Multidimensional size functions for shape comparison

Size Theory has proven to be a useful framework for shape analysis in the context of pattern recognition. Its main tool is a shape descriptor called size function. Size Theory has been mostly developed in the $1$-dimensional setting, meaning that shapes are studied with respect to functions, defined on the studied objects, with values in $R$. The potentialities of the $k$-dimensional setting, that is using functions with values in $R^k$, were not explored until now for lack of an efficient computational approach. In this paper we provide the theoretical results leading to a concise and complete shape descriptor also in the multidimensional case. This is possible because we prove that in Size Theory the comparison of multidimensional size functions can be reduced to the $1$-dimensional case by a suitable change of variables. Indeed, a foliation in half-planes can be given, such that the restriction of a multidimensional size function to each of these half-planes turns out to be a classical size function in two scalar variables. This leads to the definition of a new distance between multidimensional size functions, and to the proof of their stability with respect to that distance. Experiments are carried out to show the feasibility of the method.