Weighted Hyper-Laplacian Prior with Overlapping Group Sparsity for Image Restoration under Cauchy Noise

In this paper, we deal with the Cauchy image restoration problem under the maximum a posteriori framework. We propose a novel image prior, weighted hyper-Laplacian prior with overlapping group sparsity on the image gradient. This prior allows us to simultaneously promote the structural and pixel-level sparseness of the natural image gradient. The performance can be further improved by introducing the in-group-weights to balance the different scales of the components within each group. To tackle the corresponding optimization problem, we present a novel quadratic majorizer for majorization-minimization. We adopt the non-convex alternating direction method of multipliers as the main algorithm framework. The proposed regularizer can be reduced to the related variational regularizers including the total variation, the hyper-Laplacian, and the total variation with overlapping group sparsity. The comparative experiments with those existing gradient-based regularizers demonstrate the effectiveness of the proposed method in terms of PSNR and SSIM values.