Computing Multiscale Curve and Surface Skeletons of Genus 0 Shapes Using a Global Importance Measure

We present a practical algorithm for computing robust multiscale curve and surface skeletons of 3D objects of genus zero. Based on a model that follows an advection principle, we assign to each point on the skeleton a part of the object surface, called the collapse. The size of the collapse is used as a uniform importance measure for the curve and surface skeleton, so that both can be simplified by imposing a single threshold on this intuitive measure. The simplified skeletons are connected by default, without special precautions, due to the monotonicity of the importance measure. The skeletons possess additional desirable properties: They are centered, robust to noise, hierarchical, and provide a natural skeleton-to-boundary mapping. We present a voxel-based algorithm that is straightforward to implement and simple to use. We illustrate our method on several realistic 3D objects.

[1]  Gábor Székely,et al.  Estimating shortest paths and minimal distances on digitized three-dimensional surfaces , 1993, Pattern Recognit..

[2]  Dinesh Manocha,et al.  Homotopy-preserving medial axis simplification , 2005, SPM '05.

[3]  Deborah Silver,et al.  Curve-Skeleton Properties, Applications, and Algorithms , 2007, IEEE Trans. Vis. Comput. Graph..

[4]  Alexandru Telea,et al.  A Robust Level-Set Algorithm for Centerline Extraction , 2003, VisSym.

[5]  Olaf Kübler,et al.  Hierarchic Voronoi skeletons , 1995, Pattern Recognit..

[6]  Joachim Giesen,et al.  Surface reconstruction based on a dynamical system † , 2002, Comput. Graph. Forum.

[7]  Kaleem Siddiqi,et al.  A shock grammar for recognition , 1996, Proceedings CVPR IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[8]  Martin Rumpf,et al.  A Continuous Skeletonization Method Based on Level Sets , 2002, VisSym.

[9]  Alexandru Telea,et al.  Skeletonization and Distance Transforms of 3D Volumes Using Graphics Hardware , 2006, DGCI.

[10]  Chris Pudney,et al.  Distance-Ordered Homotopic Thinning: A Skeletonization Algorithm for 3D Digital Images , 1998, Comput. Vis. Image Underst..

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

[12]  James C. Mullikin,et al.  The vector distance transform in two and three dimensions , 1992, CVGIP Graph. Model. Image Process..

[13]  Benjamin B. Kimia,et al.  A formal classification of 3D medial axis points and their local geometry , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[15]  Thomas A. Funkhouser,et al.  The Princeton Shape Benchmark , 2004, Proceedings Shape Modeling Applications, 2004..

[16]  Tosiyasu L. Kunii,et al.  Surface coding based on Morse theory , 1991, IEEE Computer Graphics and Applications.

[17]  Ming Wan,et al.  Distance-field based skeletons for virtual navigation , 2001, Proceedings Visualization, 2001. VIS '01..

[18]  Alexandru Telea,et al.  Skeleton-based Hierarchical Shape Segmentation , 2007, IEEE International Conference on Shape Modeling and Applications 2007 (SMI '07).

[19]  Tamal K. Dey,et al.  Defining and computing curve-skeletons with medial geodesic function , 2006, SGP '06.

[20]  Robert Strzodka,et al.  Generalized distance transforms and skeletons in graphics hardware , 2004, VISSYM'04.

[21]  Taku Komura,et al.  Topology matching for fully automatic similarity estimation of 3D shapes , 2001, SIGGRAPH.

[22]  Edwin R. Hancock,et al.  Correcting Curvature-Density Effects in the Hamilton–Jacobi Skeleton , 2006, IEEE Transactions on Image Processing.

[23]  Joachim Giesen,et al.  The flow complex: a data structure for geometric modeling , 2003, SODA '03.

[24]  J. Rossignac,et al.  Plumber: a method for a multi-scale decomposition of 3D shapes into tubular primitives and bodies , 2004, SM '04.

[25]  Thomas A. Funkhouser,et al.  The Princeton Shape Benchmark (Figures 1 and 2) , 2004, Shape Modeling International Conference.

[26]  Attila Kuba,et al.  Directional 3D Thinning Using 8 Subiterations , 1999, DGCI.

[27]  Dinesh Manocha,et al.  Efficient computation of a simplified medial axis , 2003, SM '03.

[28]  Alexandru Telea,et al.  An Augmented Fast Marching Method for Computing Skeletons and Centerlines , 2002, VisSym.

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

[30]  Deepak R. Kenchammana-Hosekote,et al.  Volume animation using the skeleton tree , 1998, IEEE Symposium on Volume Visualization (Cat. No.989EX300).

[31]  Deborah Silver,et al.  Curve-skeleton applications , 2005, VIS 05. IEEE Visualization, 2005..

[32]  Tiow Seng Tan,et al.  Decomposing polygon meshes for interactive applications , 2001, I3D '01.

[33]  Kaleem Siddiqi,et al.  Flux driven automatic centerline extraction , 2005, Medical Image Anal..

[34]  Grégoire Malandain,et al.  Euclidean skeletons , 1998, Image Vis. Comput..

[35]  Nils J. Nilsson,et al.  A Formal Basis for the Heuristic Determination of Minimum Cost Paths , 1968, IEEE Trans. Syst. Sci. Cybern..

[36]  Edgar A. Ramos,et al.  Medial axis approximation and unstable flow complex , 2006, SCG '06.

[37]  Hans-Christian Hege,et al.  Fast visualization of plane-like structures in voxel data , 2002, IEEE Visualization, 2002. VIS 2002..

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

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

[40]  Alfred M. Bruckstein,et al.  Pruning Medial Axes , 1998, Comput. Vis. Image Underst..

[41]  Alexandru Telea,et al.  Quantitative comparison of tolerance-based feature transforms , 2006, VISAPP.

[42]  Narendra Ahuja,et al.  Shape Representation Using a Generalized Potential Field Model , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[43]  Ayellet Tal,et al.  Hierarchical mesh decomposition using fuzzy clustering and cuts , 2003, ACM Trans. Graph..

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

[45]  Balasubramanian Raman,et al.  Computing hierarchical curve-skeletons of 3D objects , 2005, The Visual Computer.

[46]  Markus H. Gross,et al.  Signed distance transform using graphics hardware , 2003, IEEE Visualization, 2003. VIS 2003..