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Alpha shapes

Alpha shapes. Reporter: Lincong Fang 10th Jan, 2007. What is the shape?. What are alpha shapes?. Alpha Controls the desired level of detail. What are alpha shapes?. What are alpha shapes?. Used for. Shape Modelling Creates shapes out of point sets Gives a hierarchy of shapes.

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Alpha shapes

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  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?

  5. 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 Not alpha exposed Alpha exposed

  16. Definition of alpha shapes

  17. Alpha shape

  18. Observations

  19. Definition of alpha complex

  20. Alpha complex

  21. Alpha complex

  22. Observations

  23. Observations

  24. Algorithm

  25. Limitations • Best alpha? • Maybe there is no satisfying alpha.

  26. 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

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

  28. 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

  29. 3D alpha shape

  30. Outer shells of alpha shapes

  31. Exterior face enumeration

  32. Simplicial surface generation

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

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

  35. Anisotropic alpha shapes

  36. Density determination

  37. 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.

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

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

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

  41. Density estimation

  42. Scaling algorithm

  43. Surface reconstruction

  44. Weighted alpha shapes Herbert Edelsbrunner Technical report 1992

  45. Weighted alpha shapes

  46. Alpha exposed x q p

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