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
