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Geodesics on Implicit Surfaces and Point Clouds

Geodesics on Implicit Surfaces and Point Clouds. Guillermo Sapiro Electrical and Computer Engineering University of Minnesota guille@ece.umn.edu. Supported by NSF, ONR, NGA, NIH. IPAM 2005. Overview. Motivation Distances and Geodesics Implicit surfaces Point clouds. Key Motivation.

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Geodesics on Implicit Surfaces and Point Clouds

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  1. Geodesics on Implicit Surfaces and Point Clouds Guillermo Sapiro Electrical and Computer Engineering University of Minnesota guille@ece.umn.edu 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. ^ ^ T4=T0+4Δ 10 11 12 ^ ^ T5=T0+5Δ 13 ^ T0 1 2 3 4 Pr Pmin ^ ^ T1=T0+Δ 5 ^ ^ T2=T0+2Δ ^ ^ T3=T0+3Δ 6 7 8 9 O(N) Implementation of the Fast Marching Algorithm (Yatziv, Bartesaghi, S.) • 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 • Embed M: • Extend I outside M: M I 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. 2002):

  39. Examples (VRML)

  40. Examples

  41. Intrinsic Voronoi of Point Clouds

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

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

  44. Error increases with the number of samples!

  45. 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

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

  47. End of part 1 Thank You

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