Flux invariants for shape

We consider the average outward flux through a Jordan curve of the gradient vector field of the Euclidean distance function to the boundary of a 2D shape. Using an alternate form of the divergence theorem, we show that in the limit as the area of the region enclosed by such a curve shrinks to zero, this measure has very different behaviors at medial points than at non-medial ones, providing a theoretical justification for its use in the Hamilton-Jacobi skeletonization algorithm of Siddiqi et al. (2002). We then specialize to the case of shrinking circular neighborhoods and show that the average outward flux measure also reveals the object angle at skeletal points. Hence, formulae for obtaining the boundary curves, their curvatures, and other geometric quantities of interest, can be written in terms of the average outward flux limit values at skeletal points. Thus this measure can be viewed as a Euclidean invariant for shape description: it can be used to both detect the skeleton from the Euclidean distance function, as well as to explicitly reconstruct the boundary from it. We illustrate our results with several numerical simulations.

[1]  Luc M. Vincent,et al.  Efficient computation of various types of skeletons , 1991, Medical Imaging.

[2]  J. O´Rourke,et al.  Computational Geometry in C: Arrangements , 1998 .

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

[4]  J. N. Mather,et al.  Distance from a submanifold in Euclidean space , 1981 .

[5]  Markus Ilg,et al.  Voronoi skeletons: theory and applications , 1992, Proceedings 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[6]  T. Pavlidis.  Algorithms for Graphics and Image Processing , 1981, Springer Berlin Heidelberg.

[7]  Luc Vincent,et al.  Euclidean skeletons and conditional bisectors , 1992, Other Conferences.

[8]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

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

[10]  Robert L. Ogniewicz,et al.  Discrete Voronoi skeletons , 1992 .

[11]  Steven G. Krantz,et al.  Distance to Ck hypersurfaces , 1981 .

[12]  Ching Y. Suen,et al.  Thinning Methodologies - A Comprehensive Survey , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  AurenhammerFranz.  Voronoi diagramsa survey of a fundamental geometric data structure , 1991 .

[14]  W. E. Hartnett,et al.  Shape Recognition, Prairie Fires, Convex Deficiencies and Skeletons , 1968 .

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

[16]  Azriel Rosenfeld,et al.  Sequential Operations in Digital Picture Processing , 1966, JACM.

[17]  Ming-Kuei Hu,et al.  Visual pattern recognition by moment invariants , 1962, IRE Trans. Inf. Theory.

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

[19]  E. Isaacson,et al.  Numerical Analysis for Applied Science , 1997 .

[20]  Kaleem Siddiqi,et al.  Contour Fragment Grouping and Shared, Simple Occluders , 1999, Comput. Vis. Image Underst..

[21]  J. Marsden,et al.  A mathematical introduction to fluid mechanics , 1979 .

[22]  Farzin Mokhtarian,et al.  Robust Image Corner Detection Through Curvature Scale Space , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[23]  J. Damon.  Smoothness and geometry of boundaries associated to skeletal structures, II: Geometry in the Blum case , 2004, Compositio Mathematica.

[24]  Edwin R. Hancock,et al.  A skeletal measure of 2D shape similarity , 2001, Comput. Vis. Image Underst..

[25]  Grégoire Malandain,et al.  Squelettes euclidiens d'objets discrets n-dimensionnels , 1996 .

[26]  Philip N. Klein,et al.  Recognition of Shapes by Editing Shock Graphs , 2001, ICCV.

[27]  Jeffrey C. Lagarias,et al.  Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions , 1998, SIAM J. Optim..

[28]  Ugo Montanari,et al.  A Method for Obtaining Skeletons Using a Quasi-Euclidean Distance , 1968, J. ACM.

[29]  Kaleem Siddiqi,et al.  Robust and efficient skeletal graphs , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

[30]  F. W. Warner.  Foundations of Differentiable Manifolds and Lie Groups , 1971 .

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

[32]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

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

[34]  Farzin Mokhtarian,et al.  Silhouette-Based Isolated Object Recognition through Curvature Scale Space , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[35]  George D. Stetten,et al.  Automated Identification and Measurement of Objects via Populations of Medial Primitives, with Application to Real Time 3D Echocardiography , 1999, IPMI.

[36]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[37]  Farzin Mokhtarian,et al.  A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[38]  Gabriella Sanniti di Baja,et al.  A One-Pass Two-Operation Process to Detect the Skeletal Pixels on the 4-Distance Transform , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[39]  Yun Xia,et al.  Skeletonization via the Realization of the Fire Front's Propagation and Extinction in Digital Binary Shapes , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

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

[41]  Ugo Montanari,et al.  Continuous Skeletons from Digitized Images , 1969, JACM.

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

[43]  P. Danielsson.  Euclidean distance mapping , 1980 .

[44]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

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

[46]  Michael Leyton,et al.  Symmetry-curvature duality , 1987, Comput. Vis. Graph. Image Process..

[47]  Donald D. Hoffman,et al.  Parsing silhouettes: The short-cut rule , 1999, Perception & psychophysics.

[48]  Alastair R. Allen,et al.  Fish species recognition by shape analysis of images , 1990, Pattern Recognit..

[49]  Gabriella Sanniti di Baja,et al.  Finding local maxima in a pseudo-Euclidian distance transform , 1988, Comput. Vis. Graph. Image Process..

[50]  Jan J. Koenderink,et al.  Solid shape , 1990 .

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

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

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