1. Alpha shapes Reporter: Lincong Fang 10th Jan, 2007

2. What is the shape?

3. What are alpha shapes? Alpha Controls the desired level of detail.

4. What are alpha shapes?

6. Used for Shape Modelling Creates shapes out of point sets Gives a hierarchy of shapes. Has been used for detecting pockets in proteins. For reverse engineering

7. Papers H. Edelsbrunner, D. G. Kirkpatrick and R.Seidel. On the shape of a set of points in the plane. IEEE ToIT 1983. H. Edelsbrunner and E. P. Mucke. Three dimensional alpha shapes, ToG 1994. Kaspar Fischer, Introduction to alpha shapes.

8. Authors Herbert Edelsbrunner: Chief technical consultant for Geomagic Arts & Science Professor of Computer Science at Duke University The only computer scientist ever to be awarded the prestigious National Science Foundation Alan T. Waterman Award

9. Authors David G. Kirkpatrick Professor, University of British Columbia Raimund Seidel Professor, Chair of Theoretical Computer Science, Saarland University Ernst P Mucke Univ. of Illinois, Urbana-Champaign Kaspar Fischer ETH Zurich, Switzerland

10. How to get alpha shapes The space generated by point pairs that can be touched by an empty disc of radius alpha.

11. General position No 4 points of S lie on a common plane No 5 points lie on a common sphere For any fixed alpha, the smallest sphere through any 2, 3 or 4 points of S has a radius different from alpha.

12. k-simplex

13. Simplicial Complex Delaunay triangulations are simplicial complexes.

14. Delaunay Triangulations

15. Alpha-ball and alpha exposed Alpha exposed Not alpha exposed

16. Definition of alpha shapes

17. Alpha shape

18. Observations

19. Definition of alpha complex

20. Alpha complex

22. Observations

24. Algorithm

27. Best alpha? Alpha Controls the desired level of detail.

28. Limitations Best alpha? Maybe there is no satisfying alpha.

29. Papers Baining Guo, Jai Menon, Brian Willette, Surface reconstruction using alpha shapes, Computer Graphics 1997. M. Teichmann and M. Capps., Surface reconstruction with anisotropic density-scaled alpha-shapes, IEEE Visualization ’ 98. Xiaolong Xu, Koichi Harada, Automatic surface reconstruction with alpha-shape method, Visual Computer 2003

30. Surface reconstruction using alpha shapes Baining Guo, Jai Menon, Brian Willette Computer Graphics 1997

31. Authors Baining Guo The research manager of the graphics group at Microsoft Research Asia. Jai Menon IBM Fellow Brian Willette Department of Computer Science, University of Colorado, USA

32. 3D alpha shape

33. Outer shells of alpha shapes

34. Exterior face enumeration

35. Simplicial surface generation

36. Surface reconstruction with anisotropic density-scaled alpha- shapes M. Teichmann and M. Capps. IEEE Visualization ’ 98.

37. Authors Marek Teichmann Lab. for Computer Science, MIT Michael Capps President of Epic Games

39. Anisotropic alpha shapes

40. Density determination

41. Scaling algorithm Reduces the size of the alpha ball in areas where point density is high. Undesired effect of passing very large convex hull triangles.

42. Scaling algorithm Removes triangles which connect high-density and low density areas. Undesirable effects in non-nuiformly distributed point-sets

44. Automatic surface reconstruction with alpha-shape method Xiaolong Xu, Koichi Harada Visual Computer 2003

45. Authors Xiaolong Xu Ph.D candidate of Hiroshima, University, Japan Koichi Harada Professor for Hiroshima University

46. Density estimation

47. Scaling algorithm

48. Surface reconstruction

49. Weighted alpha shapes Herbert Edelsbrunner Technical report 1992

50. Weighted alpha shapes

51. Alpha exposed p q x

52. Weighted alpha shape

53. Benefit In biology and chemistry, modeling molecular structures. In reconstructing a surface from point data not uniform. Enforce certain edges or faces.

54. A surface reconstruction algorithm using weighted alpha shapes Si Hyung Park, Seoung Soo Lee, Jong Hwa Kim FSKD 2005

55. Authors Si Hyung Park Voronoi Diagram Research Center, Hanyang University, Korea Seoung Soo Lee, Jong Hwa Kim CAESIT, Konkuk University, Korea

56. Calculation of weights

59. Constant and weighted alpha

61. Peel off

62. The conformal alpha shape filtration Frederic Cazals, Joachim Giesen, Mark Pauly, Afra Zomorodian Visual Computer 2006

63. Authors Frederic Cazals Research Director, INRIA Sophia-Antipolis Joachim Giesen, Mark Pauly Computer Science, ETH Zurich, Switzerland Afra Zomorodian Assistant Professor in the Department of Computer Science at Dartmouth College

64. Conformal alpha shapes

66. Surface reconstruction