Sampling conditions and topological guarantees for shape reconstruction algorithms

1. Sampling conditions and topological guarantees for shape reconstruction algorithms Andre Lieutier, DassaultSytemes Thanks to Dominique Attali for someslides (the niceones) Thanksto Dominique Attali, Fréderic Chazal, David Cohen-Steiner for joint work

2. Shape Reconstruction UNKNOWN INPUT OUTPUT Surface of physical object Triangulation Sample in R3 • geometrically accurate • topologically correct

3. Shape Reconstruction(or manifold learning) INPUT OUTPUT Unordered sequence of images varying in pose and lighting Low-dimensional complex

4. Shape Reconstruction(or manifold learning) UNKNOWN INPUT OUTPUT Space with small intrinsec dimension Simplicial complex Sample in Rd • geometrically accurate • topologically correct

5. Algorithms in 2D heuristics to select a subset of the Delaunay triangulation

6. Algorithms in 3D heuristics to select a subset of the Delaunay triangulation

7. A Simple Algorithm UNKNOWN INPUT OUTPUT Shape Sample -offset = union of balls with radius centered on the sample

8. A Simple Algorithm UNKNOWN INPUT OUTPUT Shape Sample From Nerve Theorem: -offset -complex

9. A Simple Algorithm Shape Sample OUTPUT

11. Reconstruction theorem

12. Reconstruction theorem Sampling conditions [Niyogi Smale Weinberger 2004]

14. Beyond the reach : WFS and m-reach

15. 1 m wfs m-reach reach Beyond the reach : WFS and m-reach wfs m m

16. Previous best known result for faithful reconstruction of set with positive m-reach(Chazal, Cohen-Steiner,Lieutier 2006)

17. Previous best known result for faithful reconstruction of set with positive m-reach(Chazal, Cohen-Steiner,Lieutier 2006)

18. Best known result for faithful reconstruction of set with positive m-reach

19. Previous best known result for faithful reconstruction of set with positive m-reach(Chazal, Cohen-Steiner,Lieutier 2006) Under the conditions of the theorem, a simple offset of the sampleis a faithfulreconsruction

20. Recoveringhomology (Cohen-Steiner,Edelsbrunner,Harer 2006) (Chazal, Lieutier 2006)

21. Recoveringhomology (Cohen-Steiner,Edelsbrunner,Harer 2006) (Chazal, Lieutier 2006)

22. Recoveringhomology (Cohen-Steiner,Edelsbrunner,Harer 2006) (Chazal, Lieutier 2006)

23. Recoveringhomology (Cohen-Steiner,Edelsbrunner,Harer 2006) (Chazal, Lieutier 2006)

24. Recoveringhomology (Cohen-Steiner,Edelsbrunner,Harer 2006) (Chazal, Lieutier 2006)

25. Rips Complex

26. Samplng condition for Cech and Rips(D. Attali, A. Lieutier 2011) [NSW04] [CCL06]

27. Questions ?

29. Cech / Rips Rips and Cech complexes generally don’t share the same topology, but ...

30. Cech / Rips

31. Possesses a spirituous cycle that we want to kill ! Cech / Rips

32. Cech / Rips Had there been a point close to the center, it would have distroy spirituous cycles appearing in the Rips, without changing the Cech.

33. Convexity defects function

34. Large m-reach=> smallconvexitydefectfunctions

35. Density authorized [NSW04] [CCL06]

36. Questions ?

37. Cech Complex

38. Nerve Theorem

39. Persistent homology

40. Persistent homology

41. Persistent homology

42. Persistent homology

43. Persistent homology

44. Persistent homology