Shearletbased regularization in statistical inverse learning with an application to xray tomography
Abstract
Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational regularization framework, convergence rates have recently been proved for a class of convex, phomogeneous regularizers with p ∈ (1, 2], in the symmetric Bregman distance. Following this path, we take a further step towards the study of sparsitypromoting regularization and extend the aforementioned convergence rates to work with ℓ ^{ p }norm regularization, with p ∈ (1, 2), for a special class of nontight Banach frames, called shearlets, and possibly constrained to some convex set. The p = 1 case is approached as the limit case (1, 2) ∋ p → 1, by complementing numerical evidence with a (partial) theoretical analysis, based on arguments from Γconvergence theory. We numerically validate our theoretical results in the context of xray tomography, under random sampling of the imaging angles, using both simulated and measured data. This application allows to effectively verify the theoretical decay, in addition to providing a motivation for the extension to shearletbased regularization.
 Publication:

Inverse Problems
 Pub Date:
 May 2022
 DOI:
 10.1088/13616420/ac59c2
 arXiv:
 arXiv:2112.12443
 Bibcode:
 2022InvPr..38e4001B
 Keywords:

 convex regularization;
 statistical learning;
 xray tomography;
 wavelets;
 shearlets;
 convergence rates;
 Bregman distance;
 Mathematics  Statistics Theory
 EPrint:
 doi:10.1088/13616420/ac59c2