On directional scaling matrices in dimension d=2

Scaling matrices are the key ingredient in subdivision schemes and multiresolution analysis, because they fix the way to refine the given data and to manipulate them. It is known that the absolute value of the scaling matrix determinant gives the number of disjoint cosets which is strictly connected with the number of filters needed to analyse a signal and then to computational complexity. Among the classical scaling matrices, we find the family of shearlet matrices that have many interesting properties that make them attractive when dealing with anisotropic problems. Their drawback is the relatively large determinant. The aim of this paper is to find a system of scaling matrices with the same good properties of shearlet matrices but with lower determinant.