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 thecomparison of multidimensional size functions can be reduced tothe $1$-dimensional case by a suitable change of variables.Indeed, a foliation in half-planes can be given, suchthat the restriction of a multidimensional size function to eachof these half-planes turns out to be a classical size function intwo scalar variables. This leads to the definition of a newdistance between multidimensional size functions, and to the proofof their stability with respect to that distance. Experiments are carried out to show the feasibility of the method.