Meshless parameterization and surface reconstruction

1. Meshless parameterization and surface reconstruction Reporter: Lincong Fang 16th May, 2007

2. Parameterization • Problem: Given a surface S in R3, find a one-to-one function f : D-> R3, D R2, such that the image of D is S. D f S

3. Surface Reconstruction • Problem: Given a set of unorganized points, approximate the underlying surface.

4. Related Works • Surface reconstruction • Delaunay / Voronoi based • Implicit methods • Provable • Parameterization for organized point set f

5. Mesh Parameterization • There are many papers

6. Meshless Parameterization f

7. Papers • Meshless parameterization and surface reconstruction • Michael S. Floater, Martin Reimers, CAGD 2001 • Meshless parameterization and B-spline surface approximation • Michael S. Floater, in The Mathematics of Surfaces IX, Springer-Verlag (2000) • Efficient Triangulation of point clouds using floater parameterization • Tim Volodine, Dirk Roose, Denis Vanderstraeten, Proc. of the Eighth SIAM Conference on Geometric Design and Computing • Triangulating point clouds with spherical topology • Kai Hormann, Martin Reimers, Proceedings of. Curve and Surface Design, 2002 • Meshing point clouds using spherical parameterization • M. Zwicker, C. Gotsman, Eurographics 2004 • Meshing genus-1 point clouds using discrete one-forms • Geetika Tewari, Craig Gotsman, Steven J. Gortler, Computers & Graphics 2006 • Meshless thin-shell simulation based on global conformal parameterization • Xiaohu Guo, Xin Li, Yunfan Bao, Xianfeng Gu, Hong Qin, IEEE ToV and CG 2006

8. Basic Idea • Given X=(x1, x2,…, xn) in R3, compute U = (u1, u2,…, un) in R2 • Triangulate U • Obtain both a triangulation and a parameterization for X

9. Compute U • Assumptions • X are samples from a 2D surface • Topology is known • Desirable property • Points closed by in U are close by in X

10. Meshless parameterization and surface reconstruction • Authors: • Michael S. Floater • Martin Reimers • Computer Aided Geometric Design 2001 Main reference : Parameterization and smooth approximation of surface triangulations, Michael S. Floater, CAGD 1997

12. Convex Contraints • Boundary condition : map boundary of X to points on a unit circle • If xj’s are neighbors of xi then require ui to be a strictly convex combination of uj’s • Solve resulting linear system Au = b

13. Identify Boundary • Use natural boundary • (given as part of the data) • Choose a boundary manually • Compute boundary • Identify boundary points • Order boundary points : curve reconstruction

14. Compute Boundary • Identify boundary points • Order boundary points

15. Neighbors and Weights • Ball neighborhoods • Radius is fixed • K nearest neighborhoods • Weights • Uniform weights • Reciprocal distance weights • Shape preserving weights

16. Uniform Weights • Uniform weights : (minimizing ) • IfNi ∪{i} = Nk ∪{k}, then ui=uk

17. Reciprocal Distance Weights • Weights: • Observation: • Minimizing • Chord parameterization for curves • Distinct parameter points • Well behaved triangulation

18. Shape Preserving Weights

19. Experiments

20. CPU Usage • Reciprocal distance weights • Shape preserving weights

21. Effect of Noise Noise added Reciprocal distance weight No Noise

22. Meshless parameterization and B-spline surface approximation • Author: • Michael S. Floater • in The Mathematics of Surfaces IX, R. Cipolla and R. Martin (eds.), Springer-Verlag (2000)

23. Meshless Parameterization Point set Meshless parameterization

24. Triangulation Delaunay triangulation Surface triangulation

25. Reparameterization Shape-preserving parameterization Spline surface

26. Retriangulation Delaunay retriangulation Surface retriangulation

27. Example Point set Triangulation Spline surface

28. Example Point set Triangulation Spline surface

29. Efficient triangulation of point clouds using Floater Parameterization • Authors: • Tim Volodine • Dirk Roose • Denis Vanderstraeten • Proc. of the Eighth SIAM Conference on Geometric Design and Computing Main reference : Mean value coordinates, Michael S. Floater, CAGD 2003

30. Boundary Extraction Boundary points :

31. Order Boundary Points

32. Mean Value Weight

33. Experiments

34. Triangulating point clouds with spherical topology • Authors: • Kai Hormann • Martin Reimers • Proceedings of. Curve and Surface Design 2002

35. Spherical Topology

36. Partition Shortest path Correspond to the edges of D 12 nearest neighbors Point set

37. Partition

38. Reconstruction of one subset

39. Optimization Optimizing 3D triangulations using discrete curvature analysis Dyn N., K. Hormann, S.-J. Kim, and D. Levin

40. Meshing point clouds using spherical parameterization • Authors: • Matthias Zwicker • Craig Gotsman • Eurographics Symposium on Point-Based Graphics 2004 Main references : Fundamentals of spherical parameterization for 3d meshes Gotsman C., Gu X., Sheffer A. SiG 2003 Computing conformal structures of surfaces Gu X., Yau S.-T. Communications in Information and Systems 2002

41. Spherical parameterization

42. Spherical Parameterization

43. O(n2) Complexity

44. Meshing genus-1 point clouds using discrete one-forms • Authors: • Geetika Tewari • Craig Gotsman • Steven J. Gortler • Computers & Graphics 2006 Main references : Computing conformal structures of surfaces Gu X., Yau S.-T. Communications in Information and Systems 2002 Discrete one-forms on meshes and applications to 3D mesh parameterization Gortler SJ, Gotsman C, Thurston D. CAGD 2006

46. Discrete one-forms

47. Discrete one-forms

48. Seamless local parameterization

49. MCB Cycles on a KNNG One Forms on Arbitrary Graph MCB : Minimal cycle basis O(E3) time Trivial cycle Nontrivial cycle

50. One-forms on the KNNG