1. Geodesics on Implicit Surfaces and Point Clouds
Guillermo Sapiro
Electrical and Computer Engineering
University of Minnesota
Supported by NSF, ONR, NGA, NIH
IPAM 2005

2. Overview Motivation Distances and Geodesics Implicit surfaces
Point clouds

3. Key Motivation
We can’t do much if we do not know the distance between two points!

4. Representation Motivation
Implicit surfaces
Facilitate fundamental computations
Natural representation for many algorithms (e.g., medical imaging)
Part of the computation very often (distance functions)
Point clouds
Natural representation for 3D scanners
Natural representation for manifold learning
Dimensionality independent
Pure geometry (no artificial meshes, etc)

5. Motivation: A Few Examples

6. Tracking (Bartesaghi, S.)

7. 3D segmentation (Bartesaghi, S. , Subramaniam)

8. Colorization (Yatziv, S.)

9. Example - Video

10. Motivation: What is a Geodesic?

11. Motivation: What is a Geodesic?

12. Background: Distance and Geodesic Computation via Dijkstra
Complexity: O(n log n)
Advantage: Works in any dimension and with any geometry (graphs)
Problems:
Not consistent
Unorganized points?
Noise?
Implicit surfaces?

13. Background: Distance and Geodesic Computation via Dijkstra
Complexity: O(n log n)
Advantage: Works in any dimension and with any geometry (graphs)
Problems:
Not consistent
Unorganized points?
Noise?
Implicit surfaces?

14. Background: Distance and Geodesic Computation via Dijkstra
Complexity: O(n log n)
Advantage: Works in any dimension and with any geometry (graphs)
Problems:
Not consistent
Unorganized points?
Noise?
Implicit surfaces?

15. Background: Distance and Geodesic Computation via Dijkstra
Complexity: O(n log n)
Advantage: Works in any dimension and with any geometry (graphs)
Problems:
Not consistent
Unorganized points?
Noise?
Implicit surfaces?

16. Background: Distance and Geodesic Computation via Dijkstra
Complexity: O(n log n)
Advantage: Works in any dimension and with any geometry (graphs)
Problems:
Not consistent
Unorganized points?
Noise?
Implicit surfaces?

17. Background: Distance Functions as Hamilton-Jacobi Equations
g = weight on the hyper-surface
The g-weighted distance function between two points p and x on the hyper-surface S is:

18. d
Slope = 1

19. Background: Computing Distance Functions as Hamilton-Jacobi Equations
Solved in O(n log n) by Tsitsiklis, by Sethian, and by Helmsen, only for Euclidean spaces and Cartesian grids
Solved only for acute 3D triangulations by Kimmel and Sethian

20. O(N) Implementation of the Fast Marching Algorithm (Yatziv, Bartesaghi, S.)
T2=T0+2Δ
T1=T0+Δ
T3=T0+3Δ
T4=T0+4Δ
T5=T0+5Δ T0
Pmin Pr 1 2 3 4 5 6 7 8 9 10 11 12 13 ^
Untidy Priority queue
Cyclic Bucket sort
Calendar queue
Rounding off priorities
Error bounded O(h)
Size of array can be small and unrelated to N

21. A real time example (following Cohen-Kimmel)

22. The Problem How to compute intrinsic distances and geodesics for
General dimensions
Implicit surfaces
Unorganized noisy points (hyper-surfaces just given by examples)

23. Our Approach
We have to solve

24. Basic Idea

25. Basic Idea

26. Basic Idea
Theorem (Memoli-S.):

27. Basic idea

28. Why is this good?

29. Implicit Form Representation
Figure from G. Turk

30. Data extension M I
Embed M:
Extend I outside M: I I

31. Examples

32. Examples

33. Examples

34. Examples

35. Unorganized points

36. Unorganized points (cont.)

37. Unorganized points

38. Randomly sampled manifolds (with noise)
Theorem (Memoli - S ):

39. Examples (VRML)

40. Examples

41. Intrinsic Voronoi of Point Clouds

42. Intermezzo: de Silva, Tenenbaum, et al...

44. Intermezzo: Tenenbaum, de Silva, et al...
Main Problem:
Doesn’t address noisy examples/measurements: Much less robust to noise!

45. Error increases with the number of samples!

47. Intermezzo: de Silva, Tenenbaum, et al...
Problems:
Doesn’t address noisy examples/measurements: Much less robust to noise!
Only convex surfaces
Uses Dijkestra (back to non consistency)
Doesn’t work for implicit surface representations

48. Concluding remarks
A general computational framework for distance functions and geodesics.
Implicit hyper-surfaces and un-organized points

49. End of part 1
Thank You