Approximate medial axis as a voronoi subcomplex

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. In this paper we present a new algorithm that approximates the medial axis straight from the Voronoi diagram in a scale and density independent manner with convergence guarantees. The advantage is that, unlike for others, one does not need to fine tune any parameter for this algorithm. We present extensive experimental evidences in support of our claims.

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

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

[3]  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).

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

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

[6]  David G. Kirkpatrick,et al.  On the shape of a set of points in the plane , 1983, IEEE Trans. Inf. Theory.

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

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

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

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

[11]  Marc Levoy,et al.  A volumetric method for building complex models from range images , 1996, SIGGRAPH.

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

[13]  Tony DeRose,et al.  Surface reconstruction from unorganized points , 1992, SIGGRAPH.

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

[15]  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).

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

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

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

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

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

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

[22]  Jean-Daniel Boissonnat,et al.  Geometric structures for three-dimensional shape representation , 1984, TOGS.

[23]  Chandrajit L. Bajaj,et al.  Sampling and reconstructing manifolds using alpha-shapes , 1997, CCCG.

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

[25]  Chandrajit L. Bajaj,et al.  Automatic reconstruction of surfaces and scalar fields from 3D scans , 1995, SIGGRAPH.

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

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

[28]  Edward J. Coyle,et al.  Arbitrary Topology Shape Reconstruction from Planar Cross Sections , 1996, CVGIP Graph. Model. Image Process..

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

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

[31]  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.

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

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

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

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

[36]  Micha Sharir,et al.  Piecewise-Linear Interpolation between Polygonal Slices , 1996, Comput. Vis. Image Underst..

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