1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space...

UNC, Stat & OR

Metrics in Curve Space

Note on SRVF representation:

Can show: Warp Invariance

Follows from Jacobean calculation

UNC, Stat & OR

Metrics in Curve Quotient Space

Above was Invariance for Individual

Curves

Now extend to:

Equivalence Classes of Curves

I.e. Orbits as Data Objects

I.e. Quotient Space

UNC, Stat & OR

More Data Objects

Final Curve Warps:

• Warp Each Data Curve,

• To Template Mean,

• Denote Warp Functions

Gives (Roughly Speaking):

Vertical Components

(Aligned Curves)

Horizontal Components

UNC, Stat & OR

More Data Objects

Final Curve Warps:

Vertical Components

(Aligned Curves)

Data Objects II

~ Kendall’s Shapes

UNC, Stat & OR

More Data Objects

Final Curve Warps:

Vertical Components

(Aligned Curves)

Data Objects III

~ Chang’s Transfo’s

UNC, Stat & OR

Toy Example

ConventionalPCAProjections

PowerSpreadAcrossSpectrum

UNC, Stat & OR

Toy Example

ConventionalPCAScores

Views of1-d CurveBendingThrough4 Dim’ns’

UNC, Stat & OR

Toy Example

ConventionalPCAScores

PatternsAre“Harmonics”In Scores

UNC, Stat & OR

Toy Example

Scores PlotShows DataAre “1”Dimensional

So NeedImprovedPCA Decomp.

UNC, Stat & OR

Toy Example

AlignedCurvePCAProjections

All Var’nIn 1st

Component

UNC, Stat & OR

Toy Example

Warps,PCProjections

Mostly1st PC,But 2nd

Helps Some

UNC, Stat & OR

Toy Example

WarpCompon’ts(+ Mean)Applied toTemplateMean

UNC, Stat & OR

PNS on SRVF Sphere

Toy Example

Tangent Space

(on Horiz. Var’n)

Thanks to Xiaosun Lu

UNC, Stat & OR

PNS on SRVF Sphere

Toy Example

PNS Projections

(Fewer Modes)

UNC, Stat & OR

PNS on SRVF Sphere

Toy Example

Tangent Space

Note: 3 Comp’s

Needed for This

UNC, Stat & OR

PNS on SRVF Sphere

Toy Example

PNS Projections

Only 2 for This

UNC, Stat & OR

TIC testbed

Special Feature: Answer Key of Known Peaks

Goal:FindWarpsTo AlignThese

UNC, Stat & OR

TIC testbed

Fisher – Rao Alignment

UNC, Stat & OR

Non - Euclidean Data Spaces

What is “Strongly Non-Euclidean” Case?

Trees as Data

UNC, Stat & OR

Non - Euclidean Data Spaces

What is “Strongly Non-Euclidean” Case?

Trees as Data

Special Challenge:

• No Tangent Plane

• Must Re-Invent

Data Analysis

UNC, Stat & OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

UNC, Stat & OR

From Graph Theory:

• Graph is set of nodes and edges

UNC, Stat & OR

From Graph Theory:

• Graph is set of nodes and edges• Tree has root and direction

UNC, Stat & OR

From Graph Theory:

• Graph is set of nodes and edges• Tree has root and direction

Data Objects: set of treesThanks to Burcu Aydin

UNC, Stat & OR

General Graph:

Thanks to Sean Skwerer

UNC, Stat & OR

Special Case Called “Tree”

• Directed

• Acyclic

UNC, Stat & OR

• Directed

• Acyclic

Graphical note:

Sometimes “grow

Others “grow down”

UNC, Stat & OR

• Directed

• Acyclic

0 Terminology:

UNC, Stat & OR

• Directed

• Acyclic

Terminology:

Children

Parent

UNC, Stat & OR

Motivating Example:

• From Dr. Elizabeth Bullitt• Dept. of Neurosurgery, UNC

UNC, Stat & OR

Motivating Example:

• Blood Vessel Trees in Brains

• Segmented from MRAs

UNC, Stat & OR

Motivating Example:

• Blood Vessel Trees in Brains

• Segmented from MRAs

• Study population of trees

Forest of Trees

UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

MRI (T1) view

Single Slice

From 3-d Image

UNC, Stat & OR

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

UNC, Stat & OR

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

UNC, Stat & OR

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

UNC, Stat & OR

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

UNC, Stat & OR

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

UNC, Stat & OR

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

UNC, Stat & OR

Marron’s brain:

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

UNC, Stat & OR

Marron’s brain:

From MRA

Reconstruct trees

Rotate to view

UNC, Stat & OR

Marron’s brain:

From MRA

Reconstruct trees

Rotate to view

UNC, Stat & OR

Marron’s brain:

From MRA

Reconstruct trees

Rotate to view

UNC, Stat & OR

Marron’s brain:

From MRA

Reconstruct trees

Rotate to view

UNC, Stat & OR

Marron’s brain:

From MRA

Reconstruct trees

Rotate to view

UNC, Stat & OR

Marron’s brain:

From MRA

Reconstruct trees

Rotate to view

UNC, Stat & OR

Now look over many people (data

objects)

Structure of population (understand

variation?)

PCA in strongly non-Euclidean Space???

, ... ,,

UNC, Stat & OR

Examples of Potential Specific Goals

, ... ,,

UNC, Stat & OR

(not accessible by traditional methods)

• Predict Stroke Tendency (Collateral

Circulation)

, ... ,,

UNC, Stat & OR

Circulation)

• Screen for Loci of Pathology

, ... ,,

UNC, Stat & OR

Circulation)

• Explore how age affects connectivity

, ... ,,

UNC, Stat & OR

Circulation)

• Explore how age affects connectivity

, ... ,,

UNC, Stat & OR

Big Picture: 4 Approaches

1. Purely Combinatorial

2. Euclidean Orthant (Phylogenetics)

3. Dyck Path

4. Persistent Homologies Topological

UNC, Stat & OR

3. Dyck Path

UNC, Stat & OR

Purely Combinatorial Data Analyses

(Study Connectivity Only)

Wang and Marron (2007)

Aydin et al (2009)

Wang et al (2012)

Aydin et al (2012)

Alfaro et al (2014)

UNC, Stat & OR

D-L Visualization of Trees

Challenge:VisualDisplay ofFull TreeStructure

UNC, Stat & OR

Challenge:VisualDisplay ofFull TreeStructure

Actually Goes toLevel 17(Truncated in View)

UNC, Stat & OR

Approach: Different Coordinate System

Aydin, et al (2011)

UNC, Stat & OR

Idea: Focus on Important Aspects (of

Nodes)

• Level of Node

• Number of Children

UNC, Stat & OR

Idea: Focus on Important Aspects (of

Nodes)

• Level of Node

• Number of Children

Using These as Coordinates

UNC, Stat & OR

D-LView

UNC, Stat & OR

D-LView

UNC, Stat & OR

D-LView

UNC, Stat & OR

D-LView

MuchDeeper ThanEarlyView

UNC, Stat & OR

D-LView

# Des-cend-ants(logscale)

UNC, Stat & OR

D-LView

ColorCodesBranchThick-ness

UNC, Stat & OR

D-LView

RevealsStrangeStruc-ture

UNC, Stat & OR

D-LView

RevealsStrangeStruc-ture

LinkingErrors

UNC, Stat & OR

D-LView

FixedVersion

UNC, Stat & OR

2. Euclidean Orthant

(Phylogenetics)

3. Dyck Path

UNC, Stat & OR

Euclidean Orthant Approach

People:

• Scott Provan

• Sean Skwerer

• Megan Owen

• Ezra Miller

• Martin Styner

• Ipek Oguz

UNC, Stat & OR

Setting: Connectivity & Length

UNC, Stat & OR

Background: Phylogenetic Trees

UNC, Stat & OR

Important Concept from Evolutionary Biology

UNC, Stat & OR

Phylogenetic Trees

Idea: Study

“Common

Ancestry”

Via a tree

Species are leaves

thanks to Susan Holmes

UNC, Stat & OR

Phylogenetic Trees

Very Early Reference:

E. Schröder (1870),

Zeit. für. Math. Phys.,

15, 361-376.

thanks to Susan Holmes

UNC, Stat & OR

Phylogenetic Trees

Important Reference:

Billera L, Holmes S, & Vogtmann K (2001)

UNC, Stat & OR

Phylogenetic Trees

Important Reference:

Billera L, Holmes S, & Vogtmann K (2001)

Put Large Field on Firm Mathematical Basis

UNC, Stat & OR

Major Restriction: Need common leaves

UNC, Stat & OR

Big Payoff: Data space nearly Euclidean

sort of Euclidean

UNC, Stat & OR

Approach:

• Find common cortical landmarks (Oguz)

corresponding across cases

UNC, Stat & OR

Approach:

Oguz (2009)

UNC, Stat & OR

Approach:

• Treat as pseudo – leaves

by projecting to points on tree

UNC, Stat & OR

Labeled n-Trees

5-tree

e.g. n = 5

UNC, Stat & OR

Labeled n-Trees

leaves - fixed a priori

5-tree

UNC, Stat & OR

Labeled n-Trees

leaves - fixed a priori

root5-tree

UNC, Stat & OR

Labeled n-Trees

leaves - fixed a priori labeled {0,1, . . . ,n}

root5-tree

UNC, Stat & OR

Labeled n-Trees

Internal (nonleaf) vertices degree ≥ 3

root5-tree

UNC, Stat & OR

Labeled n-Trees

# edges from node

root5-tree

UNC, Stat & OR

Labeled n-Trees

Internal (nonleaf) vertices degree ≥ 3(note: not labelled)

root5-tree

UNC, Stat & OR

Labeled n-Trees

edge e has nonneg. length

|e1|=3

|e2|=4

|e3|=6

5-tree

UNC, Stat & OR

Labeled n-Trees

Note: To study‘topology’,(i.e. tree structure)

|e1|=3

|e2|=4

|e3|=6

5-tree

UNC, Stat & OR

Labeled n-Trees

Note: To study‘topology’,(i.e. tree structure)

Enough to consideronly lengths of internal edges

|e1|=3

|e2|=4

|e3|=6

5-tree

UNC, Stat & OR

Labeled n-Trees

Terminology:Leaf edges called ‘pendants’

(Care about lengths of pendants in brain arteries)

|e1|=3

|e2|=4

|e3|=6

5-tree

UNC, Stat & OR

Toy Examples

Same tree, since same internal edges

UNC, Stat & OR

Toy Examples

Different tree, since different connections

UNC, Stat & OR

Toy Examples

A valid tree, called “Star tree” or “0 tree” (since all internal edge lengths are 0)

UNC, Stat & OR

Tree Space Examples, T-4

𝐸𝑑𝑔𝑒 { 𝐼 𝑛𝑖𝑓 |𝑒|>0𝑂𝑢𝑡 𝑖𝑓 |𝑒|=0

Set of mutually compatible splits tree

Thanks to Megan Owen

UNC, Stat & OR

Three quadrants meeting at common axis

UNC, Stat & OR

Star Tree = 0 Tree

UNC, Stat & OR

Star Tree(point)

UNC, Stat & OR

Star Tree(point)

1-edge Trees(line)

UNC, Stat & OR

Star Tree(point)

1-edge Trees(line)

2-edge Trees(planes)

UNC, Stat & OR

MathematicalConstruct:

Manifold StratifiedSpace

UNC, Stat & OR

Manifolds (planes)of Different

Dimensions Glued Together

UNC, Stat & OR

‘Connectivity’ of T-4

UNC, Stat & OR

Cone Structure

(Reflects edge lengths > 0)

UNC, Stat & OR

Cone Structure Star (0) Tree

(At Origin)

UNC, Stat & OR

Cone Structure Star (0) Tree Single Edge

(On Boundary Lines)

UNC, Stat & OR

Cone Structure Star (0) Tree Single Edge

Trees Full (2 Edge)

(On Planes)

UNC, Stat & OR

Each LineConnects to 3 planes(# compatible

edges)

UNC, Stat & OR

Each LineConnects to 3 planes(# compatible

edges)

UNC, Stat & OR

Geodesic Paths

Given 2 trees,

Shortest path

Some math:

• Can show unique in this space

Both geodesics & shortest

UNC, Stat & OR

Geodesic Examples, T-4

VeryInterestingGeometry

UNC, Stat & OR

Recall:ManifoldStratifiedSpace

UNC, Stat & OR

2-d Strata(Orthants)

UNC, Stat & OR

2-d Strata(Orthants)

Glued With1-d Strata(Lines)

UNC, Stat & OR

0-d Stratum(Origin = Star Tree)

UNC, Stat & OR

3-OrthantGeodesic (angle < 180o)

UNC, Stat & OR

Cone PathGeodesic (angle > 180o)

UNC, Stat & OR

EndpointsIn SameOrthants

UNC, Stat & OR

Geodesic Paths

Given 2 trees,

Shortest path between is called the

geodesic

Fast Computation (polynomial

time):

Owen & Provan (2011)

UNC, Stat & OR

The Geodesic Path Algorithm

Owen & Provan (2011)’s polynomial

algorithm for finding geodesics between n-

UNC, Stat & OR

trees:

Start with the cone path connecting the

two trees through the origin (“star tree”).

UNC, Stat & OR

Geodesic Paths in Tree Space

UNC, Stat & OR

trees:

two trees through the origin (“star tree”). Successively “slide” the path into

successively larger sets of orthants, each

time decreasing the length of the path.

UNC, Stat & OR

trees:

two trees through the origin (“star tree”). Successively “slide” the path into

successively larger sets of orthants, each

time decreasing the length of the path. Stop when shortest path is found.

UNC, Stat & OR

Geodesics for Artery Trees

, ... ,,

UNC, Stat & OR

To illustrate geodesics

Study trees along geodesic,

From Case 2

To Case 3

Common Edges

UNC, Stat & OR

March Along 2 to 3 Geodesic

UNC, Stat & OR

Leaf Node Numbers

UNC, Stat & OR

Interior Edge Lengths

UNC, Stat & OR

Distance From Root

UNC, Stat & OR

Show Interior Edges Midway Between Children

UNC, Stat & OR

Unfortunate Consequence: Crossing Branches General Problem Embedding 3-d Trees in 2-d

UNC, Stat & OR

Count of # of Nodes

UNC, Stat & OR

Count of # of Nodes, as Function of Geodesic Step

UNC, Stat & OR

Also Total Branch Length

UNC, Stat & OR

Summarize Lessons:

• Very few Common Edges (only 5)

• Most edges swap out

UNC, Stat & OR

Summarize Lessons:

• Edges get much shorter (bottom

Recall this Later

UNC, Stat & OR

Summarize Lessons:

• Edges get much shorter (bottom

Recall this Later

• # of edges roughly constant (middle

UNC, Stat & OR

Reference for More:

Skwerer et al (2013)

Some Related Probability Theory:

Hotz et al (2013)

UNC, Stat & OR

3. Dyck Path

4. Persistent Homologies

UNC, Stat & OR

Persistent Homology Approach

Topological Data Analysis

Bendich et al (2014)

Gave Deepest Results to Date

UNC, Stat & OR

Carry Away Concept

OODA is more than a “framework”

It Provides a Focal Point

Highlights Pivotal Choices:

What should be the Data Objects?

How should they be Represented?

1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space...

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