A new algorithm for computing the 2-dimensional matching distance between size functions

Size Theory has proven to be a useful geometrical/topological approach to shape analysis and comparison. Originally introduced by considering 1-dimensional properties of shapes, described by means of real-valued functions, it has been subsequently generalized to take into account multidimensional properties coded by functions valued in Rk. In the context of Size Theory, this generalization has led to introduce a shape descriptor called k-dimensional size function, and a distance to compare size functions, namely the k-dimensional matching distance. This paper proposes a novel computational framework to deal with the 2-dimensional case of Size Theory. More precisely, some new theoretical results about approximating the 2-dimensional matching distance are presented, leading to the formulation of an algorithm for its computation (up to an arbitrary error threshold).