Structural properties of shape-spaces

Abstract

General properties of distance functions and of affinity functions are discussed in this paper. Reasons are given why a distance function for ℜn based shape-spaces should be a metric. Several distance functions that are used in shape-spaces are examined and it is shown that not all of them are metrics. It is shown which impact the type of the distance function has on the shape-space, in particular on the form of recognition or affinity regions in the shape-space. Affinity functions should be defined in such a way that they determine an affinity region with positive values inside that region and zero or negative values outside. The form of an affinity function depends on the type of the underlying distance function. This is demonstrated with several examples. © Springer-Verlag Berlin Heidelberg 2006.