A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration

Recently, the minimization of a sum of two convex functions has received considerable interest in a variational image restoration model. In this paper, we propose a general algorithmic framework for solving a separable convex minimization problem from the point of view of fixed point algorithms based on proximity operators (Moreau 1962 C. R. Acad. Sci., Paris I 255 2897-99). Motivated by proximal forward-backward splitting proposed in Combettes and Wajs (2005 Multiscale Model. Simul. 4 1168-200) and fixed point algorithms based on the proximity operator ((FPO)-O-2) for image denoising (Micchelli et al 2011 Inverse Problems 27 45009-38), we design a primal-dual fixed point algorithm based on the proximity operator ((PDFPO kappa)-O-2 for kappa is an element of [0, 1)) and obtain a scheme with a closed-form solution for each iteration. Using the firmly nonexpansive properties of the proximity operator and with the help of a special norm over a product space, we achieve the convergence of the proposed (PDFPO kappa)-O-2 algorithm. Moreover, under some stronger assumptions, we can prove the global linear convergence of the proposed algorithm. We also give the connection of the proposed algorithm with other existing first-order methods. Finally, we illustrate the efficiency of (PDFPO kappa)-O-2 through some numerical examples on image supper-resolution, computerized tomographic reconstruction and parallel magnetic resonance imaging. Generally speaking, our method (PDFPO)-O-2 (kappa = 0) is comparable with other state-of-the-art methods in numerical performance, while it has some advantages on parameter selection in real applications.