Medial Axis Approximation with Constrained Centroidal Voronoi Diagrams On Discrete Data

In this paper, we present a novel method for me- dial axis approximation based on Constrained Centroidal Voronoi Diagram of discrete data (image, volume). The pro- posed approach is based on the shape boundary subsampling by a clustering approach which generates a Voronoi Dia- gram well suited for Medial Axis extraction. The resulting Voronoi Diagram is further filtered so as to capture the cor- rect topology of the medial axis. The resulting medial axis appears largely invariant with respect to typical noise con - ditions in the discrete data. The method is tested on various synthetic as well as real images. We also show an applica- tion of the approximate medial axis to the sizing field for triangular and tetrahedral meshing.

[1]  Edouard Thiel,et al.  Chordal Axis on Weighted Distance Transforms , 2006, DGCI.

[2]  Frédéric Chazal,et al.  A Sampling Theory for Compact Sets in Euclidean Space , 2009, Discret. Comput. Geom..

[3]  Nadia Magnenat-Thalmann,et al.  Knowledge-based extraction of control skeletons for animation , 2007, IEEE International Conference on Shape Modeling and Applications 2007 (SMI '07).

[4]  S. J. Owen,et al.  3D discrete skeleton generation by wave propagation on PR-octree for finite element mesh sizing , 2004, SM '04.

[5]  Pierre Alliez,et al.  Eurographics Symposium on Geometry Processing (2007) Voronoi-based Variational Reconstruction of Unoriented Point Sets , 2022 .

[6]  Yaorong Ge,et al.  On the Generation of Skeletons from Discrete Euclidean Distance Maps , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  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.

[8]  David Coeurjolly,et al.  Optimal Separable Algorithms to Compute the Reverse Euclidean Distance Transformation and Discrete Medial Axis in Arbitrary Dimension , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  P. Basser,et al.  Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. , 1996, Journal of magnetic resonance. Series B.

[10]  Mark Pauly,et al.  Medial axis approximation from inner Voronoi balls: a demo of the Mesecina tool , 2007, SCG '07.

[11]  Marshall W. Bern,et al.  A new Voronoi-based surface reconstruction algorithm , 1998, SIGGRAPH.

[12]  Sunghee Choi,et al.  The power crust , 2001, SMA '01.

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

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

[15]  Rémy Prost,et al.  Generic Remeshing of 3D Triangular Meshes with Metric-Dependent Discrete Voronoi Diagrams , 2008, IEEE Transactions on Visualization and Computer Graphics.

[16]  Frédéric Chazal,et al.  A Sampling Theory for Compact Sets in Euclidean Space , 2006, SCG '06.

[17]  Luciano da Fontoura Costa,et al.  On Voronoi Diagrams and Medial Axes , 2002, Journal of Mathematical Imaging and Vision.

[18]  Vladimir A. Kovalevsky,et al.  Finite topology as applied to image analysis , 1989, Comput. Vis. Graph. Image Process..

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

[20]  Frédéric Chazal,et al.  The "lambda-medial axis" , 2005, Graph. Model..

[21]  Zheng Qin,et al.  Stratified helix information of medial-axis-points matching for 3D model retrieval , 2007, MIR '07.

[22]  Tamal K. Dey,et al.  Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee , 2003, Algorithmica.

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

[24]  Françoise Peyrin,et al.  Shape description of three-dimensional images based on medial axis , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[25]  M. Yvinec,et al.  Variational tetrahedral meshing , 2005, SIGGRAPH 2005.

[26]  Matthew L. Baker,et al.  Computing a Family of Skeletons of Volumetric Models for Shape Description , 2006, GMP.