Natural pseudo-distance and optimal matching between reduced size functions

We study a dissimilarity measure between shapes, expressed by the natural pseudodistance between size pairs, where a shape is viewed as a topological space endowed with a real-valued continuous function. Measuring dissimilarity amounts to minimizing the change in the functions due to the application of homeomorphisms between topological spaces, with respect to the L∞-norm. A new class of shape descriptors, called reduced size functions, is introduced. They are obtained by slightly modifying the classical concept of size function. We show that reduced size functions can be compared by solving an optimal matching problem between countable point sets. In this way we obtain a distance between reduced size functions that is shown to be stable. This allows us to provide a new lower bound for the dissimilarity measure between shapes.