Locally Correct Frechet Matchings

TL;DR

This paper introduces locally correct Frechet matchings, a subset of Frechet matchings that better capture curve similarity, along with algorithms to compute them efficiently.

Contribution

It proposes the concept of locally correct Frechet matchings, proves their existence, and provides algorithms with different complexities for their computation.

Findings

Existence of at least one locally correct Frechet matching for two polygonal curves.

An O(N^3 log N) algorithm to compute a locally correct Frechet matching.

An O(N^2) algorithm for discrete Frechet matching.

Abstract

The Frechet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Frechet distance a Frechet matching. There are often many different Frechet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Frechet matchings to "natural" matchings and to this end introduce locally correct Frechet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N^3 log N) algorithm to compute it, where N is the total number of edges in both curves. We also present an O(N^2) algorithm to compute a locally correct discrete Frechet matching.