1. Flow Complex Joachim Giesen Friedrich-Schiller-Universität Jena

2. Points

3. Surface reconstruction

4. Proteins: feature extraction

5. The Flow Complex joint work with Matthias John

6. Distance function

7. x d(x) x

8. Distance function

9. Gradient flow

10. Critical points maxima saddle points

11. Flow and critical points

13. Stable manifolds

14. Flow complex

15. Back to three dimensions

16. Stable manifolds

17. Surface Reconstruction (first attempt) joint work with Matthias John

18. Surface Reconstruction Flow complex Surface reconstruction

19. Pairing and cancellation Pairing of maxima and saddle points

20. Pairing and cancellation Pairing of maxima and saddle points Cancellation of pair with minimal difference between distance values

21. Pairing and cancellation Pairing of maxima and saddle points Cancellation of pair with minimal difference between distance values

22. Pairing and cancellation Pairing of maxima and saddle points Cancellation of pair with minimal difference between distance values Until “topologically” correct surface

23. Pairing and cancellation Pairing of maxima and saddle points Cancellation of pair with minimal difference between distance values Until “topologically” correct surface

24. Pairing and cancellation Pairing of maxima and saddle points Cancellation of pair with minimal difference between distance values Until “topologically” correct surface

25. Pairing and cancellation Pairing of maxima and saddle points Until “topologically” correct surface Cancellation of pair with minimal difference between distance values

26. Pairing and cancellation Result is a (possibly pinched) closed surface

27. Experimental results Buddha 144,647 pts Hip 132,538 pts

28. Experimental results Dragon 100,250 pts Noise added

29. Pockets in Proteins joint work Matthias John

30. Pockets in proteins Weighted flow complex Pockets in molecules

31. Power distance Let (p,w) be a weighted point. Power distance: |x-p|² - w x √w p

32. Distance to weighted points

33. The weighted flow complex The weighted flow complex is also defined as the collection of stable manifolds.

34. Pockets in proteins

35. Growing balls model

36. Pockets in proteins Topological events correspond to critical points of the distance function Pocket: connected component of union of stable manifolds of positive critical points

37. Visualization Pocket visualization: stable manifolds of negative critical points in the boundary Mouth: (connected component of) stable manifolds of positive critical points in the boundary of a pocket

38. Examples Void (no mouth) Ordinary pocket (one mouth) Tunnel (two or more)

39. Examples Alphatoxin

40. Surface Reconstruction joint work with Tamal Dey, Edgar Ramos and Bardia Sadri

41. For a dense sample of a smooth surface the critical points are either close to the surface or close to the medial axis of the surface. Theorem

42. Medial axis Distance function is not differentiable on medial axis.

43. Sampling condition

44. Theorem For a dense  -sample of a smooth surface the reconstruction is homeomorphic and geometrically close to the original surface.

45. Medial Axis Approximation joint work with Edgar Ramos and Bardia Sadri

46. Gradient flow

47. Unstable manifolds of medial axis critical points.

48. For a dense  -sample of a smooth surface the union of the unstable manifolds of medial axis critical points is homotopy equivalent to the medial axis. Theorem

49. The medial axis core

50. Shape Segmentation / Matching joint work with Tamal Dey and Samrat Goswami

51. Gradient flow and critical points Anchor hulls and drivers of the flow.

52. Segmentation (2D)

53. Segmentation (3D)

54. Matching (2D)

55. Matching (3D)

56. Flow Shapes and Alpha Shapes joint work with Matthias John and Tamal Dey

57. Flow Shapes Flow Shapes: inserting the stable manifolds in order of increasing values of the distance function at the critical points

58. Flow Shapes Flow Shapes: inserting the stable manifolds in order of increasing values of the distance function at the critical points

59. Flow Shapes Flow Shapes: inserting the stable manifolds in order of increasing values of the distance function at the critical points

60. Flow Shapes Flow Shapes: inserting the stable manifolds in order of increasing values of the distance function at the critical points

61. Flow Shapes Flow Shapes: inserting the stable manifolds in order of increasing values of the distance function at the critical points

62. Flow Shapes: inserting the stable manifolds in order of increasing values of the distance function at the critical points Flow Shapes Finite Sequence C¹…Cⁿ of cell complexes. C¹ = P (point set) Cⁿ = Flow complex

63. Alpha Shapes Alpha Shapes: Delaunay complex restricted to a union of balls centered at the sample points

64. Alpha Shapes Alpha Shapes: Delaunay complex restricted to a union of balls centered at the sample points

65. Alpha Shapes Alpha Shapes: Delaunay complex restricted to a union of balls centered at the sample points

66. Alpha Shapes Alpha Shapes: Delaunay complex restricted to a union of balls centered at the sample points

67. Alpha Shapes Alpha Shapes: Delaunay complex restricted to a union of balls centered at the sample points Finite Sequence C¹…Cⁿ´ of cell complexes, n´ ≥ n. C¹ = P (point set) Cⁿ´ = Delaunay triangulation

68. Theorem For every α ≥ 0 the flow shape corresponding to the distance value α and the alpha shape corresponding to balls of radius α are homotopy equivalent.

69. Comparison of the shapes Flow shapeAlpha shape

70. Comparison of the shapes Flow shapeAlpha shape

71. Comparison of the shapes Flow shapeAlpha shape

72. Comparison of the shapes Flow shapeAlpha shape

73. The End Thank you!