Meshless parameterization and surface reconstruction Reporter: Lincong Fang 16th May, 2007

Parameterization  Problem: Given a surface S in R 3, find a one-to-one function f : D-> R 3, D R 2, such that the image of D is S. f D S

Surface Reconstruction  Problem: Given a set of unorganized points, approximate the underlying surface.

Related Works  Surface reconstruction  Delaunay / Voronoi based  Implicit methods  Provable  Parameterization for organized point set f

Mesh Parameterization  There are many papers

Meshless Parameterization f

Papers  Meshless parameterization and surface reconstruction  Michael S. Floater, Martin Reimers, CAGD 2001  Meshless parameterization and B-spline surface approximation  Michael S. Floater, in The Mathematics of Surfaces IX, Springer-Verlag (2000)  Efficient Triangulation of point clouds using floater parameterization  Tim Volodine, Dirk Roose, Denis Vanderstraeten, Proc. of the Eighth SIAM Conference on Geometric Design and Computing  Triangulating point clouds with spherical topology  Kai Hormann, Martin Reimers, Proceedings of. Curve and Surface Design, 2002  Meshing point clouds using spherical parameterization  M. Zwicker, C. Gotsman, Eurographics 2004  Meshing genus-1 point clouds using discrete one-forms  Geetika Tewari, Craig Gotsman, Steven J. Gortler, Computers & Graphics 2006  Meshless thin-shell simulation based on global conformal parameterization  Xiaohu Guo, Xin Li, Yunfan Bao, Xianfeng Gu, Hong Qin, IEEE ToV and CG 2006

Basic Idea  Given X=(x 1, x 2,…, x n ) in R 3, compute U = (u 1, u 2,…, u n ) in R 2  Triangulate U  Obtain both a triangulation and a parameterization for X

Compute U  Assumptions  X are samples from a 2D surface  Topology is known  Desirable property  Points closed by in U are close by in X

 Michael S. Floater  Professor at the Department of Informatics (IFI) of the University of Oslo, and member of the Center of Mathematics for Applications(CMA), Norway.  Editor of the journal Computer Aided Geometric Design.

 Martin Reimers  Postdoctor  CMA, University of Olso, Norway

 Meshless parameterization and surface reconstruction  Authors:  Michael S. Floater Michael S. Floater  Martin Reimers Martin Reimers  Computer Aided Geometric Design 2001 Main reference : Parameterization and smooth approximation of surface triangulations, Michael S. Floater, CAGD 1997

Convex Contraints  Boundary condition : map boundary of X to points on a unit circle  If x j ’s are neighbors of x i then require u i to be a strictly convex combination of u j ’s  Solve resulting linear system Au = b

Identify Boundary  Use natural boundary  (given as part of the data)  Choose a boundary manually  Compute boundary  Identify boundary points  Order boundary points : curve reconstruction

Compute Boundary  Identify boundary points  Order boundary points

Neighbors and Weights  Ball neighborhoods  Radius is fixed  K nearest neighborhoods  Weights  Uniform weights  Reciprocal distance weights  Shape preserving weights

Uniform Weights  Uniform weights : (minimizing )  If N i ∪ {i} = N k ∪ {k}, then u i =u k

Reciprocal Distance Weights  Weights:  Observation:  Minimizing  Chord parameterization for curves  Distinct parameter points  Well behaved triangulation

Shape Preserving Weights

Experiments

CPU Usage  Reciprocal distance weights  Shape preserving weights

Effect of Noise No Noise Noise added Reciprocal distance weight

 Meshless parameterization and B-spline surface approximation  Author:  Michael S. Floater Michael S. Floater  in The Mathematics of Surfaces IX, R. Cipolla and R. Martin (eds.), Springer- Verlag (2000)

Meshless Parameterization Point setMeshless parameterization

Triangulation Delaunay triangulationSurface triangulation

Reparameterization Shape-preserving parameterization Spline surface

Retriangulation Delaunay retriangulationSurface retriangulation

Example Point set Triangulation Spline surface

Example Point setTriangulationSpline surface

 Tim Volodine  PhD student, research assistant  K.U. Leuven, Belgium

 Dirk Roose  Professor  Department of Computer Science, Faculty of Applied Sciences, Head of the research group Scientific Computing  K.U.Leuven, Belgium

 Denis Vanderstraeten  Director of Research and IPR at Metris  J2EE Business Analyst / Software Engineer  Belgium

 Efficient triangulation of point clouds using Floater Parameterization  Authors:  Tim Volodine Tim Volodine  Dirk Roose Dirk Roose  Denis Vanderstraeten Denis Vanderstraeten  Proc. of the Eighth SIAM Conference on Geometric Design and Computing Main reference : Mean value coordinates, Michael S. Floater, CAGD 2003

Boundary Extraction Boundary points :

Order Boundary Points

Mean Value Weight

Experiments

 Kai Hormann  Assistant professor  Department of informatics, Computer graphics group  Clausthal University of Technology, Germany

 Triangulating point clouds with spherical topology  Authors:  Kai Hormann Kai Hormann  Martin Reimers Martin Reimers  Proceedings of. Curve and Surface Design 2002

Spherical Topology

Partition Point set 12 nearest neighbors Shortest path Correspond to the edges of D

Partition

Reconstruction of one subset

Optimization Optimizing 3D triangulations using discrete curvature analysis Dyn N., K. Hormann, S.-J. Kim, and D. Levin

 Matthias Zwicker  Assistant Professor  Computer Graphics Laboratory  University of California, San Diego, USA

 Craig Gotsman  Professor  Department of Computer Science  Harvard University

 Meshing point clouds using spherical parameterization  Authors:  Matthias Zwicker Matthias Zwicker  Craig Gotsman Craig Gotsman  Eurographics Symposium on Point-Based Graphics 2004 Main references : Fundamentals of spherical parameterization for 3d meshes Gotsman C., Gu X., Sheffer A. SiG 2003 Computing conformal structures of surfaces Gu X., Yau S.-T. Communications in Information and Systems 2002

Spherical parameterization

Spherical Parameterization

O(n 2 ) Complexity

 Geetika Tewari  Graduate Student  Computer Science, Division of Engineering and Applied Sciences  Harvard University

 Steven J. Gortler  Co-Director of Undergraduate Studies in Computer Science  Harvard University

 Meshing genus-1 point clouds using discrete one-forms  Authors:  Geetika Tewari Geetika Tewari  Craig Gotsman Craig Gotsman  Steven J. Gortler Steven J. Gortler  Computers & Graphics 2006 Main references : Computing conformal structures of surfaces Gu X., Yau S.-T. Communications in Information and Systems 2002 Discrete one-forms on meshes and applications to 3D mesh parameterization Gortler SJ, Gotsman C, Thurston D. CAGD 2006

Discrete one-forms

Seamless local parameterization

MCB : Minimal Cycle Basis

MCB Cycles on a KNNG MCBMCB : Minimal cycle basis Trivial cycle Nontrivial cycle O(E 3 ) time One Forms on Arbitrary Graph

One-forms on the KNNG

Parameterize subgraphs

Example

Summary  Disk topology  Fast and efficient  Complex topology  Slow  Other Methods  More applications  Surface fitting  Ect.