Image Restoration using Group Sparse Representation via Weighted Nuclear Norm Minimization

As the matrix formed by nonlocal similar patches in a natural image is of a low rank, the nuclear norm minimization (NNM) has been widely studied for image processing. Since the singular values have clear meanings and should be treated differently, NNM regularizes each of them equally, which often restricts its capability and flexibility. Recent advances have suggested that the weighted nuclear norm minimization (WNNM) has shown great potential in different image restoration studies, where singular values are assigned different value. However, it still lacks a mathematical derivation why the weighted nuclear norm is more appropriate than the nuclear norm. In this paper, we proposed a new scheme for image restoration using group sparse representation via weighted nuclear norm minimization (GSR-WNNM). We show mathematically the advantage of WNNM, from a group sparse representation perspective, where GSR offers a powerful mechanism of combining local sparsity and nonlocal self-similarity of images simultaneously in a unified framework. Then, an effective dictionary for each group is learned from the reconstructed image itself rather a large number of natural image dataset, ensuring a low computational complexity. Moreover, to further improve the computational efficiency of the proposed method, we have developed an implementation of fast convergence via the alternating direction method of multipliers (ADMM). Experimental results have shown that the proposed GSR-WNNM method significantly outperforms the state-of-the-art methods both quantitatively and qualitatively.