Application of second generation wavelets to blind spherical deconvolution

TL;DR

This paper introduces an adaptive method using second generation wavelets for blind spherical deconvolution, effectively handling errors in both the signal and kernel within a non-parametric framework.

Contribution

It extends second generation spherical wavelets to blind deconvolution, providing a theoretically sound and adaptive algorithm that improves upon existing methods.

Findings

The proposed method achieves better $L^p$ loss performance.

It adapts to target function sparsity and smoothness.

Demonstrates improvement over blockwise-SVD algorithm.

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 ($p\geq 1$) on Besov spaces on the sphere. We hereby extend the application of second generation spherical wavelets to the blind deconvolution framework. 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.