Locally correct Fréchet matchings

The Fréchet distance is a metric to compare two curves, which is based on monotone matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Fréchet matchings to “natural” matchings and to this end introduce locally correct Fréchet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N³log⁡N) algorithm to compute it, where N is the number of edges in both curves. We also present an O(N²) algorithm to compute a locally correct discrete Fréchet matching.