Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encountered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Further, we propose a preconditioned method, of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography (CT) image reconstruction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality.
Recently, the l p -norm regularization minimization problem (P λ p ) has attracted great attention in compressed sensing. However, the l p -norm x p p in problem (P λ p ) is nonconvex and non-Lipschitz for all p ∈ (0, 1), and there are not many optimization theories and methods are proposed to solve this problem. In fact, it is NP-hard for all p ∈ (0, 1) and λ > 0. In this paper, we study two modified l p regularization minimization problems to approximate the NPhard problem (P λ p ). Inspired by the good performance of Half algorithm and 2/3 algorithm in some sparse signal recovery problems, two iterative thresholding algorithms are proposed to solve the problems (P λ p,1/2,ǫ ) and (P λ p,2/3,ǫ ) respectively. Numerical results show that our algorithms perform effectively in finding the sparse signal in some sparse signal recovery problems for some proper p ∈ (0, 1).
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