Rotation invariant, Riesz bases of directional wavelets

This article addresses the issue of designing bases for L2(R2) 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 L2(R2). 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.