Non-Euclidean Analysis of Blood Vessel Trees: A Manifold Approach

1. Object Orie’d Data Analysis, Last Time • Discrimination for manifold data (Sen) • Simple Tangent plane SVM • Iterated TANgent plane SVM • Manifold SVM • Interesting point: Analysis done really in the manifold • Not just in projected tangent plane • Deeper than Principal Geodesic Analysis? • Manifold version of DWD?

2. Mildly Non-Euclidean Spaces Useful View of Manifold Data: Tangent Space Center: Frechét Mean Reason for terminology “mildly non Euclidean”

3. 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 trees

4. Strongly Non-Euclidean Spaces Motivating Example: • Blood Vessel Trees in Brains • From Dr. Elizabeth Bullitt • Dept. of Neurosurgery, UNC • Segmented from MRAs • Study population of trees (Forest?)

5. Blood vessel tree data • Marron’s brain: • MRI view • Single Slice • From 3-d Image

6. Blood vessel tree data • Marron’s brain: • MRA view • “A” for Angiography” • Finds blood vessels • (show up as white) • Track through 3d

7. Blood vessel tree data • Marron’s brain: • MRA view • “A” for Angiography” • Finds blood vessels • (show up as white) • Track through 3d

8. Blood vessel tree data • Marron’s brain: • MRA view • “A” for Angiography” • Finds blood vessels • (show up as white) • Track through 3d

9. Blood vessel tree data • Marron’s brain: • MRA view • “A” for Angiography” • Finds blood vessels • (show up as white) • Track through 3d

10. Blood vessel tree data • Marron’s brain: • MRA view • “A” for Angiography” • Finds blood vessels • (show up as white) • Track through 3d

11. Blood vessel tree data • Marron’s brain: • MRA view • “A” for Angiography” • Finds blood vessels • (show up as white) • Track through 3d

12. Blood vessel tree data • Marron’s brain: • From MRA • Segment tree • of vessel segments • Using tube tracking • Bullitt and Aylward (2002)

13. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

14. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

15. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

16. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

17. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

18. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

19. Blood vessel tree data , , ... , Now look over many people (data objects) Structure of population (understand variation?) PCA in strongly non-Euclidean Space???

20. Blood vessel tree data • The tree team: • Very Interdsciplinary • Neurosurgery: • Bullitt, Ladha • Statistics: • Wang, Marron • Optimization: • Aydin, Pataki

21. Blood vessel tree data , , ... , • Possible focus of analysis: • Connectivity structure only (topology) • Location, size, orientation of segments • Structure within each vessel segment

22. Blood vessel tree data • Present Focus: • Topology only • Already challenging • Later address others • Then add attributes • To tree nodes • And extend analysis

23. Blood vessel tree data • Recall from above: • Marron’s brain: • Focus on back • Connectivity only

24. Blood vessel tree data • Present Focus: • Topology only • Raw data as trees • Marron’s • reduced tree • Back tree only

25. Blood vessel tree data Topology only E.g. Back Trees Full Population Study as movie Understand variation?

26. Strongly Non-Euclidean Spaces Statistics on Population of Tree-Structured Data Objects? • Mean??? • Analog of PCA??? Strongly non-Euclidean, since: • Space of trees not a linear space • Not even approximately linear (no tangent plane)

27. Mildly Non-Euclidean Spaces Useful View of Manifold Data: Tangent Space Center: Frechét Mean Reason for terminology “mildly non Euclidean”

28. Strongly Non-Euclidean Spaces Mean of Population of Tree-Structured Data Objects? Natural approach: Fréchet mean Requires a metric (distance) On tree space

29. Strongly Non-Euclidean Spaces Appropriate metrics on tree space: Wang and Marron (2007) • Depends on: • Tree structure • And nodal attributes • Won’t go further here • But gives appropriate Fréchet mean

30. Strongly Non-Euclidean Spaces Appropriate metrics on tree space: Wang and Marron (2007) • For topology only (studied here): • Use Hamming Distance • Just number of nodes not in common • Gives appropriate Fréchet mean

31. Strongly Non-Euclidean Spaces PCA on Tree Space? • Recall Conventional PCA: Directions that explain structure in data • Data are points in point cloud • 1-d and 2-d projections allow insights about population structure

32. Illust’n of PCA View: PC1 Projections

33. Illust’n of PCA View: Projections on PC1,2 plane

34. Source Batch Adj: PC 1-3 & DWD direction

35. Source Batch Adj: DWD Source Adjustment

36. Strongly Non-Euclidean Spaces PCA on Tree Space? Key Ideas: • Replace 1-d subspace that best approximates data • By 1-d representation that best approximates data Wang and Marron (2007) define notion of Treeline (in structure space)

37. Strongly Non-Euclidean Spaces PCA on Tree Space: Treeline • Best 1-d representation of data Basic idea: • From some starting tree • Grow only in 1 “direction”

38. Strongly Non-Euclidean Spaces PCA on Tree Space: Treeline • Best 1-d representation of data Problem: Hard to compute • In particular: to solve optimization problem Wang and Marron (2007) • Maximum 4 vessel trees • Hard to tackle serious trees (e.g. blood vessel trees)

39. Strongly Non-Euclidean Spaces PCA on Tree Space: Treeline Problem: Hard to compute Solution: Burḉu Aydin & Gabor Pataki (linear time algorithm) (based on clever “reformulation” of problem) Description coming in Participant Presentation

40. PCA for blood vessel tree data • PCA on Tree Space: Treelines • Interesting to compare: • Population of Left Trees • Population of Right Trees • Population of Back Trees • And to study 1st, 2nd, 3rd & 4th treelines

41. PCA for blood vessel tree data Study “Directions” 1, 2, 3, 4 For sub- populations B, L, R (interpret later)

42. Strongly Non-Euclidean Spaces PCA on Tree Space: Treeline Next represent data as projections • Define as closest point in tree line (same as Euclidean PCA) • Have corresponding score (length of projection along line) • And analog of residual (distance from data point to projection)

43. PCA for blood vessel tree data Raw Data & Treelines, PC1, PC2, PC3:

44. PCA for blood vessel tree data Raw Data & Treelines, PC1, PC2, PC3: Projections, Scores, Residuals

45. PCA for blood vessel tree data Raw Data & Treelines, PC1, PC2, PC3: Cumulative Scores, Residuals

46. PCA for blood vessel tree data Now look deeper at “Directions” 1, 2, 3, 4 For sub- populations B, L, R

47. PCA for blood vessel tree data • Notes on Treeline Directions: • PC1 always to left • BACK has most variation to right (PC2) • LEFT has more varia’n to 2nd level (PC2) • RIGHT has more var’n to 1st level (PC2) • See these in the data?

48. PCA for blood vessel tree data • Notes: • PC1 - left • BACK - right • LEFT 2nd lev • RIGHT 1st lev • See these??

49. PCA for blood vessel tree data Individual (each PC separately) Scores Plot

50. PCA for blood vessel tree data Identify this person