Efficient and robust computation of an approximated medial axis

The medial axis can be viewed as a compact representation for an arbitrary model; it is an essential geometric structure in many applications. A number of practical algorithms for its computation have been aimed at speeding up its computation and at addressing its instabilities. In this paper we propose a new algorithm to compute the medial axis with arbitrary precision. It exhibits several desirable properties not previously combined in a practical and efficient algorithm. First, it allows for a tradeoff between computation time and accuracy, making it well-suited for applications in which an approximation of the medial axis suffices, but computational efficiency is of particular concern. Second, it is output sensitive: the computation complexity of the algorithm does not depend on the size of the representation of a model, but on the size of the representation of the resulting medial axis. Third, the densities of the approximated medial axis points in different areas are adaptive to local free space volumes, based on the assumption that a coarser approximation in wide open area can still suffice the requirements of the applications. We present theoretical results, bounding the error introduced by the approximation process. The algorithm has been implemented and experimental results are presented that illustrate its computational efficiency and robustness.

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

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

[3]  Ching Y. Suen,et al.  Thinning Methodologies - A Comprehensive Survey , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

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

[5]  Victor J. Milenkovic,et al.  Robust Construction of the Voronoi Diagram of a Polyhedron , 1993, CCCG.

[6]  Patrick Shen-Pei Wang,et al.  Analytical Comparison of Thinning Algorithms , 1993, Int. J. Pattern Recognit. Artif. Intell..

[7]  Ingemar Ragnemalm,et al.  The Euclidean distance transform in arbitrary dimensions , 1992, Pattern Recognit. Lett..

[8]  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.

[9]  Alan L. Yuille,et al.  FORMS: A flexible object recognition and modelling system , 1995, Proceedings of IEEE International Conference on Computer Vision.

[10]  Nicholas M. Patrikalakis,et al.  Computation of the Medial Axis Transform of 3-D polyhedra , 1995, Symposium on Solid Modeling and Applications.

[11]  M. Overmars,et al.  Approximating generalized Voronoi diagrams in any dimension , 1995 .

[12]  George M. Turkiyyah,et al.  Computation of 3D skeletons using a generalized Delaunay triangulation technique , 1995, Comput. Aided Des..

[13]  Dinesh Manocha,et al.  OBBTree: a hierarchical structure for rapid interference detection , 1996, SIGGRAPH.

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

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

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

[17]  David H. Eberly,et al.  Zoom-Invariant Vision of Figural Shape: The Mathematics of Cores , 1996, Comput. Vis. Image Underst..

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

[19]  Howie Choset,et al.  Incremental Construction of the Generalized Voronoi Diagram , the Generalized Voronoi Graph , and the Hierarchical Generalized Voronoi Graph , 1999 .

[20]  Dinesh Manocha,et al.  Fast Proximity Queries with Swept Sphere Volumes , 1999 .

[21]  Kaleem Siddiqi,et al.  The Hamilton-Jacobi skeleton , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

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

[23]  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).

[24]  Lydia E. Kavraki,et al.  A framework for using the workspace medial axis in PRM planners , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[25]  Dinesh Manocha,et al.  Fast computation of generalized Voronoi diagrams using graphics hardware , 1999, SIGGRAPH.

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

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

[28]  M. Allali.  Compression on the digital unit sphere , 2001 .

[29]  Sunghee Choi,et al.  The power crust, unions of balls, and the medial axis transform , 2001, Comput. Geom..

[30]  M. Tanemura,et al.  Random Sequential Covering of a Sphere with Identical Spherical Caps , 2001 .

[31]  Hyeong In Choi,et al.  The Medial Axis Transform , 2002, Handbook of Computer Aided Geometric Design.

[32]  Dinesh Manocha,et al.  Efficient computation of a simplified medial axis , 2003, SM '03.

[33]  Pattern Recognition Letters , 1995 .

[34]  Tamal K. Dey,et al.  Approximate medial axis as a voronoi subcomplex , 2002, SMA '02.

[35]  Oliver Brock,et al.  Adapting the sampling distribution in PRM planners based on an approximated medial axis , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.