Abstract
In this paper we explore the local geometry of the medial axis (MA) and shocks (SH), and their structural changes under deformations, by viewing these symmetries as subsets of the symmetry set (SS) and present two results. First, we establish that the local form of the medial axis must generically be one of three cases, which we denote by the A notation explained below (here, it merely serves as a reference to sections of the paper): endpoints (A 3), interior points (A 1 2), and junctions (A 1 3). The local form of shocks is then derived from a sub-classification of these points into six types. Second, we address the (classical) instabilities of the MA, i.e., abrupt changes in the representation arising from slight changes in shape, as when a new branch appears with slight protrusion. The identification of these ‘transitions’ is clearly crucial in robust object recognition. We show that for the medial axis only two such instabilities are generically possible: (i) when four branches come together (A 1 4), and (ii) when a new branch grows out of an existing one (A 1 A 3). Similarly, there are six cases of shock instabilities, derived as sub-classifications of the MA instabilities. We give an explicit example of a dent forming in an ellipse where many of the transitions described in the paper can be seen to appear.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asada, H. and Brady, M. 1983. The curvature primal sketch. IEEE PAMI, 8:2-14.
Blum, H. 1973. Biological shape and visual science. J. Theor. Biol., 38:205-287.
Bogaevsky, I.A. 2002. Perestroikas of shocks and singularities of minimum functions. Physica D: Nonlinear Phenomena, 173(1-2), 1-29.
Bogaevsky, I.A. 1990. Metamorphoses of singularities of minimum functions, and bifurcations of shock waves of the Burgers equation with vanishing viscosity. St. Petersburg (Leningrad) Math. J., 1(4):807-823.
Bruce, J. and Giblin, P. 1992. Curves and Singularities. Second Edition. Cambridge University Press.
Bruce, J., Giblin, P., and Gibson, C. 1985. Symmetry sets. Proceedings of the Royal Society of Edinburgh, 101A:163-186.
Bruce, J.W. and Giblin, P.J. 1986. Growth, motion and 1-parameter families of symmetry sets. Proceedings of the Royal Society of Edinburgh, 104A:179-204.
Buchner, M. 1978. The structure of the cutlocus in dimension less than or equal to six. Compositio Mathematica, 37:103-119.
Geiger, D., Liu, T.-L., and Kohn, R.V. 1998. Representation and selfsimilarity of shapes. In ICCV1998, Sixth International Conference on Computer Vision, Bombay, India, 1998, IEEE Computer Society Press.
Giblin, P.J. and Kimia, B.B. 1999. On the intrinsic reconstruction of shape from its symmetries. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Fort Collins, Colorado, USA, IEEE Computer Society Press, June 23-25 1999, pp. 79-84, (to appear) in IEEE Transactions on Pattern Analysis and Machine Intelligence.Sixth International Conference on Computer Vision, Bombay, India, 1998, IEEE Computer Society Press.
Klein, P., Tirthapura, S., Sharvit, D., and Kimia, B. 2000. A tree-edit distance algorithm for comparing simple, closed shapes. In Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), San Francisco, California, pp. 696-704.
Leyton, M. 1987. Symmetry-curvature duality. Computer Vision, Graphics, and Image processing, 38:327-341.
Sebastian, T., Klein, P., and Kimia, B. 2000. An edit distance approach to shape matching. Technical Report LEMS 183, LEMS, Brown University.
Sebastian, T.B., Klein, P.N., and Kimia, B.B. 2001. Recognition of shapes by editing shock graphs. In Proceedings of the Eighth International Conference on Computer Vision, Vancouver, Canada, IEEE Computer Society Press, pp. 755-762.
Sharvit, D., Chan, J., Tek, H., and Kimia, B.B. 1998Symmetry-based indexing of image databases. Journal of Visual Communication and Image Representation, 9(4):366- 380.
Siddiqi, K. and Kimia, B.B. 1996. A shock grammar for recognition.In Proceedings of the Conference on Computer Vision and Pattern Recognition, pp. 507-513.
Siddiqi, K., Shokoufandeh, A., Dickinson, S., and Zucker, S. 1998. Shock graphs and shape matching. In ICCV1998 Sixth International Conference on ComputerVision, Bombay, India, 1998, IEEE Computer Society Press, pp. 222-229.
Tek, H. and Kimia, B.B. 1999. A discerete wave propagation method for the exact recovery of bisectors as shocks. Technical Report LEMS 181, LEMS, Brown University.
Tek, H. and Kimia, B.B. 1999. Symmetry maps of free-form curve segments via wave propagation. In Proceedings of the Fifth International Conference on Computer Vision, KerKyra, Greece, IEEE Computer Society Press, pp. 362-369.
Tirthapura, S., Sharvit, D., Klein, P., and Kimia, B.B. 1998. Indexing based on edit-distance matching of shape graphs. In SPIE Inter.Symposium on Voice, Video, and Data Communications, Boston, pp. 25-36.
Yomdin, Y. 1981. On the local structure of a generic central set. Compositio Mathematica, 43:225-238.
Zhu, S.C. and Yuille, A.L. 1996. FORMS: A flexible object recognition and modeling system. Intl. J. of Computer Vision, 20(3):187- 212.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Giblin, P.J., Kimia, B.B. On the Local Form and Transitions of Symmetry Sets, Medial Axes, and Shocks. International Journal of Computer Vision 54, 143–157 (2003). https://doi.org/10.1023/A:1023761518825
Issue Date:
DOI: https://doi.org/10.1023/A:1023761518825