Interior Medial Axis Transform computation of 3D objects bound by free-form surfaces

This paper presents an algorithm for generating the Interior Medial Axis Transform (iMAT) of 3D objects with free-form boundaries. The algorithm proposed uses the exact representation of the part and generates an approximate rational spline description of the iMAT. The algorithm generates the iMAT by a tracing technique that marches along the object's boundary. The level of approximation is controlled by the choice of the step size in the tracing procedure. Criteria based on distance and local curvature of boundary entities are used to identify the junction points and the search for these junction points is done in an efficient way. The algorithm works for multiply-connected objects as well. Results of the implementation are provided.

[1]  Benjamin B. Kimia,et al.  The Medial Scaffold of 3D Unorganized Point Clouds , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  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).

[3]  Nina Amenta,et al.  The medial axis of a union of balls , 2001, Comput. Geom..

[4]  B. Gurumoorthy,et al.  A tracing algorithm for constructing medial axis transform of 3D objects bound by free-form surfaces , 2005, International Conference on Shape Modeling and Applications 2005 (SMI' 05).

[5]  B. Gurumoorthy,et al.  Constructing medial axis transform of extruded and revolved 3D objects with free-form boundaries , 2005, Comput. Aided Des..

[6]  Guido Gerig,et al.  A Continuous 3-D Medial Shape Model With Branching , 2006 .

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

[8]  William C. Regli,et al.  2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling , 2009 .

[9]  Jarek Rossignac,et al.  Proceedings of the third ACM symposium on Solid modeling and applications , 1995, SIGGRAPH 1995.

[10]  James N. Damon,et al.  Tree Structure for Contractible Regions in ℝ3 , 2007, International Journal of Computer Vision.

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

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

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

[14]  Franz Aurenhammer,et al.  Medial axis computation for planar free-form shapes , 2009, Comput. Aided Des..

[15]  Tamal K. Dey,et al.  Approximate medial axis for CAD models , 2003, SM '03.

[16]  Gershon Elber,et al.  Rational bisectors of CSG primitives , 1999, SMA '99.

[17]  Gershon Elber,et al.  The bisector surface of rational space curves , 1998, TOGS.

[18]  Hans-Peter Seidel,et al.  Proceedings of the seventh ACM symposium on Solid modeling and applications , 2002 .

[19]  Gershon Elber,et al.  Computing Rational Bisectors , 1999, IEEE Computer Graphics and Applications.

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

[21]  Franz-Erich Wolter,et al.  Local and global geometric methods for analysis, interrogation, reconstruction, modification and design of shape , 2000, Proceedings Computer Graphics International 2000.

[22]  Gershon Elber,et al.  A computational model for nonrational bisector surfaces: curve-surface and surface-surface bisectors , 2000, Proceedings Geometric Modeling and Processing 2000. Theory and Applications.

[23]  Gershon Elber,et al.  Bisector curves of planar rational curves , 1998, Comput. Aided Des..

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

[25]  Jeff Erickson,et al.  Nice Point Sets Can Have Nasty Delaunay Triangulations , 2001, SCG '01.

[26]  Cecil G. Armstrong,et al.  Adaptive Shape-Sensitive Meshing of the Medial Axis , 2002, Engineering with Computers.

[27]  Ari Rappoport,et al.  Computing the Voronoi diagram of a 3-D polyhedron by separate computation of its symbolic and geometric parts , 1999, SMA '99.

[28]  D. Ross.  Computer-aided design , 1961, CACM.

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

[30]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

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

[32]  Kaleem Siddiqi,et al.  Medial Representations: Mathematics, Algorithms and Applications , 2008 .

[33]  Gershon Elber,et al.  Precise Voronoi cell extraction of free-form rational planar closed curves , 2005, SPM '05.

[34]  Cecil G. Armstrong,et al.  Modelling requirements for finite-element analysis , 1994, Comput. Aided Des..

[35]  Jin J. Chou.  Voronoi diagrams for planar shapes , 1995, IEEE Computer Graphics and Applications.

[36]  Ralph R. Martin,et al.  Proceedings of the 9th IMA Conference on the Mathematics of Surfaces, University of Cambridge, UK, September 4-7, 2000 , 2000, IMA Conference on the Mathematics of Surfaces.

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

[38]  H. N. Gürsoy,et al.  An automatic coarse and fine surface mesh generation scheme based on medial axis transform: Part i algorithms , 1992, Engineering with Computers.

[39]  Gabriella Sanniti di Baja,et al.  (3, 4)-weighted Skeleton Decomposition for Pattern Representation and Description , 1994, Pattern Recognit..

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

[41]  Dinesh Manocha,et al.  Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling, Beijing, China, June 4-6, 2007 , 2007, Symposium on Solid and Physical Modeling.

[42]  Yu-Chi Chang,et al.  Medial Axis Transform (MAT) of General 2D Shapes and 3D Polyhedra for Engineering Applications , 2001 .

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

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

[45]  Elaine Cohen,et al.  Tracing ridges on B-Spline surfaces , 2009, Symposium on Solid and Physical Modeling.

[46]  Nicholas M. Patrikalakis,et al.  An automatic coarse and fine surface mesh generation scheme based on medial axis transform: Part II implementation , 1992, Engineering with Computers.

[47]  Gershon Elber,et al.  Precise Voronoi Cell Extraction of Free-Form Planar Piecewise C1-Continuous Closed Rational Curves , 2007, Int. J. Comput. Geom. Appl..

[48]  Kunwoo Lee,et al.  Computing the medial surface of a 3-D boundary representation model , 1997 .

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

[50]  B. Gurumoorthy,et al.  Constructing medial axis transform of planar domains with curved boundaries , 2003, Comput. Aided Des..

[51]  Gábor Renner,et al.  A divide and conquer algorithm for medial surface calculation of planar polyhedra , 2007, Comput. Aided Des..

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

[53]  Peter J. Giblin,et al.  Symmetry Sets and Medial Axes in Two and Three Dimensions , 2000, IMA Conference on the Mathematics of Surfaces.

[54]  Dinesh Manocha,et al.  Exact computation of the medial axis of a polyhedron , 2004, Comput. Aided Geom. Des..