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

2. Parameterization  Problem: Given a surface S in R 3, find a one-to-one function f : D-> R 3, D R 2, such that the image of D is S. f D 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=(x 1, x 2,…, x n ) in R 3, compute U = (u 1, u 2,…, u n ) in R 2  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.  Michael S. Floater  Professor at the Department of Informatics (IFI) of the University of Oslo, and member of the Center of Mathematics for Applications(CMA), Norway.  Editor of the journal Computer Aided Geometric Design.

11.  Martin Reimers  Postdoctor  CMA, University of Olso, Norway

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

14. Convex Contraints  Boundary condition : map boundary of X to points on a unit circle  If x j ’s are neighbors of x i then require u i to be a strictly convex combination of u j ’s  Solve resulting linear system Au = b

15. 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

16. Compute Boundary  Identify boundary points  Order boundary points

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

18. Uniform Weights  Uniform weights : (minimizing )  If N i ∪ {i} = N k ∪ {k}, then u i =u k

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

20. Shape Preserving Weights

21. Experiments

22. CPU Usage  Reciprocal distance weights  Shape preserving weights

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

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

25. Meshless Parameterization Point setMeshless parameterization

26. Triangulation Delaunay triangulationSurface triangulation

27. Reparameterization Shape-preserving parameterization Spline surface

28. Retriangulation Delaunay retriangulationSurface retriangulation

29. Example Point set Triangulation Spline surface

30. Example Point setTriangulationSpline surface

31.  Tim Volodine  PhD student, research assistant  K.U. Leuven, Belgium

32.  Dirk Roose  Professor  Department of Computer Science, Faculty of Applied Sciences, Head of the research group Scientific Computing  K.U.Leuven, Belgium

33.  Denis Vanderstraeten  Director of Research and IPR at Metris  J2EE Business Analyst / Software Engineer  Belgium

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

35. Boundary Extraction Boundary points :

36. Order Boundary Points

37. Mean Value Weight

38. Experiments

39.  Kai Hormann  Assistant professor  Department of informatics, Computer graphics group  Clausthal University of Technology, Germany

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

41. Spherical Topology

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

43. Partition

44. Reconstruction of one subset

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

46.  Matthias Zwicker  Assistant Professor  Computer Graphics Laboratory  University of California, San Diego, USA

47.  Craig Gotsman  Professor  Department of Computer Science  Harvard University

48.  Meshing point clouds using spherical parameterization  Authors:  Matthias Zwicker Matthias Zwicker  Craig Gotsman 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

49. Spherical parameterization

50. Spherical Parameterization

51. O(n 2 ) Complexity

52.  Geetika Tewari  Graduate Student  Computer Science, Division of Engineering and Applied Sciences  Harvard University

53.  Steven J. Gortler  Co-Director of Undergraduate Studies in Computer Science  Harvard University

54.  Meshing genus-1 point clouds using discrete one-forms  Authors:  Geetika Tewari Geetika Tewari  Craig Gotsman Craig Gotsman  Steven J. Gortler 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

56. Discrete one-forms

58. Seamless local parameterization

59. MCB : Minimal Cycle Basis

60. MCB Cycles on a KNNG MCBMCB : Minimal cycle basis Trivial cycle Nontrivial cycle O(E 3 ) time One Forms on Arbitrary Graph

61. One-forms on the KNNG

62. Parameterize subgraphs

63. Example

64. Summary  Disk topology  Fast and efficient  Complex topology  Slow  Other Methods  More applications  Surface fitting  Ect.