Level set diagrams of polyhedral objects

Shape descriptors and feature-based representations are of primary interests in the area of solid modeling. They allow us for easier storage, recognition and general treatments of objects. Axial structures such as skeletons are popular shape descriptors which have been widely studied. Most of the studies focus on a particular type of skeleton called the Medial Axis. Medial Axes can be extracted from discrete volumetric data as well as boundary-based representations. In the later case, however, no algorithm is known to perform well and accurately. We propose a new paradigm for constructing one dimensional axial structures associated with a polyhedral object. These structures, called the level set diagrams, are associated with scalar functions defined over the set of vertices of a polyhedron. We study in details the level set diagram associated with the shortest path distance to a source point. This particular association fits nicely into a theoretical framework and presents interesting properties for the purpose of shape description.

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

[2]  Bruce G. Baumgart.  A polyhedron representation for computer vision , 1975, AFIPS '75.

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

[4]  Michael Henle,et al.  A combinatorial introduction to topology , 1978 .

[5]  Kevin Weiler,et al.  Edge-Based Data Structures for Solid Modeling in Curved-Surface Environments , 1985, IEEE Computer Graphics and Applications.

[6]  Joseph S. B. Mitchell,et al.  The Discrete Geodesic Problem , 1987, SIAM J. Comput..

[7]  Frederic Fol Leymarie,et al.  Snakes and Skeletons , 1989 .

[8]  Philip J. Schneider,et al.  An algorithm for automatically fitting digitized curves , 1990 .

[9]  John A. Goldak,et al.  Constructing discrete medial axis of 3-D objects , 1991, Int. J. Comput. Geom. Appl..

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

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

[12]  Mark de Berg,et al.  Trekking in the Alps Without Freezing or Getting Tired , 1993, ESA.

[13]  Paul H. C. Eilers.  Smoothing and Interpolation with Finite Differences , 1994, Graphics Gems.

[14]  Sabine Coquillart,et al.  Axial deformations: an intuitive deformation technique , 1994, Comput. Aided Des..

[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]  Francis Lazarus.  Courbes, cylindres et métamorphoses pour l'image de synthèse. (Curves, cylinders and metamorphoses for computer graphics) , 1995 .

[17]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

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

[19]  Valerio Pascucci,et al.  Contour trees and small seed sets for isosurface traversal , 1997, SCG '97.

[20]  Tosiyasu L. Kunii,et al.  A feature-based approach for smooth surfaces , 1997, SMA '97.

[21]  A. Fournier,et al.  Growing and Animating Polygonal Models of Animals , 1997, Comput. Graph. Forum.

[22]  John Hart.  Morse Theory for Implicit Surface Modeling , 1997, VisMath.

[23]  Mark de Berg,et al.  Trekking in the Alps Without Freezing or Getting Tired , 1993, Algorithmica.

[24]  Dominique Attali,et al.  Computing and Simplifying 2D and 3D Continuous Skeletons , 1997, Comput. Vis. Image Underst..

[25]  Hans-Christian Hege.  Mathematical visualization: algorithms, applications and numerics , 1998 .

[26]  Anne Verroust-Blondet,et al.  Extracting skeletal curves from 3D scattered data , 1999, Proceedings Shape Modeling International '99. International Conference on Shape Modeling and Applications.