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 Solid Modeling

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

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

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

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

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 [2] P. Alliez, D. Cohen-Steiner, Y. Tong, M. Desbrun Voronoi-based variational reconstruction of unoriented point sets. SGP 2007.

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

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

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

Pole graph G

The negatively weighted edges of the pole graph

Pole graph G The positively weighted edges of pole graph

Weights

Super node->G’

Pole matrix

Remaining tetrahedra

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

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

More results

Efficacy input points Tetrahedralize:13.5 minutes 157 minutes 265minutes

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

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

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

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

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

Overview Point set Tensor estimation Implicit function + contouring

Tensor estimation

Normal estimation(PCA)

Voronoi PCA

Noise-free case

Noise-free vs noisy

Noisy case

Implicit function Tensors

Delaunay refinement

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

Rationale Anisotropic tensors: favor alignment Isotropic tensors: favor smoothness

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

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, V vertices { v i } E edges { e i } Tensor C F=(f 1,f 2,…,f v ) t

Solver

Generalized eigenvalue problem maxEigenvector (PWL function)

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

Contouring F=(f 1,f 2,…,f v ) t

Sparse sampling

Noise

Nested components

Comparison PoissonGEP Poisson reconstruction

Comparison Poisson reconstruction

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

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