Medial zones: Formulation and applications

The popularity of medial axis in shape modeling and analysis comes from several of its well known fundamental properties. For example, medial axis captures the connectivity of the domain, has a lower dimension than the space itself, and is closely related to the distance function constructed over the same domain. We propose the new concept of a medial zone of an n-dimensional semi-analytic domain @W that subsumes the medial axis MA(@W) of the same domain as a special case, and can be thought of as a 'thick' skeleton having the same dimension as that of @W. We show that by transforming the exact, non-differentiable, distance function of domain @W into approximate but differentiable distance functions, and by controlling the geodesic distance to the crests of the approximate distance functions of domain @W, one obtains families of medial zones of @W that are homeomorphic to the domain and are supersets of MA(@W). We present a set of natural properties for the medial zones MZ(@W) of @W and discuss practical approaches to compute both medial axes and medial zones for 3-dimensional semi-analytic sets with rigid or evolving boundaries. Due to the fact that the medial zones fuse some of the critical geometric and topological properties of both the domain itself and of its medial axis, re-formulating problems in terms of medial zones affords the 'best of both worlds' in applications such as geometric reasoning, robotic and autonomous navigation, and design automation.

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

[2]  Ata A. Eftekharian,et al.  Shape and Topology Optimization With Medial Zones , 2011, DAC 2011.

[3]  Vadim Shapiro,et al.  An approach to systematic part design , 1997 .

[4]  Krishnan Suresh,et al.  Automating the CAD/CAE dimensional reduction process , 2003, SM '03.

[5]  Horea T. Ilies,et al.  Distance functions and skeletal representations of rigid and non-rigid planar shapes , 2009, Comput. Aided Des..

[6]  Hwan Pyo Moon,et al.  MATHEMATICAL THEORY OF MEDIAL AXIS TRANSFORM , 1997 .

[7]  James N. Damon,et al.  Determining the Geometry of Boundaries of Objects from Medial Data , 2005, International Journal of Computer Vision.

[8]  Leonidas J. Guibas,et al.  An efficient algorithm for finding the CSG representation of a simple polygon , 1988, SIGGRAPH.

[9]  Marisol Delgado,et al.  A survey of bond graphs : Theory, applications and programs , 1991 .

[10]  Philip N. Klein,et al.  Recognition of shapes by editing their shock graphs , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[11]  Vadim Shapiro.  A Convex Deficiency Tree Algorithm for Curved Polygons , 2001, Int. J. Comput. Geom. Appl..

[12]  R. Bing.  The Geometric Topology of 3-Manifolds , 1983 .

[13]  Peter Sampl.  Semi-Structured Mesh Generation Based on Medial Axis , 2000, IMR.

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

[15]  Vadim Shapiro,et al.  Construction and optimization of CSG representations , 1991, Comput. Aided Des..

[16]  Peter Sampl.  Medial Axis Construction in Three Dimensions and its Application to Mesh Generation , 2001, Engineering with Computers.

[17]  Alan E. Middleditch,et al.  Convex Decomposition of Simple Polygons , 1984, TOGS.

[18]  Richard H. Crawford,et al.  Three-dimensional halfspace constructive solid geometry tree construction from implicit boundary representations , 2003, SM '03.

[19]  F. Chazal,et al.  Stability and Finiteness Properties of Medial Axis and Skeleton , 2004 .

[20]  Christoph von Tycowicz,et al.  Eigenmodes of Surface Energies for Shape Analysis , 2010, GMP.

[21]  Gábor Székely,et al.  Multiscale Medial Loci and Their Properties , 2003, International Journal of Computer Vision.

[22]  Vadim Shapiro,et al.  Separation for boundary to CSG conversion , 1993, TOGS.

[23]  Robert Kohn,et al.  Representation and Self-Similarity of Shapes , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[24]  Robert Kohn,et al.  Representation and self-similarity of shapes , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[25]  Leonidas J. Guibas,et al.  An efficient algorithm for finding the CSG representation of a simple polygon , 1988, Algorithmica.

[26]  Martin Aigner,et al.  Robust Computation of Foot Points on Implicitly Defined Curves , 2005 .

[27]  Jean-Daniel Boissonnat,et al.  Stability and Computation of Medial Axes - a State-of-the-Art Report , 2009, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration.

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

[29]  Wooi-Boon Goh,et al.  Strategies for shape matching using skeletons , 2008, Comput. Vis. Image Underst..

[30]  André Lieutier,et al.  Any open bounded subset of Rn has the same homotopy type than its medial axis , 2003, SM '03.

[31]  Benjamin B. Kimia,et al.  The Role of Propagation and Medial Geometry in Human Vision , 2002, Biologically Motivated Computer Vision.

[32]  Ali Shokoufandeh,et al.  Shock Graphs and Shape Matching , 1998, International Journal of Computer Vision.

[33]  Alan C. Bovik,et al.  Handbook of Image and Video Processing (Communications, Networking and Multimedia) , 2005 .

[34]  Michael Yu Wang,et al.  Shape and topology optimization of compliant mechanisms using a parameterization level set method , 2007, J. Comput. Phys..

[35]  Vadim Shapiro,et al.  Shape optimization with topological changes and parametric control , 2007 .

[36]  Eitan Grinspun,et al.  Discrete laplace operators: no free lunch , 2007, Symposium on Geometry Processing.

[37]  Daniel Cohen-Or,et al.  Medial axis based solid representation , 2004, SM '04.

[38]  Vadim Shapiro,et al.  On the role of geometry in mechanical design , 1989 .

[39]  Kenji Shimada,et al.  Skeleton-based computational method for the generation of a 3D finite element mesh sizing function , 2004, Engineering with Computers.

[40]  Helmut Pottmann,et al.  Geometry of the Squared Distance Function to Curves and Surfaces , 2002, VisMath.

[41]  Jerry D. Gibson,et al.  Handbook of Image and Video Processing , 2000 .

[42]  Jayant Shah,et al.  Gray skeletons and segmentation of shapes , 2005, Comput. Vis. Image Underst..

[43]  Vadim Shapiro,et al.  Semi-analytic geometry with R-functions , 2007, Acta Numerica.

[44]  Sven J. Dickinson,et al.  Canonical Skeletons for Shape Matching , 2006, 18th International Conference on Pattern Recognition (ICPR'06).