1. Spectral surface reconstruction Reporter: Lincong Fang 24th Sep, 2008

2. Point clouds

3. Surface reconstruction Unorganized Unoriented (no oriented normals) Non-uniform, sparse sampling Possibly with noise

4. Applications Computer Graphics Medical Imaging Computer-aided Design Solid Modeling

5. Approaches Delaunay\Voronoi based Implicit surfaces Deformable models Spectral Etc.

6. Approaches Delaunay\Voronoi based Unorganized, unoriented, non-uniform, noise

7. Approaches Implicit surfaces Unorganized, unoriented, non-uniform, noise

8. Approaches Deformable models Adrei Sharf, Thomas Lewiner, Ariel Shamir, Leif Kobbelt, Daniel Cohen–OR. Competing fronts for coarse–to–fine surface reconstruction. EG2006

9. Approaches Delaunay\Voronoi based Implicit surfaces Deformable models Spectral Etc. [1] R. Kolluri, J. Richard Shewchuk, J. F. O’Brien, Spectral surface reconstruction from noisy point clouds. SGP 2004. [2] P. Alliez, D. Cohen-Steiner, Y. Tong, M. Desbrun Voronoi-based variational reconstruction of unoriented point sets. SGP 2007.

10. Spectral surface reconstruction from noisy point clouds R. Kolluri (Google) J. Richard Shewchuk J. F. O’Brien University of Califonia, Berkeley SGP 2004

11. The eigencrust algorithm Partition the tetrahedra of a Delaunay tetrahedralization into inside/outside Identify the triangular faces that interface between the subgraphs

12. Poles Nina Amenta, Marshall Bern, Manolis Kamvysselis. A new Voronoi-based surface reconstruction algorithm. SigGraph 98

13. Pole graph G

14. The negatively weighted edges of the pole graph

15. Pole graph G The positively weighted edges of pole graph

16. Weights

17. Super node->G’

18. Pole matrix

19. Remaining tetrahedra

20. The final mesh The final mesh is the “eigencrust” The triangles where the inside and outside tetrahedra meet

21. Results If all adjacent tetrahedra are labeled the same, the point is an outlier Undersampled regions are handled without holes

22. More results

23. Efficacy 2008414 input points Tetrahedralize:13.5 minutes 157 minutes 265minutes

24. Voronoi-based variational reconstruction of unoriented point sets P. Alliez D. Cohen-Steiner Y. Tong M. Desbrun SGP 2007 (best paper award)

25. Pierre Alliez Researcher at INRIA in the GEOMETRICA group Research Geometry Processing: geometry compression, surface approximation, mesh parameterization, surface remeshing and mesh generation Avid user of the CGAL library CGAL developer

26. David Cohen-Steiner Researcher at INRIA in the GEOMETRICA team Research Approximation problems in applied geometry and topology Meshes and point clouds are of particular interest

27. Yiying Tong Assistant Professor Computer Science and Engineering Dept. at Michigan State University Research Computer simulation/animation Discrete geometric modeling Discrete differential geometry Face recognition using 3D models

28. Mathieu Desbrun Associate Professor in Computer Science and Computational Science & Engineering California Institute of Technology Research Applying discrete differential geometry to a wide range of fields and applications

29. Overview Point set Tensor estimation Implicit function + contouring

30. Tensor estimation

31. Normal estimation(PCA)

32. Voronoi PCA

33. Noise-free case

34. Noise-free vs noisy

35. Noisy case

36. Implicit function Tensors

37. Delaunay refinement

39. Variational formulation Find implicit function f such that its gradient  f best aligns to the principal component of the tensors Anisotropic Dirichlet energy Measures alignment with tensors  f Biharmonic energy Measures smoothness of  f Regularization

40. Rationale Anisotropic tensors: favor alignment Isotropic tensors: favor smoothness

41. Rationale Anisotropic tensors: favor alignment Isotropic tensors: favor smoothness Large aligned gradients + smoothness ->consistent orientation of  f

42. Solver A: Anisotropic Laplacian operator B: Isotropic Bilaplacian operator Desbrun M, Kanso E, Tong Y. Discrete differential forms for Computational modeling. In Discrete Differential Geometry. ACM SIGGRAPH Course, 2006. V vertices { v i } E edges { e i } Tensor C F=(f 1,f 2,…,f v ) t

43. Solver

44. Generalized eigenvalue problem maxEigenvector (PWL function)

45. Standard eigenvalue problem Solver: Implicitly restarted Arnoldi method (ARPACK++)

46. Contouring F=(f 1,f 2,…,f v ) t

47. Sparse sampling

48. Noise

49. Nested components

50. Comparison PoissonGEP Poisson reconstruction

51. Comparison Poisson reconstruction

52. Sforz(250K points) Out-of-core factorization 25 minutes

53. Conclusion Pros Handles unoriented point sets Handles noisy point sets Cons Slow Not easy to implement