Many-to-many feature matching for structural pattern recognition

Abstract

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 subset of nodes in one graph and a subset of nodes in the other. In this thesis we aim to develop a framework to establish many-to-many correspondences between the nodes of two noisy, vertex-labeled weighted graphs. The difficulty of providing such correspondences is due to the fact that any subset of nodes in one graph may correspond to any subset of nodes in another. To overcome this combinatorial challenge, we transform the graphs into an alternative domain in which the many-to-many graph matching becomes that of matching point sets. Our interest in transforming the many-to-many graph matching problem into that of many-to-many point matching is motivated by the fact that a number of algorithms have proven useful in establishing such correspondences in the geometric space in polynomial-time. Our goal is to use one such algorithm to approximate the solution for the original graph representations. The algorithm is based on recent developments in efficient low-distortion metric embedding of graphs into normed vector spaces. We present two such embedding algorithms, beginning with Matousek's algorithm [66], in which the dimensionality of a graph's embedding is graph-dependent. Two graphs to be matched may yield embeddings with different dimensionality, requiring a projection step to bring them to the same space. We overcome this problem by introducing a novel embedding technique, using a spherical encoding of graph structure, that embeds both graphs into a single space of prescribed dimensionality. By embedding weighted graphs into normed vector spaces, we reduce the problem of many-to-many graph matching to the problem 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. The computed mass flows yield a set of many-to-many node correspondences between the original graphs. Empirical evaluation of the algorithm on an extensive set of recognition trials, including a comparison with competing graph matching approaches, demonstrates both the robustness and efficiency of the overall approach.