Learning smooth shapes by probing

We consider the problem of discovering a smooth unknown surface S bounding an object O in R^3. The discovery process consists of moving a point probing device in the free space around O so that it repeatedly comes in contact with S. We propose a probing strategy for generating a sequence of surface samples on S from which a triangulated surface can be generated that approximates S within any desired accuracy. We bound the number of probes and the number of elementary moves of the probing device. Our solution is an extension of previous work on Delaunay refinement techniques for surface meshing. The approximating surface we generate enjoys the many nice properties of the meshes obtained by those techniques, e.g. exact topological type, normal approximation, etc.

[1]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[2]  Richard Cole,et al.  Shape from Probing , 1987, J. Algorithms.

[3]  Howie Choset,et al.  Sensor-Based Exploration: Incremental Construction of the Hierarchical Generalized Voronoi Graph , 2000, Int. J. Robotics Res..

[4]  Steven Skiena,et al.  Geometric Reconstruction Problems , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

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

[6]  Ho-Lun Cheng,et al.  Guaranteed quality triangulation of molecular skin surfaces , 2004, IEEE Visualization 2004.

[7]  Mariette Yvinec,et al.  Probing a scene of nonconvex polyhedra , 2005, Algorithmica.

[8]  Joseph F. Traub,et al.  Complexity and information , 1999, Lezioni Lincee.

[9]  Houman Borouchaki,et al.  Molecular Surface Modeling and Meshing , 2002, Engineering with Computers.

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

[11]  Gert Vegter,et al.  Meshing skin surfaces with certified topology , 2005, Ninth International Conference on Computer Aided Design and Computer Graphics (CAD-CG'05).

[12]  Alfred M. Bruckstein,et al.  Blind approximation of planar convex sets , 1994, IEEE Trans. Robotics Autom..

[13]  Herbert Edelsbrunner,et al.  Triangulating Topological Spaces , 1997, Int. J. Comput. Geom. Appl..

[14]  S. Oudot,et al.  Sampling and Meshing Surfaces with Guarantees , 2005 .

[15]  Leonidas J. Guibas,et al.  Learning smooth objects by probing , 2005, SCG.

[16]  Kenneth L. Clarkson,et al.  Building triangulations using ε-nets , 2006, STOC '06.

[17]  David P. Dobkin,et al.  Probing Convex Polytopes , 1990, Autonomous Robot Vehicles.

[18]  Howie Choset,et al.  Sensor-Based Exploration: The Hierarchical Generalized Voronoi Graph , 2000, Int. J. Robotics Res..

[19]  Sunghee Choi,et al.  A Simple Algorithm for Homeomorphic Surface Reconstruction , 2002, Int. J. Comput. Geom. Appl..

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

[21]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[22]  Tamal K. Dey,et al.  Sampling and Meshing a Surface with Guaranteed Topology and Geometry , 2007, SIAM J. Comput..

[23]  Tamal K. Dey,et al.  Provable surface reconstruction from noisy samples , 2006, Comput. Geom..

[24]  T. J. Richardson,et al.  Approximation of Planar Convex Sets from Hyperplane Probes , 1997, Discret. Comput. Geom..

[25]  H. Woxniakowski.  Information-Based Complexity , 1988 .

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

[27]  Mariette Yvinec,et al.  Non Convex Contour Reconstruction , 1990, J. Symb. Comput..

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

[29]  Jean-Daniel Boissonnat,et al.  Complexity of the delaunay triangulation of points on surfaces the smooth case , 2003, SCG '03.

[30]  Kenji Shimada,et al.  Automatic triangular mesh generation of trimmed parametric surfaces for finite element analysis , 1998, Comput. Aided Geom. Des..

[31]  Gert Vegter,et al.  Isotopic approximation of implicit curves and surfaces , 2004, SGP '04.

[32]  Günter Rote,et al.  The convergence rate of the sandwich algorithm for approximating convex functions , 1992, Computing.

[33]  T. O’Neil.  Geometric Measure Theory , 2002 .

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

[35]  Steven J. Owen,et al.  Advancing Front Surface Mesh Generation in Parametric Space Using a Riemannian Surface Definition , 1998, IMR.

[36]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[37]  Mariette Yvinec,et al.  Probing a scene of non convex polyhedra , 1989, SCG '89.