Filters for anisotropic wavelet decompositions

TL;DR

This paper introduces a simple construction for quadrature mirror filterbanks that facilitate anisotropic wavelet decompositions, enabling effective directional feature analysis in multiple dimensions.

Contribution

It provides a new, straightforward method for constructing orthogonal wavelet filterbanks suitable for anisotropic and multidimensional shearlet-like transforms.

Findings

Constructs QMF filterbanks for anisotropic wavelet analysis.

Characterizes conditions for refinable functions and wavelet existence.

Generalizes shearlet construction to arbitrary dimensions and scalings.

Abstract

Like the continous 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 discrete object between orthogonal wavelet analysis. We also 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.