Semi-Continuity of Skeletons in Two-Manifold and Discrete Voronoi Approximation

The skeleton of a 2D shape is an important geometric structure in pattern analysis and computer vision. In this paper we study the skeleton of a 2D shape in a two-manifold <inline-formula><tex-math>$\mathcal {M}$</tex-math> <alternatives><inline-graphic xlink:type="simple" xlink:href="liu-ieq1-2430342.gif"/></alternatives></inline-formula>, based on a geodesic metric. We present a formal definition of the skeleton <inline-formula><tex-math>$S(\Omega )$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="liu-ieq2-2430342.gif"/></alternatives></inline-formula> for a shape <inline-formula> <tex-math>$\Omega$</tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="liu-ieq3-2430342.gif"/></alternatives> </inline-formula> in <inline-formula><tex-math>$\mathcal {M}$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="liu-ieq4-2430342.gif"/></alternatives></inline-formula> and show several properties that make <inline-formula><tex-math>$S(\Omega )$</tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="liu-ieq5-2430342.gif"/> </alternatives></inline-formula> distinct from its Euclidean counterpart in <inline-formula><tex-math>$\mathbb {R}^2$ </tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="liu-ieq6-2430342.gif"/></alternatives></inline-formula>. We further prove that for a shape sequence <inline-formula><tex-math>$\lbrace \Omega _i\rbrace$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="liu-ieq7-2430342.gif"/></alternatives></inline-formula> that converge to a shape <inline-formula><tex-math>$\Omega$</tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="liu-ieq8-2430342.gif"/> </alternatives></inline-formula> in <inline-formula><tex-math>$\mathcal {M}$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="liu-ieq9-2430342.gif"/></alternatives></inline-formula>, the mapping <inline-formula> <tex-math>$\Omega \rightarrow \overline{S}(\Omega )$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="liu-ieq10-2430342.gif"/></alternatives></inline-formula> is lower semi-continuous. A direct application of this result is that we can use a set <inline-formula><tex-math>$P$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="liu-ieq11-2430342.gif"/></alternatives></inline-formula> of sample points to approximate the boundary of a 2D shape <inline-formula><tex-math>$\Omega$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="liu-ieq12-2430342.gif"/></alternatives></inline-formula> in <inline-formula><tex-math> $\mathcal {M}$</tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="liu-ieq13-2430342.gif"/></alternatives> </inline-formula>, and the Voronoi diagram of <inline-formula><tex-math>$P$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="liu-ieq14-2430342.gif"/></alternatives></inline-formula> inside <inline-formula><tex-math> $\Omega \subset \mathcal {M}$</tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="liu-ieq15-2430342.gif"/> </alternatives></inline-formula> gives a good approximation to the skeleton <inline-formula><tex-math>$S(\Omega )$ </tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="liu-ieq16-2430342.gif"/></alternatives></inline-formula>. Examples of skeleton computation in topography and brain morphometry are illustrated.

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

[2]  Daniel A. Keim,et al.  Medial-axis-based cartograms , 2005, IEEE Computer Graphics and Applications.

[3]  G. Anastassiou,et al.  Differential Geometry of Curves and Surfaces , 2014 .

[4]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[5]  Marshall W. Bern,et al.  Surface Reconstruction by Voronoi Filtering , 1998, SCG '98.

[6]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[7]  Paul M. Thompson,et al.  Hamilton–Jacobi Skeleton on Cortical Surfaces , 2008, IEEE Transactions on Medical Imaging.

[8]  Herbert Edelsbrunner,et al.  Weak witnesses for Delaunay triangulations of submanifolds , 2007, Symposium on Solid and Physical Modeling.

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

[10]  A. Toga,et al.  Mapping brain asymmetry , 2003, Nature Reviews Neuroscience.

[11]  Jean-Daniel Boissonnat,et al.  Natural neighbor coordinates of points on a surface , 2001, Comput. Geom..

[12]  Benjamin B. Kimia,et al.  Shapes, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space , 1995, International Journal of Computer Vision.

[13]  Kaleem Siddiqi,et al.  Hamilton-Jacobi Skeletons , 2002, International Journal of Computer Vision.

[14]  Marko Subasic,et al.  Level Set Methods and Fast Marching Methods , 2003 .

[15]  F. Chazal,et al.  The λ-medial axis , 2005 .

[16]  A. Aleksandrov,et al.  Intrinsic Geometry of Surfaces , 1967 .

[17]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Kurt Mehlhorn,et al.  Curve reconstruction: Connecting dots with good reason , 2000, Comput. Geom..

[19]  G. Matheron.  Random Sets and Integral Geometry , 1976 .

[20]  Alfred M. Bruckstein,et al.  Multivalued distance maps for motion planning on surfaces with moving obstacles , 1998, IEEE Trans. Robotics Autom..

[21]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[22]  F. Chazal,et al.  Stability and homotopy of a subset of the medial axis , 2004, SM '04.

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

[24]  Ligang Liu,et al.  Fast Wavefront Propagation (FWP) for Computing Exact Geodesic Distances on Meshes , 2015, IEEE Transactions on Visualization and Computer Graphics.

[25]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[26]  Kai Tang,et al.  Construction of Iso-Contours, Bisectors, and Voronoi Diagrams on Triangulated Surfaces , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[27]  T. Maekawa.  Computation of Shortest Paths on Free-Form Parametric Surfaces , 1996 .

[28]  Joseph S. B. Mitchell,et al.  The Discrete Geodesic Problem , 1987, SIAM J. Comput..

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

[30]  Yong-Jin Liu,et al.  Exact geodesic metric in 2-manifold triangle meshes using edge-based data structures , 2013, Comput. Aided Des..

[31]  André Lieutier,et al.  Any open bounded subset of Rn has the same homotopy type as its medial axis , 2004, Comput. Aided Des..

[32]  Ron Kimmel,et al.  Images as Embedded Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images , 2000, International Journal of Computer Vision.

[33]  Kai Tang,et al.  The complexity of geodesic Voronoi diagrams on triangulated 2-manifold surfaces , 2013, Inf. Process. Lett..

[34]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[35]  Longin Jan Latecki,et al.  Convexity Rule for Shape Decomposition Based on Discrete Contour Evolution , 1999, Comput. Vis. Image Underst..

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

[37]  Tony F. Chan,et al.  Computational differential geometry and intrinsic surface processing , 2010 .