Approximate medial axis as a voronoi subcomplex

Abstract Medial axis as a compact representation of shapes has evolved as an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to approximate them. One line of research considers the point cloud representation of the boundary surface of a solid and then attempts to compute an approximate medial axis from this point sample. It is known that the Voronoi vertices converge to the medial axis for a curve in 2D as the sample density approaches infinity. Unfortunately, the same is not true in 3D. Recently, it is discovered that a subset of Voronoi vertices called poles converge to the medial axis in 3D. However, in practice, a continuous approximation as opposed to a discrete one is sought. Recently few algorithms have been proposed which use the Voronoi diagram and its derivatives to compute this continuous approximation. These algorithms are scale or density dependent. Most of them do not have convergence guarantees, and one of them computes it indirectly from the power diagram of the poles. Recently, we proposed a new algorithm that approximates the medial axis straight from the Voronoi diagram in a scale and density independent manner with convergence guarantees. In this paper, we present several experimental results with this algorithm that support our theoretical claims and also show its effectiveness on practical data sets.

[1]  Herbert Edelsbrunner,et al.  Three-dimensional alpha shapes , 1992, VVS.

[2]  Hao Chen,et al.  An accelerated triangulation method for computing the skeletons of free-form solid models , 1997, Comput. Aided Des..

[3]  Kaleem Siddiqi,et al.  Divergence-Based Medial Surfaces , 2000, ECCV.

[4]  Christoph M. Hoffmann,et al.  How to Construct the Skeleton of CSG Objects , 1990 .

[5]  Damian J. Sheehy,et al.  Shape Description By Medial Surface Construction , 1996, IEEE Trans. Vis. Comput. Graph..

[6]  Jacques-Olivier Lachaud,et al.  Delaunay conforming iso-surface, skeleton extraction and noise removal , 2001, Comput. Geom..

[7]  Nicholas M. Patrikalakis,et al.  Automated interrogation and adaptive subdivision of shape using medial axis transform , 1991 .

[8]  Ari Rappoport,et al.  Computing Voronoi skeletons of a 3-D polyhedron by space subdivision , 2002, Comput. Geom..

[9]  J. Brandt.  Convergence and continuity criteria for discrete approximations of the continuous planar skeleton , 1994 .

[10]  V. Ralph Algazi,et al.  Continuous skeleton computation by Voronoi diagram , 1991, CVGIP Image Underst..

[11]  Jean-Daniel Boissonnat,et al.  Smooth surface reconstruction via natural neighbour interpolation of distance functions , 2000, SCG '00.

[12]  Leonidas J. Guibas,et al.  A probabilistic roadmap planner for flexible objects with a workspace medial-axis-based sampling approach , 1999, Proceedings 1999 IEEE/RSJ International Conference on Intelligent Robots and Systems. Human and Environment Friendly Robots with High Intelligence and Emotional Quotients (Cat. No.99CH36289).

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

[14]  Dinesh Manocha,et al.  Accurate computation of the medial axis of a polyhedron , 1999, SMA '99.

[15]  Philip M. Hubbard,et al.  Approximating polyhedra with spheres for time-critical collision detection , 1996, TOGS.

[16]  Wayne Niblack,et al.  Generating skeletons and centerlines from the distance transform , 1992, CVGIP Graph. Model. Image Process..

[17]  Nicholas M. Patrikalakis,et al.  An Algorithm for the Medial Axis Transform of 3D Polyhedral Solids , 1996, IEEE Trans. Vis. Comput. Graph..

[18]  Benjamin B. Kimia,et al.  A formal classification of 3D medial axis points and their local geometry , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

[19]  Sunghee Choi,et al.  The power crust , 2001, SMA '01.

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

[21]  Seth J. Teller,et al.  Assisted articulation of closed polygonal models , 1998, SIGGRAPH '98.

[22]  Tamal K. Dey,et al.  Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee , 2003, Algorithmica.

[23]  Tamal K. Dey,et al.  Decimating samples for mesh simplification , 2001, CCCG.

[24]  Franz-Erich Wolter.  Cut Locus and Medial Axis in Global Shape Interrogation and Representation , 1995 .

[25]  Herbert Edelsbrunner,et al.  Shape Reconstruction with Delaunay Complex , 1998, LATIN.

[26]  Robert L. Ogniewicz,et al.  Skeleton-space: a multiscale shape description combining region and boundary information , 1994, 1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[27]  Mark A. Ganter,et al.  Skeleton-based modeling operations on solids , 1997, SMA '97.

[28]  Tamal K. Dey,et al.  Detecting undersampling in surface reconstruction , 2001, SCG '01.

[29]  Alla Sheffer,et al.  Hexahedral Mesh Generation using the Embedded Voronoi Graph , 1999, Engineering with Computers.