Topology Preserving Graph Matching

We describe an algorithm for graph matching which preserves global topological structure using an homology preserving graph matching. We show that for simplicial homology, graph matching is equivalent to finding an optimal simplicial chain map, which can be posed as a linear program satisfying boundary commutativity, simplex face intersection and assignment constraints. The homology preserving matching provides (i) globally consistent matching with higher order simplex weights, (ii) match verification using topological invariants, and (iii) topological constraints to improve any geometric matching approach. Vision applications require ’topologically rich’ graphs, so we describe a general simplicial complex construction such that salient image structures correspond to homology groups. Finally, we evaluate performance on random graphs and demonstrate use of homology preserving subgraph matching for object matching applications.