Nonconvex Low-Rank and Total-Variation Regularized Model and Algorithm for Image Deblurring

Non-blind image deblurring aims to reconstruct the original image for noised linear convolution transform with some known kernel. If the noise is Gaussian, one might use the least square minimization with the observed image and the kernel for this task. However, in most cases, the convolution operator makes the problem very ill-posed and hard to solve directly. To this end, regularizations, which are recruited to characterize known statistical priors about the original images, are then developed to help. Among them, two frequently used ones are the low-rank and total-variation regularizations. The earlier related works employed them separately, that is, just using one of them, not both. Until several years ago, people have considered combing these two regularizations together. Existing results show that the hybrid regularized model performs much better than the single one. However, the current composite regularization just uses the convex methodology. The nonconvex implementations are still missing. Considering nonconvex regularizations can beat convex ones in various cases, in this paper, we propose a novel nonconvex hybrid model in which, the L1/2 and Schatten-1/2 norms are used. We use these two nonconvex functions due to that their proximal maps are easy to calculate even without convexity. Both L1/2 and Schatten-1/2 norms enjoy closed-form proximal maps. The proposed nonconvex hybrid regularized model is naturally a nonconvex linearly constrained problem which can be solved by the alternating directions of multipliers. However, the nonconvexity breaks the theoretical guarantees. Thus, we turn to solve the penalty problem rather than the original form. The alternating minimization method applied to the penalty then yields the proposed algorithm, in which, each substep just involves very simple computations. In our algorithm, an important parameter is the penalty parameter. If it is infinity, the penalty is then identical to the original problem. But the large penalty parameter will make the algorithm iterate slowly. Thus, to improve the speed and narrow the penalty problem and the original one, for the penalty parameter, we use a warm-up technique, that is, increasing the penalty parameter in the iterations. The convergence of the algorithm is proved under very mild assumptions, which can be easily satisfied in applications. The numerical experiments are conducted on six natural test images. The performance of the proposed algorithm verifies the convergence theory. Comparisons with other algorithms demonstrate the efficiency of our algorithm. © 2020, Science Press. All right reserved.