Abstract
It is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a (derived) “association graph.” However, it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized, i.e. are trees, since maximal cliques are not constrained to preserve the partial order. Here we provide a solution to the problem of matching two trees, by constructing the association graph using the graph-theoretic concept of connectivity. We prove that in the new formulation there is a one-to-one correspondence between maximal cliques and maximal subtree isomorphisms, and show how to solve the matching problem using simple “replicator” dynamical systems developed in theoretical biology. Such continuous solutions to discrete problems can motivate analog and biological implementations. We illustrate the power of the approach by matching articulated and deformed shapes described by shock trees.
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References
A. P. Ambler, H. G. Barrow, C. M. Brown, R. M. Burstall, and R. J. Popplestone. A versatile computer-controlled assembly system. In Proc. 3rd Int. J. Conf. Artif. Intell., pages 298–307, Stanford, CA, 1973.
D. H. Ballard and C. M. Brown. Computer Vision. Prentice-Hall, Englewood Cliffs, N.J, 1982.
H. G. Barrow and R. M. Burstall. Subgraph isomorphism, matching relational structures, and maximal cliques. Inform. Process. Lett., 4(4):83–84, 1976.
L. E. Baum and J. A. Eagon. An inequality with applications to statistical estimation for probabilistic functions of markov processes and to a model for ecology. Bull. Amer. Math. Soc., 73:360–363, 1967.
R. C. Bolles and R. A. Cain. Recognizing and locating partially visible objects: The locus-feature-focus method. Int. J. Robotics Res., 1(3):57–82, 1982.
I. M. Bomze. Evolution towards the maximum clique. J. Global Optim., 10:143–164, 1997.
I. M. Bomze, M. Pelillo, and R. Giacomini. Evolutionary approach to the maximum clique problem: Empirical evidence on a larger scale. In I. M. Bomze, T. Csendes, R. Horst, and P. M. Pardalos, editors, Developments in Global Optimization, pages 95–108, Dordrecht, The Netherlands, 1997. Kluwer.
J. F. Crow and M. Kimura. An Introduction to Population Genetics Theory. Harper & Row, New York, 1970.
R. A. Fisher. The Genetical Theory of Natural Selection. Oxford University Press, London, UK, 1930.
M. Garey and D. Johnson. Computer and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.
F. Harary. Graph Theory. Addison-Wesley, Reading, MA, 1969.
J. Hofbauer and K. Sigmund. The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, UK, 1988.
R. Horaud and T. Skordas. Stereo correspondence through feature grouping and maximal cliques. IEEE Trans. Pattern Anal. Machine Intell., 11(11):1168–1180, 1989.
B. B. Kimia, A. Tannenbaum, and S. W. Zucker. Shape, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space. Int. J. Comp. Vision, 15:189–224, 1995.
V. Losert and E. Akin. Dynamics of games and genes: Discrete versus continuous time. J. Math. Biol., 17:241–251, 1983.
Y. Lyubich, G. D. Maistrowskii, and Y. G. Ol'khovskii. Selection-induced convergence to equilibrium in a single-locus autosomal population. Problems of Information Transmission, 16:66–75, 1980.
D. Marr and K. H. Nishihara. Representation and recognition of the spatial organization of three-dimensional shapes. Proc. R. Soc. Lond. B, 200:269–294, 1978.
D. Miller and S. W. Zucker. Efficient simplex-like methods for equilibria of nonsymmetric analog networks. Neural Computation, 4(2):167–190, 1992.
D. Miller and S. W. Zucker. Computing with self-excitatory cliques: A model and an application to hyperacuity-scale computation in visual cortex. Neural Computation, 1998. To be published.
T. S. Motzkin and E. G. Straus. Maxima for graphs and a new proof of a theorem of Turán. Canad. J. Math., 17:533–540, 1965.
H. Ogawa. Labeled point pattern matching by delaunay triangulation and maximal cliques. Pattern Recognition, 19:35–40, 1986.
P. M. Pardalos and J. Xue. The maximum clique problem. J. Global Optim., 4:301–328, 1994.
M. Pelillo. Relaxation labeling networks for the maximum clique problem. J. Artif. Neural Networks, 2:313–328, 1995.
M. Pelillo. The dynamics of nonlinear relaxation labeling processes. J. Math. Imaging Vision, 7:309–323, 1997.
M. Pelillo. A unifying framework for relational structure matching. Submitted, 1997.
M. Pelillo and A. Jagota. Feasible and infeasible maxima in a quadratic program for maximum clique. J. Artif. Neural Networks, 2:411–420, 1995.
M. Pelillo, K. Siddiqi, and S. W. Zucker. Attributed tree matching and maximum weight cliques. Submitted, 1997.
F. Pla and J. A. Marchant. Matching feature points in image sequences through a region-based method. Comp. Vision Image Understanding, 66:271–285, 1997.
B. Radig. Image sequence analysis using relational structures. Pattern Recognition, 17:161–167, 1984.
H. Rom and G. Medioni. Hierarchical decomposition and axial shape description. IEEE Trans. Pattern Anal. Machine Intell., 15(10):973–981, 1993.
A. Rosenfeld, R. A. Hummel, and S. W. Zucker. Scene labeling by relaxation operations. IEEE Trans. Syst. Man Cybern., 6:420–433, 1976.
H. Samet. Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading, MA, 1990.
K. Siddiqi, A. Shokoufandeh, S. J. Dickinson, and S. W. Zucker. Shock graphs and shape matching. In Proc. Int. Conf. Gomp. Vision, pages 222–229, Bombay, India, 1998.
V. Venkateswar and R. Chellappa. Hierarchical stereo and motion correspondence using feature groupings. Int. J. Comp. Vision, 15:245–269, 1995.
J. W. Weibull. Evolutionary Game Theory. MIT Press, Cambridge, MA, 1995.
B. Yang, W. E. Snyder, and G. L. Bilbro. Matching oversegmented 3D images using association graphs. Image Vision Comput., 7:135–143, 1989.
S. Zhu and A. L. Yuille. FORMS: A flexible object recognition and modelling system. Int. J. Comp. Vision, 20(3):187–212, 1996.
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© 1998 Springer-Verlag Berlin Heidelberg
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Pelillo, M., Siddiqi, K., Zucker, S.W. (1998). Matching hierarchical structures using association graphs. In: Burkhardt, H., Neumann, B. (eds) Computer Vision — ECCV’98. ECCV 1998. Lecture Notes in Computer Science, vol 1407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054730
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DOI: https://doi.org/10.1007/BFb0054730
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