Many-to-many graph matching via metric embedding

Graph matching is an important component in many object recognition algorithms. Although most graph matching algorithms seek a one-to-one correspondence between nodes, it is often the case that a more meaningful correspondence exists between a cluster of nodes in one graph and a cluster of nodes in the other. We present a matching algorithm that establishes many-to-many correspondences between nodes of noisy, vertex-labeled weighted graphs. The algorithm is based on recent developments in efficient low-distortion metric embedding of graphs into normed vector spaces. By embedding weighted graphs into normed vector spaces, we reduce the problem of many-to-many graph matching to that of computing a distribution-based distance measure between graph embeddings. We use a specific measure, the earth mover's distance, to compute distances between sets of weighted vectors. Empirical evaluation of the algorithm on an extensive set of recognition trials demonstrates both the robustness and efficiency of the overall approach.

[1]  Leonidas J. Guibas,et al.  The Earth Mover's Distance under transformation sets , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[2]  Anupam Gupta,et al.  Cuts, Trees and ℓ1-Embeddings of Graphs* , 2004, Comb..

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

[4]  H. C. Longuet-Higgins,et al.  An algorithm for associating the features of two images , 1991, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[5]  J. M. Sek.  On embedding trees into uniformly convex Banach spaces , 1999 .

[6]  Mikkel Thorup,et al.  On the approximability of numerical taxonomy (fitting distances by tree metrics) , 1996, SODA '96.

[7]  Terry Caelli,et al.  Inexact Multisubgraph Matching Using Graph Eigenspace and Clustering Models , 2002, SSPR/SPR.

[8]  Edwin R. Hancock,et al.  Structural Graph Matching Using the EM Algorithm and Singular Value Decomposition , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[9]  Kaleem Siddiqi,et al.  Many-to-many Matching of Attributed Trees Using Association Graphs and Game Dynamics , 2001, IWVF.

[10]  J. Matousek,et al.  On embedding trees into uniformly convex Banach spaces , 1999 .

[11]  Michael E. Saks,et al.  Trees and Euclidean metrics , 1998, STOC '98.

[12]  Piotr Indyk,et al.  Algorithmic applications of low-distortion geometric embeddings , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[13]  Ali Shokoufandeh,et al.  On the Representation and Matching of Qualitative Shape at Multiple Scales , 2002, ECCV.

[14]  Leonidas J. Guibas,et al.  The Earth Mover's Distance as a Metric for Image Retrieval , 2000, International Journal of Computer Vision.

[15]  David K. Smith.  Network Flows: Theory, Algorithms, and Applications , 1994 .

[16]  Edward M. Riseman,et al.  How Easy is Matching 2D Line Models Using Local Search? , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

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

[18]  Robert M. Haralick,et al.  Structural Descriptions and Inexact Matching , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[19]  J. Bourgain.  On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .

[20]  Steven Gold,et al.  A Graduated Assignment Algorithm for Graph Matching , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[21]  Avinash C. Kak,et al.  3-D Object Recognition Using Bipartite Matching Embedded in Discrete Relaxation , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  S. Umeyama,et al.  Least-Squares Estimation of Transformation Parameters Between Two Point Patterns , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[23]  PaperNo.  Recognition of shapes by editing shock graphs , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[24]  Tyng-Luh Liu,et al.  Approximate tree matching and shape similarity , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.