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

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1 UNC, Stat & OR Metrics in Curve Space Note on SRVF representation: Can show: Warp Invariance Follows from Jacobean calculation

Transcript of 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was...

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UNC, Stat & OR

Metrics in Curve Space

Note on SRVF representation:

Can show: Warp Invariance

Follows from Jacobean calculation

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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

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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

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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

Data Objects II

~ Kendall’s Shapes

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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

Data Objects III

~ Chang’s Transfo’s

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UNC, Stat & OR

Toy Example

ConventionalPCAProjections

PowerSpreadAcrossSpectrum

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UNC, Stat & OR

Toy Example

ConventionalPCAScores

Views of1-d CurveBendingThrough4 Dim’ns’

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UNC, Stat & OR

Toy Example

ConventionalPCAScores

PatternsAre“Harmonics”In Scores

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UNC, Stat & OR

Toy Example

Scores PlotShows DataAre “1”Dimensional

So NeedImprovedPCA Decomp.

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UNC, Stat & OR

Toy Example

AlignedCurvePCAProjections

All Var’nIn 1st

Component

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UNC, Stat & OR

Toy Example

Warps,PCProjections

Mostly1st PC,But 2nd

Helps Some

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UNC, Stat & OR

Toy Example

WarpCompon’ts(+ Mean)Applied toTemplateMean

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UNC, Stat & OR

PNS on SRVF Sphere

Toy Example

Tangent Space

PCA

(on Horiz. Var’n)

Thanks to Xiaosun Lu

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UNC, Stat & OR

PNS on SRVF Sphere

Toy Example

PNS Projections

(Fewer Modes)

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UNC, Stat & OR

PNS on SRVF Sphere

Toy Example

Tangent Space

PCA

Note: 3 Comp’s

Needed for This

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UNC, Stat & OR

PNS on SRVF Sphere

Toy Example

PNS Projections

Only 2 for This

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UNC, Stat & OR

TIC testbed

Special Feature: Answer Key of Known Peaks

Goal:FindWarpsTo AlignThese

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UNC, Stat & OR

TIC testbed

Fisher – Rao Alignment

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UNC, Stat & OR

Non - Euclidean Data Spaces

What is “Strongly Non-Euclidean” Case?

Trees as Data

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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

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

Thanks to Burcu Aydin

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory:

• Graph is set of nodes and edges

Thanks to Burcu Aydin

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory:

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

Thanks to Burcu Aydin

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory:

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

Data Objects: set of treesThanks to Burcu Aydin

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

General Graph:

Thanks to Sean Skwerer

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

Special Case Called “Tree”

• Directed

• Acyclic

5

43

21

0

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

Special Case Called “Tree”

• Directed

• Acyclic

5

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0

Graphical note:

Sometimes “grow

up”

Others “grow down”

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

Special Case Called “Tree”

• Directed

• Acyclic

5

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0 Terminology:

Root

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Strongly Non-Euclidean Spaces

Special Case Called “Tree”

• Directed

• Acyclic

5

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Terminology:

Children

Of

Parent

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Strongly Non-Euclidean Spaces

Motivating Example:

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

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

Motivating Example:

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

• Blood Vessel Trees in Brains

• Segmented from MRAs

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UNC, Stat & OR

Strongly Non-Euclidean Spaces

Motivating Example:

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

• Blood Vessel Trees in Brains

• Segmented from MRAs

• Study population of trees

Forest of Trees

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

MRI (T1) view

Single Slice

From 3-d Image

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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UNC, Stat & OR

Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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UNC, Stat & OR

Blood vessel tree data

Now look over many people (data

objects)

Structure of population (understand

variation?)

PCA in strongly non-Euclidean Space???

, ... ,,

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Blood vessel tree data

Examples of Potential Specific Goals

, ... ,,

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UNC, Stat & OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

• Predict Stroke Tendency (Collateral

Circulation)

, ... ,,

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UNC, Stat & OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

• Predict Stroke Tendency (Collateral

Circulation)

• Screen for Loci of Pathology

, ... ,,

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UNC, Stat & OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

• Predict Stroke Tendency (Collateral

Circulation)

• Screen for Loci of Pathology

• Explore how age affects connectivity

, ... ,,

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UNC, Stat & OR

Blood vessel tree data

Examples of Potential Specific Goals

(not accessible by traditional methods)

• Predict Stroke Tendency (Collateral

Circulation)

• Screen for Loci of Pathology

• Explore how age affects connectivity

, ... ,,

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UNC, Stat & OR

Blood vessel tree data

Big Picture: 4 Approaches

1. Purely Combinatorial

2. Euclidean Orthant (Phylogenetics)

3. Dyck Path

4. Persistent Homologies Topological

DA

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UNC, Stat & OR

Blood vessel tree data

Big Picture: 4 Approaches

1. Purely Combinatorial

2. Euclidean Orthant (Phylogenetics)

3. Dyck Path

4. Persistent Homologies Topological

DA

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UNC, Stat & OR

Blood vessel tree data

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)

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D-L Visualization of Trees

Challenge:VisualDisplay ofFull TreeStructure

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D-L Visualization of Trees

Challenge:VisualDisplay ofFull TreeStructure

Actually Goes toLevel 17(Truncated in View)

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D-L Visualization of Trees

Approach: Different Coordinate System

Aydin, et al (2011)

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UNC, Stat & OR

D-L Visualization of Trees

Approach: Different Coordinate System

Idea: Focus on Important Aspects (of

Nodes)

• Level of Node

• Number of Children

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UNC, Stat & OR

D-L Visualization of Trees

Approach: Different Coordinate System

Idea: Focus on Important Aspects (of

Nodes)

• Level of Node

• Number of Children

Using These as Coordinates

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D-L Visualization of Trees

D-LView

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D-L Visualization of Trees

D-LView

Nodes

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D-L Visualization of Trees

D-LView

Level

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D-L Visualization of Trees

D-LView

Level

MuchDeeper ThanEarlyView

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D-L Visualization of Trees

D-LView

# Des-cend-ants(logscale)

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D-L Visualization of Trees

D-LView

ColorCodesBranchThick-ness

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D-L Visualization of Trees

D-LView

RevealsStrangeStruc-ture

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D-L Visualization of Trees

D-LView

RevealsStrangeStruc-ture

LinkingErrors

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D-L Visualization of Trees

D-LView

FixedVersion

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UNC, Stat & OR

Blood vessel tree data

Big Picture: 4 Approaches

1. Purely Combinatorial

2. Euclidean Orthant

(Phylogenetics)

3. Dyck Path

4. Persistent Homologies Topological

DA

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Euclidean Orthant Approach

People:

• Scott Provan

• Sean Skwerer

• Megan Owen

• Ezra Miller

• Martin Styner

• Ipek Oguz

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Euclidean Orthant Approach

Setting: Connectivity & Length

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UNC, Stat & OR

Euclidean Orthant Approach

Setting: Connectivity & Length

Background: Phylogenetic Trees

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Euclidean Orthant Approach

Setting: Connectivity & Length

Background: Phylogenetic Trees

Important Concept from Evolutionary Biology

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Phylogenetic Trees

Idea: Study

“Common

Ancestry”

Via a tree

Species are leaves

thanks to Susan Holmes

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Phylogenetic Trees

Very Early Reference:

E. Schröder (1870),

Zeit. für. Math. Phys.,

15, 361-376.

thanks to Susan Holmes

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Phylogenetic Trees

Important Reference:

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

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Phylogenetic Trees

Important Reference:

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

Put Large Field on Firm Mathematical Basis

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UNC, Stat & OR

Euclidean Orthant Approach

Setting: Connectivity & Length

Background: Phylogenetic Trees

Major Restriction: Need common leaves

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UNC, Stat & OR

Euclidean Orthant Approach

Setting: Connectivity & Length

Background: Phylogenetic Trees

Major Restriction: Need common leaves

Big Payoff: Data space nearly Euclidean

sort of Euclidean

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UNC, Stat & OR

Euclidean Orthant Approach

Major Restriction: Need common leaves

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UNC, Stat & OR

Euclidean Orthant Approach

Major Restriction: Need common leaves

Approach:

• Find common cortical landmarks (Oguz)

corresponding across cases

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UNC, Stat & OR

Euclidean Orthant Approach

Major Restriction: Need common leaves

Approach:

• Find common cortical landmarks (Oguz)

corresponding across cases

Oguz (2009)

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Euclidean Orthant Approach

Major Restriction: Need common leaves

Approach:

• Find common cortical landmarks (Oguz)

corresponding across cases

• Treat as pseudo – leaves

by projecting to points on tree

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Labeled n-Trees

5-tree

e.g. n = 5

Thanks to Sean Skwerer

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Labeled n-Trees

leaves - fixed a priori

5-tree

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Labeled n-Trees

leaves - fixed a priori

root5-tree

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Labeled n-Trees

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

root5-tree

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Labeled n-Trees

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

Internal (nonleaf) vertices degree ≥ 3

root5-tree

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Labeled n-Trees

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

Internal (nonleaf) vertices degree ≥ 3

# edges from node

root5-tree

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Labeled n-Trees

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

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

root5-tree

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Labeled n-Trees

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

Internal (nonleaf) vertices degree ≥ 3

edge e has nonneg. length

root

|e1|=3

|e2|=4

|e3|=6

5-tree

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Labeled n-Trees

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

root

|e1|=3

|e2|=4

|e3|=6

5-tree

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Labeled n-Trees

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

Enough to consideronly lengths of internal edges

root

|e1|=3

|e2|=4

|e3|=6

5-tree

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Labeled n-Trees

Terminology:Leaf edges called ‘pendants’

(Care about lengths of pendants in brain arteries)

root

|e1|=3

|e2|=4

|e3|=6

5-tree

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Toy Examples

=

Same tree, since same internal edges

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Toy Examples

=

Different tree, since different connections

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Toy Examples

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

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Tree Space Examples, T-4

𝐸𝑑𝑔𝑒 { 𝐼 𝑛𝑖𝑓 |𝑒|>0𝑂𝑢𝑡 𝑖𝑓 |𝑒|=0

Set of mutually compatible splits tree

Thanks to Megan Owen

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Three quadrants meeting at common axis

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Three quadrants meeting at common axis

Star Tree = 0 Tree

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Three quadrants meeting at common axis

Star Tree(point)

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Three quadrants meeting at common axis

Star Tree(point)

1-edge Trees(line)

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Three quadrants meeting at common axis

Star Tree(point)

1-edge Trees(line)

2-edge Trees(planes)

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Three quadrants meeting at common axis

MathematicalConstruct:

Manifold StratifiedSpace

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Three quadrants meeting at common axis

MathematicalConstruct:

Manifold StratifiedSpace

Manifolds (planes)of Different

Dimensions Glued Together

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Three quadrants meeting at common axis

MathematicalConstruct:

Manifold StratifiedSpace

Manifolds (planes)of Different

Dimensions Glued Together

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Three quadrants meeting at common axis

MathematicalConstruct:

Manifold StratifiedSpace

Manifolds (planes)of Different

Dimensions Glued Together

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‘Connectivity’ of T-4

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‘Connectivity’ of T-4

Cone Structure

(Reflects edge lengths > 0)

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‘Connectivity’ of T-4

Cone Structure Star (0) Tree

(At Origin)

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‘Connectivity’ of T-4

Cone Structure Star (0) Tree Single Edge

Trees

(On Boundary Lines)

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‘Connectivity’ of T-4

Cone Structure Star (0) Tree Single Edge

Trees Full (2 Edge)

Trees

(On Planes)

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‘Connectivity’ of T-4

Each LineConnects to 3 planes(# compatible

edges)

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‘Connectivity’ of T-4

Each LineConnects to 3 planes(# compatible

edges)

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Geodesic Paths

Given 2 trees,

Shortest path

Some math:

• Can show unique in this space

Both geodesics & shortest

paths

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Geodesic Examples, T-4

Thanks to Megan Owen

VeryInterestingGeometry

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Geodesic Examples, T-4

Thanks to Megan Owen

Recall:ManifoldStratifiedSpace

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Geodesic Examples, T-4

Thanks to Megan Owen

Recall:ManifoldStratifiedSpace

2-d Strata(Orthants)

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Geodesic Examples, T-4

Thanks to Megan Owen

Recall:ManifoldStratifiedSpace

2-d Strata(Orthants)

Glued With1-d Strata(Lines)

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Geodesic Examples, T-4

Thanks to Megan Owen

Recall:ManifoldStratifiedSpace

0-d Stratum(Origin = Star Tree)

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Geodesic Examples, T-4

Thanks to Megan Owen

3-OrthantGeodesic (angle < 180o)

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Geodesic Examples, T-4

Thanks to Megan Owen

Cone PathGeodesic (angle > 180o)

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Geodesic Examples, T-4

Thanks to Megan Owen

EndpointsIn SameOrthants

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Geodesic Paths

Given 2 trees,

Shortest path between is called the

geodesic

Fast Computation (polynomial

time):

Owen & Provan (2011)

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The Geodesic Path Algorithm

Owen & Provan (2011)’s polynomial

algorithm for finding geodesics between n-

trees

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The Geodesic Path Algorithm

Owen & Provan (2011)’s polynomial

algorithm for finding geodesics between n-

trees:

Start with the cone path connecting the

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

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Geodesic Paths in Tree Space

Thanks to Sean Skwerer

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The Geodesic Path Algorithm

Owen & Provan (2011)’s polynomial

algorithm for finding geodesics between n-

trees:

Start with the cone path connecting the

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.

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Geodesic Paths in Tree Space

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Geodesic Paths in Tree Space

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The Geodesic Path Algorithm

Owen & Provan (2011)’s polynomial

algorithm for finding geodesics between n-

trees:

Start with the cone path connecting the

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.

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Geodesics for Artery Trees

, ... ,,

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Geodesics for Artery Trees

To illustrate geodesics

Study trees along geodesic,

From Case 2

To Case 3

Common Edges

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March Along 2 to 3 Geodesic

Thanks to Sean Skwerer

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March Along 2 to 3 Geodesic

Thanks to Sean Skwerer

Leaf Node Numbers

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March Along 2 to 3 Geodesic

Thanks to Sean Skwerer

Interior Edge Lengths

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March Along 2 to 3 Geodesic

Thanks to Sean Skwerer

Distance From Root

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March Along 2 to 3 Geodesic

Thanks to Sean Skwerer

Show Interior Edges Midway Between Children

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March Along 2 to 3 Geodesic

Thanks to Sean Skwerer

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

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

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March Along 2 to 3 Geodesic

Count of # of Nodes

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March Along 2 to 3 Geodesic

Count of # of Nodes, as Function of Geodesic Step

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March Along 2 to 3 Geodesic

Also Total Branch Length

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Geodesics for Artery Trees

Summarize Lessons:

• Very few Common Edges (only 5)

• Most edges swap out

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Geodesics for Artery Trees

Summarize Lessons:

• Very few Common Edges (only 5)

• Most edges swap out

• Edges get much shorter (bottom

plot)

Recall this Later

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Geodesics for Artery Trees

Summarize Lessons:

• Very few Common Edges (only 5)

• Most edges swap out

• Edges get much shorter (bottom

plot)

Recall this Later

• # of edges roughly constant (middle

plot)

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Euclidean Orthant Approach

Reference for More:

Skwerer et al (2013)

Some Related Probability Theory:

Hotz et al (2013)

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Blood vessel tree data

Big Picture: 4 Approaches

1. Purely Combinatorial

2. Euclidean Orthant (Phylogenetics)

3. Dyck Path

4. Persistent Homologies

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Blood vessel tree data

Persistent Homology Approach

Topological Data Analysis

Bendich et al (2014)

Gave Deepest Results to Date

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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?