Spectral surface reconstruction

Reporter: Lincong Fang24th 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 2004.[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

( , )G V E

:1.( , ), , are poles of

2.( , '),( , '), ( ', ), ( , '),

, are poles of , ', '

are poles of ', and ( , ')

is an edge of the Delaunay

tetrahedralization

E u v u v s

u u u v u v v v

u v s u v

s s s

Pole graph GThe negatively weighted edges of the pole graph

Pole graph GThe positively weighted edges of pole graph

Weights

4 4cos,u v e 4 4cos

,u v e

Super node->G’

: outside polesO super node z

, ,z v u O u v

Pole matrix

,i jij ji v vL L | |ii ijj iL L

Lx Dx ii iiD L

ix one node of G' one pole one tetrahedron

: the eigenvector associated with the

smallest eigenvalue

x

Remaining tetrahedra

Graph H: two supernodes (inside and outside)

Edge: if two unlabeled tetrahedra share a triangular face

Edge to one of the supernodes: if an unlabeled tetrahedron

shares a face with labeled tetrahedron

max

min

Weight(aspect ratio):e

e

One negative edge weight:

an edge connecting the

inside and outside supernodes

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

2008414 input pointsTetrahedralize: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 setPoint setTensor Tensor

estimationestimationImplicit functionImplicit function

+ contouring+ contouring

Tensor estimation

Normal estimation(PCA)

1 1

T

i i

ik ik

p p p p

C

p p p p

l l lC v v

0( ) : ( ) 0T x x p v

Voronoi PCA

Noise-free case

Noise-free vs noisy

Noisy case

Implicit function

TensorsTensors Implicit functionImplicit function

Delaunay refinement

Delaunay refinement

Variational formulation

Find implicit function f such that its gradient f best aligns to the principal component of the tensors

Anisotropic Dirichlet energyMeasures alignment with tensors

Biharmonic energyMeasures smoothness of ff

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 forComputational modeling. In Discrete Differential Geometry.ACM SIGGRAPH Course, 2006.

V vertices {vi}E edges {ei}

Tensor C F=(f1,f2,…,fv)t

Solver

10 0*t

cA d d 10 0*tB d d

0: vertex/edge incident matrix( )td E V

1 1(* ) (* ) , 1...ti i

c ii iiti i

e Cei E

e e

*1 | |

(* )| |

iii

i

e

e

Generalized eigenvalue problem

(1 )t tE F AF F BF

AF BF

/ 0E F

maxEigenvector(PWL function)

Standard eigenvalueproblem

Compute Cholesky factorization of B(TAUCS)

tB LL

11

tt t t t

t

L AL G GAF LL F L AL L F L F

G L F

Solver: Implicitly restarted Arnoldi method (ARPACK++)

Contouring

F=(f1,f2,…,fv)t

Sparse sampling

Noise

Nested components

Comparison

PoissonPoisson GEPGEP

Poisson reconstruction

Comparison

Poisson reconstruction

Sforz(250K points)

Out-of-core factorization25 minutes

Conclusion

Pros Handles unoriented point sets Handles noisy point sets

Cons Slow Not easy to implement