Variational implicit surface meshing

In this paper, we propose a new algorithm to mesh implicit surfaces which produces meshes both with a good triangle aspect ratio as well as a good approximation quality. The number of vertices of the output mesh is defined by the end-user. For this goal, we perform a two-stage processing: an initialization step followed by an iterative optimization step. The initialization step consists in capturing the surface topology and allocating the vertex budget. The optimization algorithm is based on a variational vertices relaxation and triangulation update. In addition a gradation parameter can be defined to adapt the mesh sampling to the curvature of the implicit surface. We demonstrate the efficiency of the approach on synthetic models as well as real-world acquired data, and provide comparisons with previous approaches.

[1]  Martin Isenburg,et al.  Isotropic surface remeshing , 2003, 2003 Shape Modeling International..

[2]  Thomas Lewiner,et al.  Efficient Implementation of Marching Cubes' Cases with Topological Guarantees , 2003, J. Graphics, GPU, & Game Tools.

[3]  David J. Kriegman,et al.  Parameterized Families of Polynomials for Bounded Algebraic Curve and Surface Fitting , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[4]  Ho-Lun Cheng,et al.  Dynamic Skin Triangulation , 2001, SODA '01.

[5]  Tamal K. Dey,et al.  Sampling and meshing a surface with guaranteed topology and geometry , 2004, SCG '04.

[6]  Herbert Edelsbrunner,et al.  Triangulating topological spaces , 1994, SCG '94.

[7]  Joshua A. Levine,et al.  Delaunay Meshing of Isosurfaces , 2007, IEEE International Conference on Shape Modeling and Applications 2007 (SMI '07).

[8]  Joel H. Saltz,et al.  Ieee Transactions on Visualization and Computer Graphics Reconstruction of Cellular Biological Structures from Optical Microscopy Data , 2022 .

[9]  Steve Oudot,et al.  Provably good sampling and meshing of surfaces , 2005, Graph. Model..

[10]  E Chernyaev,et al.  Marching cubes 33 : construction of topologically correct isosurfaces , 1995 .

[11]  Richard K. Beatson,et al.  Reconstruction and representation of 3D objects with radial basis functions , 2001, SIGGRAPH.

[12]  Alexander A. Pasko,et al.  Geometric modeling in the analysis of trivariate functions , 1988, Comput. Graph..

[13]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[14]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[15]  Mario Botsch,et al.  Feature sensitive surface extraction from volume data , 2001, SIGGRAPH.

[16]  Hans-Peter Seidel,et al.  Multi-level partition of unity implicits , 2003, ACM Trans. Graph..

[17]  Jean-Daniel Boissonnat,et al.  Isotopic Implicit Surface Meshing , 2004, STOC '04.

[18]  Rémy Prost,et al.  Generic Remeshing of 3D Triangular Meshes with Metric-Dependent Discrete Voronoi Diagrams , 2008, IEEE Transactions on Visualization and Computer Graphics.

[19]  Denis Friboulet,et al.  Compactly Supported Radial Basis Functions Based Collocation Method for Level-Set Evolution in Image Segmentation , 2007, IEEE Transactions on Image Processing.

[20]  Jean-Marc Chassery,et al.  Approximated Centroidal Voronoi Diagrams for Uniform Polygonal Mesh Coarsening , 2004, Comput. Graph. Forum.

[21]  Sarah F. Frisken.  Constrained Elastic Surface Nets: Generating Smooth Surfaces from Binary Segmented Data , 1998, MICCAI.

[22]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[23]  Hao Zhang,et al.  Delaunay mesh construction , 2007, Symposium on Geometry Processing.

[24]  Adrian Hilton,et al.  Marching triangles: range image fusion for complex object modelling , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[25]  Samir Akkouche,et al.  Adaptive Implicit Surface Polygonization Using Marching Triangles , 2001, Comput. Graph. Forum.

[26]  Michael Garland,et al.  Surface simplification using quadric error metrics , 1997, SIGGRAPH.

[27]  Pierre Alliez,et al.  Isotropic Remeshing of Surfaces: A Local Parameterization Approach , 2003, IMR.

[28]  Mohan S. Kankanhalli,et al.  Adaptive marching cubes , 1995, The Visual Computer.

[29]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[30]  John C. Hart,et al.  Guaranteeing the topology of an implicit surface polygonization for interactive modeling , 1997, SIGGRAPH Courses.

[31]  Tao Ju,et al.  Dual contouring of hermite data , 2002, ACM Trans. Graph..

[32]  Nadia Magnenat-Thalmann,et al.  Surface from Scattered Points: A Brief Survey of Recent Developments , 2005 .

[33]  Cláudio T. Silva,et al.  High-Quality Extraction of Isosurfaces from Regular and Irregular Grids , 2006, IEEE Transactions on Visualization and Computer Graphics.

[34]  Alexander A. Pasko,et al.  Discretization of functionally based heterogeneous objects , 2003, SM '03.

[35]  Peter Lindstrom,et al.  Out-of-core simplification of large polygonal models , 2000, SIGGRAPH.

[36]  L. Paul Chew,et al.  Guaranteed-quality mesh generation for curved surfaces , 1993, SCG '93.

[37]  Yutaka Ohtake,et al.  Dual/Primal mesh optimization for polygonized implicit surfaces , 2002, SMA '02.

[38]  Joshua A. Levine,et al.  Delpsc: a delaunay mesher for piecewise smooth complexes , 2008, SCG '08.