Application of second generation wavelets to blind spherical deconvolution

Abstract

We adress the problem of spherical deconvolution in a non parametric statistical framework, where both the signal and the operator kernel are subject to error measurements. After a preliminary treatment of the kernel, we apply a thresholding procedure to the signal in a second generation wavelet basis. Under standard assumptions on the kernel, we study the theoritical performance of the resulting algorithm in terms of L p losses (p ≥ 1) on Besov spaces on the sphere. We hereby extend the application of second generation spherical wavelets to the blind deconvolution framework [16]. The procedure is furthermore adaptive with regard both to the target function sparsity and smoothness, and the kernel blurring effect. We end with the study of a concrete example, putting into evidence the improvement of our procedure on the recent blockwise-SVD algorithm [6].

A general approach to regularizing inverse problems with regional data using Slepian wavelets

“…It would be beyond the scope and size of this article to give a complete survey of such papers here. Examples of other works where wavelets have been used for inverse problems on the sphere are [13,37,40,43]. In [8,23] it was shown that a wavelet-based regularization can be constructed if the svd of the forward operator is known.…”