The computational geometry of comparing shapes

Abstract

This article is a survey on methods from computational geometry for comparing shapes that we developed within our work group at Freie Universität Berlin. In particular, we will present the ideas and complexity considerations for the computation of two distance measures, the Hausdorff distance and the Fréchet distance. Whereas the former is easier to compute, the latter better captures the similarity of shapes as perceived by human observers. We will consider shapes modelled by curves in the plane as well as surfaces in three-dimensional space. Especially, the Fréchet distance of surfaces seems computationally intractable and is of yet not even known to be computable. At least the decision problem is shown to be recursively enumerable. © 2009 Springer Berlin Heidelberg.