The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.12 - December (2009 vol.31)
pp: 2227-2242
Mikhail Zaslavskiy , Mines ParisTech and Institute Curie-INSERM, France
Francis Bach , INRIA-d'Informatique de l'Ecole Normale Supérieure, Paris
Jean-Philippe Vert , Mines ParisTech and Institute Curie-INSERM, France
ABSTRACT
We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We, therefore, construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore, perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four data sets: simulated graphs, QAPLib, retina vessel images, and handwritten Chinese characters. In all cases, the results are competitive with the state of the art.
INDEX TERMS
Graph algorithms, graph matching, convex programming, gradient methods, machine learning, classification, image processing.
CITATION
Mikhail Zaslavskiy, Francis Bach, Jean-Philippe Vert, "A Path Following Algorithm for the Graph Matching Problem", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.31, no. 12, pp. 2227-2242, December 2009, doi:10.1109/TPAMI.2008.245
REFERENCES
[1] R.S.T. Lee and J.N.K. Liu, “An Oscillatory Elastic Graph Matching Model for Recognition of Offline Handwritten Chinese Characters,” Proc. Third Int'l Conf. Knowledge-Based Intelligent Information Eng. Systems, pp. 284-287, 1999.
[2] A. Filatov, A. Gitis, and I. Kil, “Graph-Based Handwritten Digit String Recognition,” Proc. Third Int'l Conf. Document Analysis and Recognition, pp. 845-848, 1995.
[3] H.F. Wang and E.R. Hancock, “Correspondence Matching Using Kernel Principal Components Analysis and Label Consistency Constraints,” Pattern Recognition, vol. 39, no. 6, pp. 1012-1025, June 2006.
[4] B. Luo and E.R. Hancock, “Alignment and Correspondence Using Singular Value Decomposition,” Lecture Notes in Computer Science, vol. 1876, pp. 226-235, Springer, 2000.
[5] M. Carcassoni and E.R. Hancock, “Spectral Correspondence for Point Pattern Matching,” Pattern Recognition, vol. 36, pp. 193-204, 2002.
[6] C. Schellewald and C. Schnor, “Probabilistic Subgraph Matching Based on Convex Relaxation,” Lecture Notes in Computer Science, vol. 3757, pp. 171-186, Springer, 2005.
[7] R. Singh, J. Xu, and B. Berger, “Pairwise Global Alignment of Protein Interaction Networks by Matching Neighborhood Topology,” Proc. 11th Int'l Conf. Research in Computational Molecular Biology, 2007.
[8] Y. Wang, F. Makedon, J. Ford, and H. Huang, “A Bipartite Graph Matching Framework for Finding Correspondences between Structural Elements in Two Proteins,” Proc. IEEE 26th Ann. Int'l Conf. Eng. in Medicine and Biology Society, pp. 2972-2975, 2004.
[9] W.R. Taylor, “Protein Structure Comparison Using Bipartite Graph Matching and Its Application to Protein Structure Classification,” Molecular and Cellular Proteomics, vol. 1, no. 4, pp. 334-339, 2002.
[10] D.C. Schmidt and L.E. Druffel, “A Fast Backtracking Algorithm for Test Directed Graphs for Isomorphism,” J. ACM, vol. 23, no. 3, pp. 433-445, 1976.
[11] J.R. Ullmann, “An Algorithm for Subgraph Isomorphism,” J. ACM, vol. 23, no. 1, pp. 31-42, 1976.
[12] L.P. Cordella, P. Foggia, C. Sansone, and M. Vento, “Performance Evaluation of the vf Graph Matching Algorithm,” Proc. 10th Int'l Conf. Image Analysis and Processing, vol. 2, pp. 1038-1041, 1991.
[13] S. Umeyama, “An Eigendecomposition Approach to Weighted Graph Matching Problems,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 5, pp. 695-703, Sept. 1988.
[14] L.S. Shapiro and J.M. Brady, “Feature-Based Correspondence,” Image and Vision Computing, vol. 10, pp. 283-288, 1992.
[15] T. Caelli and S. Kosinov, “An Eigenspace Projection Clustering Method for Inexact Graph Matching,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 4, pp. 515-519, Apr. 2004.
[16] H. Almohamad and S. Duffuaa, “A Linear Programming Approach for the Weighted Graph Matching Problem,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, no. 5, pp. 522-525, May 1993.
[17] S. Gold and A. Rangarajan, “A Graduated Assignment Algorithm for Graph Matching,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 4, pp. 377-388, Apr. 1996.
[18] D. Cremers, T. Kohlberger, and C. Schnor, “Evaluation of Convex Optimization Techniques for the Weighted Graph-Matching Problem in Computer Vision,” Proc. 23rd DAGM-Symp. Pattern Recognition, vol. 2191, 2001.
[19] S. Belongie, J. Malik, and J. Puzicha, “Shape Matching and Object Recognition Using Shape Contexts,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 4, pp. 509-522, Apr. 2002.
[20] K. Brein, M. Remm, and E. Sonnhammer, “Inparanoid: A Comprehensive Database of Eukaryotic Orthologs,” Nucleic Acids Research, vol. 33, 2005.
[21] M.R. Garey and D.S. Johnson, Computer and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, 1979.
[22] J.M. Borwein and A.S. Lewis, Convex Analysis and Nonlinear Optimization. Springer-Verlag, 2000.
[23] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge Univ. Press, 2003.
[24] L.F. McGinnis, “Implementation and Testing of a Primal-Dual Algorithm for the Assignment Problem,” Operations Research, vol. 31, no. 2, pp. 277-291, 1983.
[25] D. Conte, P. Foggia, C. Sansone, and M. Vento, “Thirty Years of Graph Matching in Pattern Recognition,” Int'l J. Pattern Recognition and Artificial Intelligence, vol. 18, pp. 265-298, 2004.
[26] G.H. Golub and C.F.V. Loan, Matrix Computations, third ed. Johns Hopkins Univ. Press, 1996.
[27] L.F. McGinnis, “Implementation and Testing of a Primal-Dual Algorithm for the Assignment Problem,” Operations Research, vol. 31, pp. 277-291, 1983.
[28] H. Kuhn, “The Hungarian Method for the Assignment Problem,” Naval Research, vol. 2, pp. 83-97, 1955.
[29] M. Frank and P. Wolfe, “An Algorithm for Quadratic Programming,” Naval Research Logistics Quarterly, vol. 3, pp. 95-110, 1956.
[30] F.R.K. Chung, Spectral Graph Theory. Am. Math. Soc., 1997.
[31] R. Rockafeller, Convex Analysis. Princeton Univ. Press, 1970.
[32] K.M. Anstreicher and N.W. Brixius, “A New Bound for the Quadratic Assignment Problem Based on Convex Quadratic Programming,” Math. Programming, vol. 89, no. 3, pp. 341-357, 2001.
[33] A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, 1987.
[34] E. Allgower and K. Georg, Numerical Continuation Methods. Springer, 1990.
[35] J. Milnor, Topology from the Differentiable Viewpoint. Univ. Press of Virginia, 1969.
[36] D. Bertsekas, Nonlinear Programming. Athena Scientific, 1999.
[37] M.E.J. Newman, S.H. Strogatz, and D.J. Watts, “Random Graphs with Arbitrary Degree Distributions and Their Applications,” Physical Rev., vol. 64, 2001.
[38] E. Cela, “Quadratic Assignment Problem Library,” www.opt. math.tu-graz.ac.atqaplib/, 2007.
[39] T. Walter, J.-C. Klein, P. Massin, and A. Erignay, “Detection of the Median Axis of Vessels in Retinal Images,” European J. Ophthalmology, vol. 13, no. 2, 2003.
[40] F. Bookstein, “Principal Warps: Thin-Plate Splines and the Decomposition of Deformations,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 6, pp. 567-585, June 1989.
[41] T. Saito, H. Yamada, and K. Yamamoto, “On the Data Base etl9b of Handprinted Characters in jis Chinese Characters and Its Analysis,” IEICE Trans., vol. 68, no. 4, pp. 757-764, 1985.
6 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool