Filters for anisotropic wavelet decompositions

Abstract

Like the continuous shearlet transform and their relatives, discrete transformations based on the interplay between several filterbanks with anisotropic dilations provide a high potential to recover directed features in two and more dimensions. Due to simplicity, most of the directional systems constructed so far were using prediction–correction methods based on interpolatory subdivision schemes. In this paper, we give a simple but effective construction for QMF (quadrature mirror filter) filterbanks which are the usual discrete tools in orthogonal wavelet analysis. We characterize when the filterbank gives rise to the existence of refinable functions, and hence wavelets, and give a generalized shearlet construction for arbitrary dimensions and arbitrary scalings for which the filterbank construction ensures the existence of an orthogonal wavelet analysis. We also show that, under some restriction on the dilation factors, this multiple filterbank system satisfies the slope resolution property , which is a key feature in all types of directional transforms.