Using matching distance in size theory: A survey

In this survey we illustrate how the matching distance between reduced size functions can be applied for shape comparison. We assume that each shape can be thought of as a compact connected manifold with a real continuous function defined on it, that is a pair : (M,phi : M --> R), called size pair. In some sense, the function phi focuses on the properties and the invariance of the problem at hand. In this context, matching two size pairs (M, phi) and (N, psi) means looking for a homeomorphism between M and N that minimizes the difference of values taken by phi and psi on the two manifolds. Measuring the dissimilarity between two shapes amounts to the difficult task of computing the value delta = inf(f) max(P is an element of M)\phi(P) - psi(f(P))\, where f varies among all the homeomorphisms from M to N. From another point of view, shapes can be described by reduced size functions associated with size pairs. The matching distance between reduced size functions allows for a robust to perturbations comparison of shapes. The link between reduced size functions and the dissimilarity measure delta is established by a theorem, stating that the matching distance provides an easily computable lower bound for delta. Throughout this paper we illustrate this approach to shape comparison by means of examples and experiments. (C) 2007 Wiley Periodicals, Inc.