Matching hierarchical structures using association graphs

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.