Shape understanding by contour-driven retiling

Given a triangle mesh representing a closed manifold surface of arbitrary genus, a method is proposed to automatically extract the Reeb graph of the manifold with respect to the height function. The method is based on a slicing strategy that traces contours while inserting them directly in the mesh as constraints. Critical areas, which identify isolated and non-isolated critical points of the surface, are recognized and coded in the extended Reeb graph (ERG). The remeshing strategy guarantees that topological features are correctly maintained in the graph, and the tiling of ERG nodes reproduces the original shape at a minimal, but topologically correct, geometric level.

[1]  Valerio Pascucci,et al.  Fast isocontouring for improved interactivity , 1996, VVS '96.

[2]  Remco C. Veltkamp,et al.  Shape matching: similarity measures and algorithms , 2001, Proceedings International Conference on Shape Modeling and Applications.

[3]  Dong-Soo Kim,et al.  Surface slicing algorithm based on topology transition , 2001, Comput. Aided Des..

[4]  Debasish Dutta,et al.  An accurate slicing procedure for layered manufacturing , 1996, Comput. Aided Des..

[5]  Tosiyasu L. Kunii,et al.  Constructing a Reeb graph automatically from cross sections , 1991, IEEE Computer Graphics and Applications.

[6]  M. Spagnuolo,et al.  Shape understanding by contour-driven retiling , 2003, The Visual Computer.

[7]  HARRY BLUM,et al.  Shape description using weighted symmetric axis features , 1978, Pattern Recognit..

[8]  David Eppstein,et al.  On triangulating three-dimensional polygons , 1998, Comput. Geom..

[9]  Lee R. Nackman,et al.  Three-Dimensional Shape Description Using the Symmetric Axis Transform I: Theory , 1985, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  George Allen.  An introduction to solid modelling , 1984, Comput. Graph..

[11]  Michel Pocchiola,et al.  Advances in Discrete and Computational Geometry , 1998 .

[12]  Chandrajit L. Bajaj,et al.  Topology preserving data simplification with error bounds , 1998, Comput. Graph..

[13]  Herbert Edelsbrunner,et al.  Auditory Morse Analysis of Triangulated Manifolds , 1997, VisMath.

[14]  Silvia Biasotti,et al.  Shape Abstraction Using Computational Topology Techniques , 2002, Workshop on Geometric Modeling.

[15]  Tosiyasu L. Kunii,et al.  Algorithms for Extracting Correct Critical Points and Constructing Topological Graphs from Discrete Geographical Elevation Data , 1995, Comput. Graph. Forum.

[16]  Taku Komura,et al.  Topology matching for fully automatic similarity estimation of 3D shapes , 2001, SIGGRAPH.

[17]  Han-Wei Shen,et al.  A Near Optimal Isosurface Extraction Algorithm Using the Span Space , 1996, IEEE Trans. Vis. Comput. Graph..

[18]  Adam Ostaszewski,et al.  Topology : a geometric approach , 1992 .

[19]  B. Falcidieno,et al.  Invited Lecture: A Shape Abstraction Paradigm for Modeling Geometry and Semantics , 1998 .

[20]  Anne Verroust-Blondet,et al.  Level set diagrams of polyhedral objects , 1999, SMA '99.

[21]  Richard M. Leahy,et al.  Automated graph-based analysis and correction of cortical volume topology , 2001, IEEE Transactions on Medical Imaging.

[22]  Marc J. van Kreveld,et al.  On Quality Paths on Polyhedral Terrains , 1994, IGIS.

[23]  Tosiyasu L. Kunii,et al.  Surface coding based on Morse theory , 1991, IEEE Computer Graphics and Applications.

[24]  T. Banchoff.  Critical Points and Curvature for Embedded Polyhedral Surfaces , 1970 .

[25]  Marco Attene,et al.  Re-meshing techniques for topological analysis , 2001, Proceedings International Conference on Shape Modeling and Applications.

[26]  Silvia Biasotti,et al.  Extended Reeb Graphs for Surface Understanding and Description , 2000, DGCI.

[27]  Herbert Edelsbrunner,et al.  Hierarchical morse complexes for piecewise linear 2-manifolds , 2001, SCG '01.

[28]  Alex Pentland,et al.  Perceptual Organization and the Representation of Natural Form , 1986, Artif. Intell..

[29]  Jack Snoeyink,et al.  Computing contour trees in all dimensions , 2000, SODA '00.

[30]  Masaki Hilaga,et al.  Topological Modeling for Visualization , 1997 .