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Spectral surface reconstruction

Spectral surface reconstruction. Reporter: Lincong Fang 24th Sep, 2008. Point clouds. Surface reconstruction. Unorganized Unoriented (no oriented normals) Non-uniform, sparse sampling Possibly with noise. Applications. Computer Graphics Medical Imaging Computer-aided Design

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Spectral surface reconstruction

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  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, ArielShamir, 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. Pole graph G 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. Point set Tensor estimation Implicit function + contouring Overview

  30. Tensor estimation

  31. Normal estimation(PCA)

  32. Voronoi PCA

  33. Noise-free case

  34. Noise-free vs noisy

  35. Noisy case

  36. Tensors Implicit function Implicit function

  37. Delaunay refinement

  38. Delaunay refinement

  39. Variational formulation • Find implicit function f such that its gradient  f best aligns to the principal component of the tensors Biharmonic energyMeasures smoothness of f Anisotropic Dirichlet energy Measures alignment with tensors 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 V vertices {vi} E edges {ei} Tensor C F=(f1,f2,…,fv)t 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.

  43. Solver

  44. Generalized eigenvalue problem Eigenvector (PWL function) max

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

  46. Contouring F=(f1,f2,…,fv)t

  47. Sparse sampling

  48. Noise

  49. Nested components

  50. Poisson GEP Comparison Poisson reconstruction

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