Localized spherical deconvolution

TL;DR

This paper introduces a novel spherical deconvolution algorithm combining SVD inversion with thresholding, demonstrating theoretical robustness and promising practical results for recovering objects with varying regularity.

Contribution

The paper presents a new spherical deconvolution method that adapts to object regularity and inhomogeneous smoothness, with proven theoretical bounds and effective numerical performance.

Findings

Upper bounds for the procedure's behavior under $\,L_p$ loss.

Adaptation to regularity and inhomogeneous smoothness.

Numerical results show promising practical performance.

Abstract

We provide a new algorithm for the treatment of the deconvolution problem on the sphere which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. We establish upper bounds for the behavior of our procedure for any $\mathbb {L}_p$ loss. It is important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover as well as to inhomogeneous smoothness. We also perform a numerical study which proves that the procedure shows very promising properties in practice as well.