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Spectral surface reconstruction Reporter: Lincong Fang 24th Sep, 2008
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Point clouds
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Surface reconstruction Unorganized Unoriented (no oriented normals) Non-uniform, sparse sampling Possibly with noise
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Applications Computer Graphics Medical Imaging Computer-aided Design Solid Modeling
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Approaches Delaunay\Voronoi based Implicit surfaces Deformable models Spectral Etc.
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Approaches Delaunay\Voronoi based Unorganized, unoriented, non-uniform, noise
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Approaches Implicit surfaces Unorganized, unoriented, non-uniform, noise
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Approaches Deformable models Adrei Sharf, Thomas Lewiner, Ariel Shamir, Leif Kobbelt, Daniel Cohen–OR. Competing fronts for coarse–to–fine surface reconstruction. EG2006
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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.
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Spectral surface reconstruction from noisy point clouds R. Kolluri (Google) J. Richard Shewchuk J. F. O’Brien University of Califonia, Berkeley SGP 2004
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The eigencrust algorithm Partition the tetrahedra of a Delaunay tetrahedralization into inside/outside Identify the triangular faces that interface between the subgraphs
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Poles Nina Amenta, Marshall Bern, Manolis Kamvysselis. A new Voronoi-based surface reconstruction algorithm. SigGraph 98
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Pole graph G
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The negatively weighted edges of the pole graph
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Pole graph G The positively weighted edges of pole graph
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Weights
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Super node->G’
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Pole matrix
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Remaining tetrahedra
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The final mesh The final mesh is the “eigencrust” The triangles where the inside and outside tetrahedra meet
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Results If all adjacent tetrahedra are labeled the same, the point is an outlier Undersampled regions are handled without holes
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More results
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Efficacy 2008414 input points Tetrahedralize:13.5 minutes 157 minutes 265minutes
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Voronoi-based variational reconstruction of unoriented point sets P. Alliez D. Cohen-Steiner Y. Tong M. Desbrun SGP 2007 (best paper award)
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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
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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
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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
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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
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Overview Point set Tensor estimation Implicit function + contouring
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Tensor estimation
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Normal estimation(PCA)
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Voronoi PCA
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Noise-free case
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Noise-free vs noisy
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Noisy case
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Implicit function Tensors
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Delaunay refinement
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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
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Rationale Anisotropic tensors: favor alignment Isotropic tensors: favor smoothness
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Rationale Anisotropic tensors: favor alignment Isotropic tensors: favor smoothness Large aligned gradients + smoothness ->consistent orientation of f
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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
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Solver
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Generalized eigenvalue problem maxEigenvector (PWL function)
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Standard eigenvalue problem Solver: Implicitly restarted Arnoldi method (ARPACK++)
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Contouring F=(f 1,f 2,…,f v ) t
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Sparse sampling
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Noise
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Nested components
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Comparison PoissonGEP Poisson reconstruction
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Comparison Poisson reconstruction
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Sforz(250K points) Out-of-core factorization 25 minutes
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Conclusion Pros Handles unoriented point sets Handles noisy point sets Cons Slow Not easy to implement
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