1. Shape Reconstruction from Samples with Cocone Tamal K. Dey Dept. of CIS Ohio State University

2. A point cloud and reconstruction

3. Surface meshing from sample

4. A point set from satelite imaging

5. A reconstruction with and without noise

6. Why Sample Based Modeling? Sampling is easy and convenient with advanced technology Automatization (no manual intervention for meshing) Uniform approach for variety of inputs (laser scanner, probe digitizer, MRI,scientific simulations) Robust algorithms are available

7. Challenges Nonuniform data Boundaries Undersampling Large data Noise

8. Nonuniform data

9. Boundaries

10. Undersampling

11. Large data 3.4 million points

12. Cocone Cocone meets the challenges It guarantees geometrically close surface with same topological type Detects boundaries Detects undersampling Handles large data (Supercocone) Watertight surface (Tight Cocone)

13.  Sampling (ABE98) Each x has a sample within  f(x) f(x) is the distance to medial axis

14. Voronoi/Delaunay

15. Surface and Voronoi Diagram Restricted Voronoi Restricted Delaunay skinny Voronoi cell poles

16. Cocone algorithm Cocone Space spanned by vectors making angle   /8 with horizontal

17. Radius, height and neighbors p  is the farthest point from p in the cocone. radius r(p): p  radius of cocone height h(p): min distance to the poles cocone neighbors N p

18. Flatness condition Vertex p is flat if 1. Ratio condition: r(p)   h(p) 2. Normal condition:  v(p),v(q)    q with p  N q

19. Boundary detection Boundary (P, ,  ) Compute the set R of flat vertices; while  p  R and p  N q with q  R and r(p)  h(p) and  v(p),v(q)  R:=R  p; endwhile return P\R end

20. Detected Boundary Samples

22. Undersampling repaired

23. Holes are created

24. Tight Cocone Guarantee: A water tight surface no matter how the input is.

25. Tight Cocone output

26. Holes are created

27. Hole filling

28. Time

30. Large Data Delaunay takes space and time Exact computation is necessary. Doubles the time. Floating pointExact arithmetic

31. Large Data (Supercocone) Octree subdivision

32. Cracks Cracks appear in surface computed from octree boxes

33. Surface matching

34. David’s Head 2 mil points, 93 minutes

35. Lucy25 3.5 million points, 198 mints

36. Shape of arbitrary dimension

37. Tangent and Normal Polytopes T  (p) = V(p)  T(p) N  (p) = V(p)  N(p)

38. Experiments

39. Sample Decimation Original 40K points   = 0.4 8K points   = 0.33 12K points

40. Rocker   0.33 11K points Original 35K points

41. Bunny   0.4 7K points   0.33 11K points Original 35K points

42. Bunny   0.4 7K points   0.33 11K points Original 35K points

43. Triangle Aspect Ratio

44. Medial axis

46. Noise Outliers Cleaned

47. Noise (Local) This is a challenge unsolved. Perturbation by very tiny amount is tolerated by Cocone.

48. Boundaries EngineeringMedical

49. Geometric Models SportsDrug design

50. Geometric Models Entertainment Mathematical

51. Meshing

52. Boundary Detection

53. Data set Engine

54. Undersampling for Nonsmoothness

55. Modeling by Parts

56. Simplification Sample decimation vs. model decimation

57. Guarantees Topology preserved, no self intersection, feature dependent 13751 tri3100 tri

58. Multiresolution 15766 tri10202 tri 7102 tri

59. Model Analysis Feature line detection Detection of dimensionality

60. Mixed Dimensions

61. Model Reconstruction after Data Segmentation

62. Conclusions SBGM with Del/Vor diagrams has great potential Challenges are Boundaries Nonsmoothness Noise Large data Robust simplification Robust feature detection