LLp norm regularization based group sparse representation for image compressed sensing recovery

One important challenge in image compressed sensing (CS) recovery methods is to develop a sparsity inducing model which can reflect the image priors appropriately and hence yields high quality recovery results. Recent advances have suggested that group sparse representation based models, which exploit nonlocal self-similarity prior, lead to superior results in image CS recovery. In this paper, we propose a CS recovery method via group sparse representation with nonconvex LLp norm regularization (GSR-LLp). In the proposed method, nonconvex LLp norm with (0 < p <= 1) is introduced as a new sparsity metric to better promote the sparsity of the group coefficients, rather than the l(0) norm. Furthermore, the principle component analysis (PCA) is utilized to learn an adaptive orthogonal dictionary for each group. To solve the GSR-driven LLp minimization problem, an efficient algorithm based on split Bregman framework and Majorization-Minimization (MM) algorithm is developed. Moreover, the proposed model is combined with a robust M-estimate to cope with the case where measurements are corrupted by impulsive noise. In this case, we substitute the l(2) norm data fidelity with Welch m-estimate which has shown the advantage of robustness against heavy-tailed impulsive noise. We develop an efficient scheme based on split Bregman framework and half-quadratic (HQ) theory to solve the resulting optimization problem (called as RGSR-LLp). Extensive experimental results show effectiveness of the proposed methods compared with the state-of-the-arts methods in CS image recovery.