Abstract
This paper shows how the eigenstructure of the adjacency matrix can be used for the purposes of robust graph-matching. We commence from the observation that the leading eigenvector of a transition probability matrix is the steady state of the associated Markov chain. When the transition matrix is the normalised adjacency matrix of a graph, then the leading eigenvector gives the sequence of nodes of the steady state random walk on the graph. We use this property to convert the nodes in a graph into a string where the node-order is given by the sequence of nodes visited in the random walk. We match graphs represented in this way, by finding the sequence of string edit operations which minimise edit distance.
Supported by CONACYT, under grant No. 146475/151752.
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References
R. C. Wilson A. M. Finch and E. R. Hancock. An energy function and continuous edit process for graph matching. Neural Computation, 10(7):1873–1894, 1998.
K. Siddiqi A. Shokoufandeh, S. J. Dickinson and S. W. Zucker. Indexing using a spectral encoding of topological structure. In Proceedings of the Computer Vision and Pattern Recognition, 1998.
Luo Bin and E. R. Hancock. Procrustes alignment with the em algorithm. In 8th International Conference on Computer Analysis of Images and Image Patterns, pages 623–631, 1999.
H. Buke. On a relation between graph edit distance and maximum common subgraph. Pattern Recognition Letters, 18, 1997.
Fan R. K. Chung. Spectral Graph Theory. American Mathematical Society, 1997.
M. A. Eshera and K. S. Fu. A graph distance measure for image analysis. SMC, 14(3):398–408, May 1984.
S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching. PAMI, 18(4):377–388, April 1996.
R. Horaud and H. Sossa. Polyhedral object recognition by indexing. Pattern Recognition, 1995.
L. Lovász. Random walks on graphs: a survey. Bolyai Society Mathematical Studies, 2(2):1–46, 1993.
V. I. Levenshtein. Binary codes capable of correcting deletions, insertions and reversals. Sov. Phys. Dokl., 6:707–710, 1966.
Bin Luo and E. R. Hancock. Structural graph matching using the EM algorithm and singular value decomposition. To appear in IEEE Trans. on Pattern Analysis and Machine Intelligence, 2001.
B. J. Oommen and K. Zhang. The normalized string editing problem revisited. PAMI, 18(6):669–672, June 1996.
A. Sanfeliu and K. S. Fu. A distance measure between attributed relational graphs for pattern recognition. IEEE Transactions on Systems, Man and Cybernetics, 13:353–362, 1983.
G. Scott and H. Longuet-Higgins. An algorithm for associating the features of two images. In Proceedings of the Royal Society of London, number 244 in B, 1991.
L. G. Shapiro and R. M. Haralick. Relational models for scene analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 4:595–602, 82.
L. S. Shapiro and J. M. Brady. A modal approach to feature-based correspondence. In British Machine Vision Conference, 1991.
S. Ullman. Filling in the gaps. Biological Cybernetics, 25:1–6, 76.
S. Umeyama. An eigen decomposition approach to weighted graph matching problems. PAMI, 10(5):695–703, September 1988.
R. S. Varga. Matrix Iterative Analysis. Springer, second edition, 2000.
R. A. Wagner. The string-to-string correction problem. Journal of the ACM, 21(1), 1974.
J. T. L. Wang, B. A. Shapiro, D. Shasha, K. Zhang, and K. M. Currey. An algorithm for finding the largest approximatelycommon substructures of two trees. PAMI, 20(8):889–895, August 1998.
R. C. Wilson and E. R. Hancock. Structural matching by discrete relaxation. PAMI, 19(6):634–648, June 1997.
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© 2002 Springer-Verlag Berlin Heidelberg
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Robles-Kelly, A., Hancock, E.R. (2002). String Edit Distance, Random Walks and Graph Matching. In: Caelli, T., Amin, A., Duin, R.P.W., de Ridder, D., Kamel, M. (eds) Structural, Syntactic, and Statistical Pattern Recognition. SSPR /SPR 2002. Lecture Notes in Computer Science, vol 2396. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-70659-3_10
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