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

Abstract

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 (FP2O) 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 (PDFP2Oκ for κ ∈ [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 PDFP2Oκ 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 PDFP2Oκ through some numerical examples on image supper-resolution, computerized tomographic reconstruction and parallel magnetic resonance imaging. Generally speaking, our method PDFP2O (κ = 0) is comparable with other state-of-the-art methods in numerical performance, while it has some advantages on parameter selection in real applications.