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,ϕ : ℳ → ℝ), called size pair. In some sense, the function ϕ focuses on the properties and the invariance of the problem at hand. In this context, matching two size pairs (ℳ, ϕ) and (𝒩, ψ) means looking for a homeomorphism between ℳ and 𝒩 that minimizes the difference of values taken by ϕ and ψ on the two manifolds. Measuring the dissimilarity between two shapes amounts to the difficult task of computing the value δ = inf_f max_P∈ℳ |ϕ(P) - ψ(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 δ is established by a theorem, stating that the matching distance provides an easily computable lower bound for δ. Throughout this paper we illustrate this approach to shape comparison by means of examples and experiments. © 2007 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 16, 154–161, 2006 [ABSTRACT FROM AUTHOR]