Anisotropic scaling matrices and subdivision schemes

Unlike wavelet, shearlets have the capability to detect directional discontinuities together with their directions. To achieve this, the considered scaling matrices have to be not only expanding, but also anisotropic. Shearlets allow for the definition of a directional multiple multiresolution analysis, where perfect reconstruction of the filterbank can be easily ensured by choosing an appropriate interpolatory multiple subdivision scheme. A drawback of shearlets is their relative large determinant that leads to a substantial complexity. The aim of the paper is to find scaling matrices in $\Z^{d \times d}$ which share the properties of shearlet matrices, i.e. anisotropic expanding matrices with the so-called slope resolution property, but with a smaller determinant. The proposed matrices provide a directional multiple multiresolution analysis whose behaviour is illustrated by some numerical tests on images.