Coupled regularization with multiple data discrepancies

TL;DR

This paper develops a general framework for coupled regularization in inverse problems involving multiple data sources, providing stability, convergence, and practical algorithms, with applications in multi-modal imaging like MR and PET.

Contribution

It introduces a unified approach to coupled regularization for multi-source inverse problems, including convergence analysis and adaptable parameter strategies.

Findings

Improved convergence rates with tailored parameter choices.

Effective algorithms demonstrated on multi-contrast MR and MR-PET.

Theoretical results applicable to Kullback-Leibler divergence cases.

Abstract

We consider a class of regularization methods for inverse problems where a coupled regularization is employed for the simultaneous reconstruction of data from multiple sources. Applications for such a setting can be found in multi-spectral or multi-modality inverse problems, but also in inverse problems with dynamic data. We consider this setting in a rather general framework and derive stability and convergence results, including convergence rates. In particular, we show how parameter choice strategies adapted to the interplay of different data channels allow to improve upon convergence rates that would be obtained by treating all channels equally. Motivated by concrete applications, our results are obtained under rather general assumptions that allow to include the Kullback-Leibler divergence as data discrepancy term. To simplify their application to concrete settings, we further elaborate several practically relevant special cases in detail. To complement the analytical results, we also provide an algorithmic framework and source code that allows to solve a class of jointly regularized inverse problems with any number of data discrepancies. As concrete applications, we show numerical results for multi-contrast MR and joint MR-PET reconstruction.