Minimization of a sparsity promoting criterion for the recovery of complex-valued signals

Ill-conditioned inverse problems are often encountered in signal/image processing. In this respect, convex objective functions including a sparsity promoting penalty term can be used. However, most of the existing optimization algorithms were developed for real-valued signals. In this paper, we are interested in complex-valued data. More precisely, we consider a class of penalty functions for which the associated regularized minimization problem can be solved numerically by a forward-backward algorithm. Functions within this class can be used to promote the sparsity of the solution. An application to parallel Magnetic Resonance Imaging (pMRI) reconstruction where complex-valued images are reconstructed is considered.