![Page 1: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/1.jpg)
1
UNC, Stat & OR
Metrics in Curve Space
Note on SRVF representation:
Can show: Warp Invariance
Follows from Jacobean calculation
![Page 2: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/2.jpg)
2
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
![Page 3: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/3.jpg)
3
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
![Page 4: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/4.jpg)
4
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
![Page 5: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/5.jpg)
5
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
![Page 6: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/6.jpg)
6
UNC, Stat & OR
Toy Example
ConventionalPCAProjections
PowerSpreadAcrossSpectrum
![Page 7: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/7.jpg)
7
UNC, Stat & OR
Toy Example
ConventionalPCAScores
Views of1-d CurveBendingThrough4 Dim’ns’
![Page 8: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/8.jpg)
8
UNC, Stat & OR
Toy Example
ConventionalPCAScores
PatternsAre“Harmonics”In Scores
![Page 9: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/9.jpg)
9
UNC, Stat & OR
Toy Example
Scores PlotShows DataAre “1”Dimensional
So NeedImprovedPCA Decomp.
![Page 10: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/10.jpg)
10
UNC, Stat & OR
Toy Example
AlignedCurvePCAProjections
All Var’nIn 1st
Component
![Page 11: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/11.jpg)
11
UNC, Stat & OR
Toy Example
Warps,PCProjections
Mostly1st PC,But 2nd
Helps Some
![Page 12: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/12.jpg)
12
UNC, Stat & OR
Toy Example
WarpCompon’ts(+ Mean)Applied toTemplateMean
![Page 13: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/13.jpg)
13
UNC, Stat & OR
PNS on SRVF Sphere
Toy Example
Tangent Space
PCA
(on Horiz. Var’n)
Thanks to Xiaosun Lu
![Page 14: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/14.jpg)
14
UNC, Stat & OR
PNS on SRVF Sphere
Toy Example
PNS Projections
(Fewer Modes)
![Page 15: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/15.jpg)
15
UNC, Stat & OR
PNS on SRVF Sphere
Toy Example
Tangent Space
PCA
Note: 3 Comp’s
Needed for This
![Page 16: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/16.jpg)
16
UNC, Stat & OR
PNS on SRVF Sphere
Toy Example
PNS Projections
Only 2 for This
![Page 17: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/17.jpg)
17
UNC, Stat & OR
TIC testbed
Special Feature: Answer Key of Known Peaks
Goal:FindWarpsTo AlignThese
![Page 18: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/18.jpg)
18
UNC, Stat & OR
TIC testbed
Fisher – Rao Alignment
![Page 19: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/19.jpg)
19
UNC, Stat & OR
Non - Euclidean Data Spaces
What is “Strongly Non-Euclidean” Case?
Trees as Data
![Page 20: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/20.jpg)
20
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
![Page 21: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/21.jpg)
21
UNC, Stat & OR
Strongly Non-Euclidean Spaces
Trees as Data Objects
Thanks to Burcu Aydin
![Page 22: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/22.jpg)
22
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
![Page 23: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/23.jpg)
23
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
![Page 24: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/24.jpg)
24
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
![Page 25: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/25.jpg)
25
UNC, Stat & OR
Strongly Non-Euclidean Spaces
General Graph:
Thanks to Sean Skwerer
![Page 26: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/26.jpg)
26
UNC, Stat & OR
Strongly Non-Euclidean Spaces
Special Case Called “Tree”
• Directed
• Acyclic
5
43
21
0
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27
UNC, Stat & OR
Strongly Non-Euclidean Spaces
Special Case Called “Tree”
• Directed
• Acyclic
5
43
21
0
Graphical note:
Sometimes “grow
up”
Others “grow down”
![Page 28: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/28.jpg)
28
UNC, Stat & OR
Strongly Non-Euclidean Spaces
Special Case Called “Tree”
• Directed
• Acyclic
5
43
21
0 Terminology:
Root
![Page 29: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/29.jpg)
29
UNC, Stat & OR
Strongly Non-Euclidean Spaces
Special Case Called “Tree”
• Directed
• Acyclic
5
43
21
0
Terminology:
Children
Of
Parent
![Page 30: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/30.jpg)
30
UNC, Stat & OR
Strongly Non-Euclidean Spaces
Motivating Example:
• From Dr. Elizabeth Bullitt• Dept. of Neurosurgery, UNC
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31
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
![Page 32: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/32.jpg)
32
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
![Page 33: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/33.jpg)
33
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
MRI (T1) view
Single Slice
From 3-d Image
![Page 34: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/34.jpg)
34
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
![Page 35: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/35.jpg)
35
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
![Page 36: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/36.jpg)
36
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
![Page 37: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/37.jpg)
37
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
![Page 38: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/38.jpg)
38
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
![Page 39: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/39.jpg)
39
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
![Page 40: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/40.jpg)
40
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Segment tree
of vessel segments
Using tube tracking
Bullitt and Aylward (2002)
![Page 41: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/41.jpg)
41
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 42: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/42.jpg)
42
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 43: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/43.jpg)
43
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 44: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/44.jpg)
44
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 45: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/45.jpg)
45
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 46: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/46.jpg)
46
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 47: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/47.jpg)
47
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???
, ... ,,
![Page 48: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/48.jpg)
48
UNC, Stat & OR
Blood vessel tree data
Examples of Potential Specific Goals
, ... ,,
![Page 49: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/49.jpg)
49
UNC, Stat & OR
Blood vessel tree data
Examples of Potential Specific Goals
(not accessible by traditional methods)
• Predict Stroke Tendency (Collateral
Circulation)
, ... ,,
![Page 50: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/50.jpg)
50
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
, ... ,,
![Page 51: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/51.jpg)
51
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
, ... ,,
![Page 52: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/52.jpg)
52
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
, ... ,,
![Page 53: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/53.jpg)
53
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
![Page 54: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/54.jpg)
54
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
![Page 55: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/55.jpg)
55
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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56
UNC, Stat & OR
D-L Visualization of Trees
Challenge:VisualDisplay ofFull TreeStructure
![Page 57: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/57.jpg)
57
UNC, Stat & OR
D-L Visualization of Trees
Challenge:VisualDisplay ofFull TreeStructure
Actually Goes toLevel 17(Truncated in View)
![Page 58: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/58.jpg)
58
UNC, Stat & OR
D-L Visualization of Trees
Approach: Different Coordinate System
Aydin, et al (2011)
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59
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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60
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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61
UNC, Stat & OR
D-L Visualization of Trees
D-LView
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62
UNC, Stat & OR
D-L Visualization of Trees
D-LView
Nodes
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63
UNC, Stat & OR
D-L Visualization of Trees
D-LView
Level
![Page 64: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/64.jpg)
64
UNC, Stat & OR
D-L Visualization of Trees
D-LView
Level
MuchDeeper ThanEarlyView
![Page 65: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/65.jpg)
65
UNC, Stat & OR
D-L Visualization of Trees
D-LView
# Des-cend-ants(logscale)
![Page 66: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/66.jpg)
66
UNC, Stat & OR
D-L Visualization of Trees
D-LView
ColorCodesBranchThick-ness
![Page 67: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/67.jpg)
67
UNC, Stat & OR
D-L Visualization of Trees
D-LView
RevealsStrangeStruc-ture
![Page 68: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/68.jpg)
68
UNC, Stat & OR
D-L Visualization of Trees
D-LView
RevealsStrangeStruc-ture
LinkingErrors
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69
UNC, Stat & OR
D-L Visualization of Trees
D-LView
FixedVersion
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70
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
![Page 71: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/71.jpg)
71
UNC, Stat & OR
Euclidean Orthant Approach
People:
• Scott Provan
• Sean Skwerer
• Megan Owen
• Ezra Miller
• Martin Styner
• Ipek Oguz
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72
UNC, Stat & OR
Euclidean Orthant Approach
Setting: Connectivity & Length
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73
UNC, Stat & OR
Euclidean Orthant Approach
Setting: Connectivity & Length
Background: Phylogenetic Trees
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74
UNC, Stat & OR
Euclidean Orthant Approach
Setting: Connectivity & Length
Background: Phylogenetic Trees
Important Concept from Evolutionary Biology
![Page 75: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/75.jpg)
75
UNC, Stat & OR
Phylogenetic Trees
Idea: Study
“Common
Ancestry”
Via a tree
Species are leaves
thanks to Susan Holmes
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76
UNC, Stat & OR
Phylogenetic Trees
Very Early Reference:
E. Schröder (1870),
Zeit. für. Math. Phys.,
15, 361-376.
thanks to Susan Holmes
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77
UNC, Stat & OR
Phylogenetic Trees
Important Reference:
Billera L, Holmes S, & Vogtmann K (2001)
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78
UNC, Stat & OR
Phylogenetic Trees
Important Reference:
Billera L, Holmes S, & Vogtmann K (2001)
Put Large Field on Firm Mathematical Basis
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79
UNC, Stat & OR
Euclidean Orthant Approach
Setting: Connectivity & Length
Background: Phylogenetic Trees
Major Restriction: Need common leaves
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80
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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81
UNC, Stat & OR
Euclidean Orthant Approach
Major Restriction: Need common leaves
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82
UNC, Stat & OR
Euclidean Orthant Approach
Major Restriction: Need common leaves
Approach:
• Find common cortical landmarks (Oguz)
corresponding across cases
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83
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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84
UNC, Stat & OR
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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85
UNC, Stat & OR
Labeled n-Trees
5-tree
e.g. n = 5
Thanks to Sean Skwerer
![Page 86: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/86.jpg)
86
UNC, Stat & OR
Labeled n-Trees
leaves - fixed a priori
5-tree
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87
UNC, Stat & OR
Labeled n-Trees
leaves - fixed a priori
root5-tree
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88
UNC, Stat & OR
Labeled n-Trees
leaves - fixed a priori labeled {0,1, . . . ,n}
root5-tree
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89
UNC, Stat & OR
Labeled n-Trees
leaves - fixed a priori labeled {0,1, . . . ,n}
Internal (nonleaf) vertices degree ≥ 3
root5-tree
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90
UNC, Stat & OR
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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91
UNC, Stat & OR
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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92
UNC, Stat & OR
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
![Page 93: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/93.jpg)
93
UNC, Stat & OR
Labeled n-Trees
Note: To study‘topology’,(i.e. tree structure)
root
|e1|=3
|e2|=4
|e3|=6
5-tree
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94
UNC, Stat & OR
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
![Page 95: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/95.jpg)
95
UNC, Stat & OR
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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96
UNC, Stat & OR
Toy Examples
=
Same tree, since same internal edges
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97
UNC, Stat & OR
Toy Examples
=
Different tree, since different connections
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98
UNC, Stat & OR
Toy Examples
A valid tree, called “Star tree” or “0 tree” (since all internal edge lengths are 0)
![Page 99: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/99.jpg)
99
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
![Page 100: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/100.jpg)
100
UNC, Stat & OR
Three quadrants meeting at common axis
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101
UNC, Stat & OR
Three quadrants meeting at common axis
Star Tree = 0 Tree
![Page 102: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/102.jpg)
102
UNC, Stat & OR
Three quadrants meeting at common axis
Star Tree(point)
![Page 103: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/103.jpg)
103
UNC, Stat & OR
Three quadrants meeting at common axis
Star Tree(point)
1-edge Trees(line)
![Page 104: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/104.jpg)
104
UNC, Stat & OR
Three quadrants meeting at common axis
Star Tree(point)
1-edge Trees(line)
2-edge Trees(planes)
![Page 105: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/105.jpg)
105
UNC, Stat & OR
Three quadrants meeting at common axis
MathematicalConstruct:
Manifold StratifiedSpace
![Page 106: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/106.jpg)
106
UNC, Stat & OR
Three quadrants meeting at common axis
MathematicalConstruct:
Manifold StratifiedSpace
Manifolds (planes)of Different
Dimensions Glued Together
![Page 107: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/107.jpg)
107
UNC, Stat & OR
Three quadrants meeting at common axis
MathematicalConstruct:
Manifold StratifiedSpace
Manifolds (planes)of Different
Dimensions Glued Together
![Page 108: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/108.jpg)
108
UNC, Stat & OR
Three quadrants meeting at common axis
MathematicalConstruct:
Manifold StratifiedSpace
Manifolds (planes)of Different
Dimensions Glued Together
![Page 109: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/109.jpg)
109
UNC, Stat & OR
‘Connectivity’ of T-4
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110
UNC, Stat & OR
‘Connectivity’ of T-4
Cone Structure
(Reflects edge lengths > 0)
![Page 111: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/111.jpg)
111
UNC, Stat & OR
‘Connectivity’ of T-4
Cone Structure Star (0) Tree
(At Origin)
![Page 112: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/112.jpg)
112
UNC, Stat & OR
‘Connectivity’ of T-4
Cone Structure Star (0) Tree Single Edge
Trees
(On Boundary Lines)
![Page 113: 1 UNC, Stat & OR Metrics in Curve Space. 2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:](https://reader037.vdocument.in/reader037/viewer/2022103006/56649f145503460f94c29204/html5/thumbnails/113.jpg)
113
UNC, Stat & OR
‘Connectivity’ of T-4
Cone Structure Star (0) Tree Single Edge
Trees Full (2 Edge)
Trees
(On Planes)
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UNC, Stat & OR
‘Connectivity’ of T-4
Each LineConnects to 3 planes(# compatible
edges)
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115
UNC, Stat & OR
‘Connectivity’ of T-4
Each LineConnects to 3 planes(# compatible
edges)
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116
UNC, Stat & OR
Geodesic Paths
Given 2 trees,
Shortest path
Some math:
• Can show unique in this space
Both geodesics & shortest
paths
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117
UNC, Stat & OR
Geodesic Examples, T-4
Thanks to Megan Owen
VeryInterestingGeometry
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118
UNC, Stat & OR
Geodesic Examples, T-4
Thanks to Megan Owen
Recall:ManifoldStratifiedSpace
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119
UNC, Stat & OR
Geodesic Examples, T-4
Thanks to Megan Owen
Recall:ManifoldStratifiedSpace
2-d Strata(Orthants)
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120
UNC, Stat & OR
Geodesic Examples, T-4
Thanks to Megan Owen
Recall:ManifoldStratifiedSpace
2-d Strata(Orthants)
Glued With1-d Strata(Lines)
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121
UNC, Stat & OR
Geodesic Examples, T-4
Thanks to Megan Owen
Recall:ManifoldStratifiedSpace
0-d Stratum(Origin = Star Tree)
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122
UNC, Stat & OR
Geodesic Examples, T-4
Thanks to Megan Owen
3-OrthantGeodesic (angle < 180o)
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123
UNC, Stat & OR
Geodesic Examples, T-4
Thanks to Megan Owen
Cone PathGeodesic (angle > 180o)
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124
UNC, Stat & OR
Geodesic Examples, T-4
Thanks to Megan Owen
EndpointsIn SameOrthants
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125
UNC, Stat & OR
Geodesic Paths
Given 2 trees,
Shortest path between is called the
geodesic
Fast Computation (polynomial
time):
Owen & Provan (2011)
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126
UNC, Stat & OR
The Geodesic Path Algorithm
Owen & Provan (2011)’s polynomial
algorithm for finding geodesics between n-
trees
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127
UNC, Stat & OR
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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UNC, Stat & OR
Geodesic Paths in Tree Space
Thanks to Sean Skwerer
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129
UNC, Stat & OR
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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UNC, Stat & OR
Geodesic Paths in Tree Space
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131
UNC, Stat & OR
Geodesic Paths in Tree Space
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132
UNC, Stat & OR
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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UNC, Stat & OR
Geodesics for Artery Trees
, ... ,,
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134
UNC, Stat & OR
Geodesics for Artery Trees
To illustrate geodesics
Study trees along geodesic,
From Case 2
To Case 3
Common Edges
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135
UNC, Stat & OR
March Along 2 to 3 Geodesic
Thanks to Sean Skwerer
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136
UNC, Stat & OR
March Along 2 to 3 Geodesic
Thanks to Sean Skwerer
Leaf Node Numbers
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137
UNC, Stat & OR
March Along 2 to 3 Geodesic
Thanks to Sean Skwerer
Interior Edge Lengths
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138
UNC, Stat & OR
March Along 2 to 3 Geodesic
Thanks to Sean Skwerer
Distance From Root
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139
UNC, Stat & OR
March Along 2 to 3 Geodesic
Thanks to Sean Skwerer
Show Interior Edges Midway Between Children
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140
UNC, Stat & OR
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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145
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March Along 2 to 3 Geodesic
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146
UNC, Stat & OR
March Along 2 to 3 Geodesic
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UNC, Stat & OR
March Along 2 to 3 Geodesic
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148
UNC, Stat & OR
March Along 2 to 3 Geodesic
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149
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March Along 2 to 3 Geodesic
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150
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March Along 2 to 3 Geodesic
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151
UNC, Stat & OR
March Along 2 to 3 Geodesic
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UNC, Stat & OR
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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155
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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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UNC, Stat & OR
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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161
UNC, Stat & OR
March Along 2 to 3 Geodesic
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March Along 2 to 3 Geodesic
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163
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March Along 2 to 3 Geodesic
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164
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March Along 2 to 3 Geodesic
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165
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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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169
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March Along 2 to 3 Geodesic
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170
UNC, Stat & OR
March Along 2 to 3 Geodesic
Count of # of Nodes
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171
UNC, Stat & OR
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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173
UNC, Stat & OR
Geodesics for Artery Trees
Summarize Lessons:
• Very few Common Edges (only 5)
• Most edges swap out
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174
UNC, Stat & OR
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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175
UNC, Stat & OR
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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UNC, Stat & OR
Euclidean Orthant Approach
Reference for More:
Skwerer et al (2013)
Some Related Probability Theory:
Hotz et al (2013)
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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
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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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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?