Proximity operators for a class of hybrid sparsity + entropy priors application to dosy NMR signal reconstruction

Inverse problems arising from Laplace transform inversion are ill-posed, and require suitable regularization strategies. Although the maximum entropy regularization approach usually appears as an adequate strategy due to its ability to recover regular positive valued signals, it was observed to lead to poor reconstruction results when the sought signal contains narrow peaks. In that case, a sparsity promoting penalty such as the h norm, combined with a positivity constraint, is more suitable. In order to derive a flexible resolution method, hybrid approaches combining both entropy and sparsity regularization strategies should be envisaged. However, the choice of an efficient optimization algorithm remains a challenging task. Among available optimization techniques, proximal methods have shown their efficiency in solving large scale possibly nonsmooth problems. This paper provides an extensive list of new proximity operators for the sum of entropy and sparsity penalties. The applicability of these results is illustrated by means of experiments, in the context of DOSY NMR signal reconstruction.