An Algorithm for the Medial Axis Transform of 3D Polyhedral Solids

The medial axis transform (MAT) is a representation of an object which has been shown to be useful in design, interrogation, animation, finite element mesh generation, performance analysis, manufacturing simulation, path planning and tolerance specification. In this paper, an algorithm for determining the MAT is developed for general 3D polyhedral solids of arbitrary genus without cavities, with nonconvex vertices and edges. The algorithm is based on a classification scheme which relates different pieces of the medial axis (MA) to one another, even in the presence of degenerate MA points. Vertices of the MA are connected to one another by tracing along adjacent edges, and finally the faces of the axis are found by traversing closed loops of vertices and edges. Representation of the MA and its associated radius function is addressed, and pseudocode for the algorithm is given along with recommended optimizations. A connectivity theorem is proven to show the completeness of the algorithm. Complexity estimates and stability analysis for the algorithms are presented. Finally, examples illustrate the computational properties of the algorithm for convex and nonconvex 3D polyhedral solids with polyhedral holes.

[1]  Ugo Montanari,et al.  Continuous Skeletons from Digitized Images , 1969, JACM.

[2]  T. Broadbent.  Complex Variables , 1970, Nature.

[3]  H. Blum.  Biological shape and visual science (part I) , 1973 .

[4]  Franco P. Preparata,et al.  The Medial Axis of a Simple Polygon , 1977, MFCS.

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

[6]  F. Bookstein.  The line-skeleton , 1979 .

[7]  P. Danielsson.  Euclidean distance mapping , 1980 .

[8]  D. T. Lee,et al.  Medial Axis Transformation of a Planar Shape , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Lee R. Nackman,et al.  Curvature relations in three-dimensional symmetric axes , 1982, Comput. Graph. Image Process..

[10]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[11]  Gene H. Golub,et al.  Matrix computations , 1983 .

[12]  L. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi , 1985, TOGS.

[13]  Azriel Rosenfeld,et al.  Axial representations of shape , 1986, Computer Vision Graphics and Image Processing.

[14]  Vijay Srinivasan,et al.  Voronoi Diagram for Multiply-Connected Polygonal Domains I: Algorithm , 1987, IBM J. Res. Dev..

[15]  Andrew Zisserman,et al.  Using a mixed wave/ diffusion process to elicit the symmetry set , 1989, Image Vis. Comput..

[16]  C. Hoffmann,et al.  A geometric investigation of the skeleton of CSG objects , 1990 .

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

[18]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

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

[20]  Anil K. Jain,et al.  Medial axis representation and encoding of scanned documents , 1991, J. Vis. Commun. Image Represent..

[21]  T. Tam,et al.  2D finite element mesh generation by medial axis subdivision , 1991 .

[22]  Martin Held,et al.  On the Computational Geometry of Pocket Machining , 1991, Lecture Notes in Computer Science.

[23]  Cecil Armstrong,et al.  Automatic Generation of Well Structured Meshed Using Medial Axis and Surface Subdivision , 1991, DAC 1991.

[24]  L. Nackman,et al.  Automatic mesh generation using the symmetric axis transformation of polygonal domains , 1992, Proc. IEEE.

[25]  C. Hoffmann.  Computer Vision, Descriptive Geometry, and Classical Mechanics , 1992 .

[26]  Franz-Erich Wolter,et al.  Curvature computations for degenerate surface patches , 1992, Comput. Aided Geom. Des..

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

[28]  Ching-Shoei Chiang.  The Euclidean distance transform , 1992 .

[29]  James H. Davenport,et al.  Voronoi diagrams of set-theoretic solid models , 1992, IEEE Computer Graphics and Applications.

[30]  J. Brandt.  Describing a solid with the three-dimensional skeleton , 1992 .

[31]  Fritz B. Prinz,et al.  Continuous skeletons of discrete objects , 1993, Solid Modeling and Applications.

[32]  Nicholas M. Patrikalakis,et al.  Computation of the solutions of nonlinear polynomial systems , 1993, Comput. Aided Geom. Des..

[33]  Nicholas M. Patrikalakis,et al.  Computation of singularities and intersections of offsets of planar curves , 1993, Comput. Aided Geom. Des..

[34]  Debasish Dutta,et al.  On the Skeleton of Simple CSG Objects , 1993 .

[35]  Kokichi Sugihara.  Approximation of Generalized Voronoi Diagrams by Ordinary Voronoi Diagrams , 1993, CVGIP Graph. Model. Image Process..

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

[37]  P. J. Vermeer.  Medial axis transform to boundary representation conversion , 1994 .

[38]  Damian J. Sheehy,et al.  Numerical Computations of Medial Surface Vertices , 1994, IMA Conference on the Mathematics of Surfaces.

[39]  Debasish Dutta,et al.  Boundary surface recovery from skeleton curves and surfaces , 1995, Comput. Aided Geom. Des..

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

[41]  Damian J. Sheehy,et al.  Computing the medial surface of a solid from a domain Delaunay triangulation , 1995, Symposium on Solid Modeling and Applications.

[42]  M. Sabin,et al.  Hexahedral mesh generation by medial surface subdivision: Part I. Solids with convex edges , 1995 .

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

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

[45]  Ling Chen,et al.  SIMD hypercube algorithm for complete Euclidean distance transform , 1995, Proceedings 1st International Conference on Algorithms and Architectures for Parallel Processing.

[46]  Evan C. Sherbrooke.  3-D shape interrogation by medial axis transform , 1996 .