A Majorize-Minimize Memory Gradient Method for Complex-Valued Inverse Problems

Complex-valued data are encountered in many application areas of signal and image processing. In the context of optimization of functions of real variables, subspace algorithms have recently attracted much in-terest, owing to their efficiency for solving large-size problems while simultaneously offering theoretical convergence guarantees. The goal of this paper is to show how some of these methods can be success-fully extended to the complex case. More precisely, we investigate the properties of the proposed complex-valued Majorize-Minimize Mem-ory Gradient (3MG) algorithm. Important practical applications of these results arise in inverse problems. Here, we focus on image recon-struction in Parallel Magnetic Resonance Imaging (PMRI). The linear operator involved in the observation model then includes a subsam-pling operator over the k-space (spatial Fourier domain) the choice of which is analyzed through our numerical results. In addition, sensi-tivity matrices associated with the multiple coil channels come into play. Comparisons with existing optimization methods confirm the good performance of the proposed algorithm