1 / 49

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.

yetty
Télécharger la présentation

Geodesics on Implicit Surfaces and Point Clouds

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related