spheres Analysis (6)
  • Principal Nested Spheres Analysis (7)
  • Principal Nested Spheres Analysis (8)
  • Principal Nested Spheres Analysis (9)
  • Principal Nested Spheres Analysis (10)
  • Digit 3 data Principal variations of shape
  • An Interesting Question
  • An Interesting Question (2)
  • An Interesting Question (3)
  • Manifold Learning
  • 1st Principal Curve
  • 1st Principal Curve (2)
  • 1st Principal Curve (3)
  • An Interesting Question (4)
  • An Interesting Question (5)
  • An Interesting Question (6)
  • An Interesting Question (7)
  • An Interesting Question (8)
  • An Interesting Question (9)
  • An Interesting Question (10)
  • An Interesting Question (11)
  • General View of Backwards PCA
  • General View of Backwards PCA (2)
  • General View of Backwards PCA (3)
  • General View of Backwards PCA (4)
  • General View of Backwards PCA (5)
  • General View of Backwards PCA (6)
  • General View of Backwards PCA (7)
  • General View of Backwards PCA (8)
  • General View of Backwards PCA (9)
  • General View of Backwards PCA (10)
  • General View of Backwards PCA (11)
  • General View of Backwards PCA (12)
  • General View of Backwards PCA (13)
  • General View of Backwards PCA (14)
  • General View of Backwards PCA (15)
  • Variation on Landmark Based Shape
  • Variation on Landmark Based Shape (2)
  • Variation on Landmark Based Shape (3)
  • Variation on Landmark Based Shape (4)
  • Variation on Landmark Based Shape (5)
  • Non - Euclidean Data Spaces
  • Non - Euclidean Data Spaces (2)
  • Strongly Non-Euclidean Spaces
  • Strongly Non-Euclidean Spaces (2)
  • Strongly Non-Euclidean Spaces (3)
  • Strongly Non-Euclidean Spaces (4)
  • Strongly Non-Euclidean Spaces (5)
  • Strongly Non-Euclidean Spaces (6)
  • Strongly Non-Euclidean Spaces (7)
  • Blood vessel tree data
  • Blood vessel tree data (2)
  • Blood vessel tree data (3)
  • Blood vessel tree data (4)
  • Blood vessel tree data (5)
  • Blood vessel tree data (6)
  • Blood vessel tree data (7)
  • Blood vessel tree data (8)
  • Blood vessel tree data (9)
  • Blood vessel tree data (10)
  • Blood vessel tree data (11)
  • Blood vessel tree data (12)
  • Blood vessel tree data (13)
  • Blood vessel tree data (14)
  • Blood vessel tree data (15)
  • Blood vessel tree data (16)
  • Blood vessel tree data (17)
  • Blood vessel tree data (18)
  • Blood vessel tree data (19)
  • Blood vessel tree data (20)
  • Blood vessel tree data (21)

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