Rotation invariant, Riesz bases of directional wavelets
Known directional wavelets are generated by a single mother wavelet in the Schwartz space \(S(\mathbb R^2)\). This article addresses the issue of designing bases for \(L^2(\mathbb R^2)\) that are generated by translations, rotations and dilations of a single mother wavelet. The results are generated with a new approach. The proposed construction simplifies previous ones by setting an odd number of directions and by choosing properly the phase of the Fourier transform of a single mother wavelet. A large part of the article is devoted to the proof of theorems that give sufficient conditions for a single mother wavelet to generate a Riesz sequence and a Riesz basis for \(L^2(\mathbb R^2)\). An example of a Riesz sequence whose restriction to each scale is orthonormal is given. The theoretical results are confirmed by numerical experiments where a discrete directional wavelet transform is introduced.
scientific article; zbMATH DE number 6967583
scientific article