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,f: M->R), called size pair. In some sense, the function f focuses on the properties and the invariance of the problem at hand. In this context, matching two size pairs (M,f) and (N,g) means looking for a homeomorphism between M and N that minimizes the difference of values taken by f and g on the two manifolds. Measuring the dissimilarity between two shapes amounts to the difficult task of computing the value d=inf_h max_P|f(P)-g(h(P))|, where h 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 d is established by a theorem, stating that the matching distance provides an easily computable lower bound for d. Throughout this paper we illustrate this approach to shape comparison by means of examples and experiments.