Rotation invariant, Riesz bases of directional wavelets

Abstract

This article addresses the issue of designing bases for L 2 ( R 2 ) that are generated by translations, rotations and dilations of a single mother wavelet ψ . We show how this construction can be simplified by setting an odd number of directions and by choosing properly the phase of the Fourier transform of ψ . A large part of the article is devoted to the proof of theorems that give sufficient conditions for ψ to generate a Riesz sequence and a Riesz basis for L 2 ( R 2 ) . An example of Riesz sequence whose restriction to each scale is orthonormal is set. Theoretical results are confirmed by numerical experiments where a discrete directional wavelet transform is introduced.