Correcting Curvature-Density Effects in the Hamilton–Jacobi Skeleton

The Hamilton–Jacobi approach has proven to be a powerful and elegant method for extracting the skeleton of two-dimensional (2-D) shapes. The approach is based on the observation that the normalized flux associated with the inward evolution of the object boundary at nonskeletal points tends to zero as the size of the integration area tends to zero, while the flux is negative at the locations of skeletal points. Nonetheless, the error in calculating the flux on the image lattice is both limited by the pixel resolution and also proportional to the curvature of the boundary evolution front and, hence, unbounded near endpoints. This makes the exact location of endpoints difficult and renders the performance of the skeleton extraction algorithm dependent on a threshold parameter. This problem can be overcome by using interpolation techniques to calculate the flux with subpixel precision. However, here, we develop a method for 2-D skeleton extraction that circumvents the problem by eliminating the curvature contribution to the error. This is done by taking into account variations of density due to boundary curvature. This yields a skeletonization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the Hamilton–Jacobi method.

[1]  H. Blum.  Biological shape and visual science (part I) , 1973 .

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

[3]  Steven W. Zucker,et al.  On the evolution of the skeleton , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[4]  Denis V. Ivanov,et al.  An efficient integer-based skeletonization algorithm , 2000, Comput. Graph..

[5]  Arthur W. Toga,et al.  Efficient Skeletonization of Volumetric Objects , 1999, IEEE Trans. Vis. Comput. Graph..

[6]  J. Crank,et al.  A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type , 1947 .

[7]  Michel Schmitt,et al.  Some examples of algorithm analysis in computational geometry by means of mathematical morphological techniques , 1988, Geometry and Robotics.

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

[9]  Gabriella Sanniti di Baja,et al.  Euclidean skeleton via centre-of-maximal-disc extraction , 1993, Image Vis. Comput..

[10]  Gabriella Sanniti di Baja,et al.  A Width-Independent Fast Thinning Algorithm , 1985, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[11]  Gabriella Sanniti di Baja,et al.  Skeletonization algorithm running on path-based distance maps , 1996, Image Vis. Comput..

[12]  Kaleem Siddiqi,et al.  Flux invariants for shape , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

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

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

[15]  Kaleem Siddiqi,et al.  Matching Hierarchical Structures Using Association Graphs , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[16]  Ali Shokoufandeh,et al.  View-based 3-D object recognition using shock graphs , 2002, Object recognition supported by user interaction for service robots.

[17]  Kaleem Siddiqi,et al.  Divergence-Based Medial Surfaces , 2000, ECCV.

[18]  Gabriella Sanniti di Baja,et al.  Ridge points in Euclidean distance maps , 1992, Pattern Recognit. Lett..

[19]  P. Choyke,et al.  Gray-scale skeletonization of small vessels in magnetic resonance angiography , 2000, IEEE Transactions on Medical Imaging.

[20]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[21]  Benjamin B. Kimia,et al.  Symmetry Maps of Free-Form Curve Segments via Wave Propagation , 2004, International Journal of Computer Vision.

[22]  Philip N. Klein,et al.  Indexing based on edit-distance matching of shape graphs , 1998, Other Conferences.

[23]  Geoffrey McLennan,et al.  Three-dimensional path planning for virtual bronchoscopy , 2004, IEEE Transactions on Medical Imaging.

[24]  Ali Shokoufandeh,et al.  Indexing using a spectral encoding of topological structure , 1999, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149).

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

[26]  Kaleem Siddiqi,et al.  The Hamilton-Jacobi skeleton , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[27]  Alfred M. Bruckstein,et al.  Skeletonization via Distance Maps and Level Sets , 1995, Comput. Vis. Image Underst..

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

[29]  Benjamin B. Kimia,et al.  On the evolution of curves via a function of curvature , 1992 .

[30]  PaperNo.  Recognition of shapes by editing shock graphs , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[31]  Mie Sato,et al.  Penalized-Distance Volumetric Skeleton Algorithm , 2001, IEEE Trans. Vis. Comput. Graph..

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

[33]  Frederic Fol Leymarie,et al.  Simulating the Grassfire Transform Using an Active Contour Model , 1992, IEEE Trans. Pattern Anal. Mach. Intell..