Compressive sensing via nonlocal low-rank tensor regularization

The aim of Compressing sensing (CS) is to acquire an original signal, when it is sampled at a lower rate than Nyquist rate previously. In the framework of CS, the original signal is often assumed to be sparse and correlated in some domain. Recently, nonlocal low-rank regularization (NLR) approach has obtained the-state-of-the-art results in CS recovery which exploits both structured sparsity of similar patches and nonconvexity of rank minimization. However, it still suffers from two problems. First, the NLR approach can not preserve the original geometrical structure of image patches and ignores the relationship between pixels because it deals with the vector form of image patches and the matrix form of patch groups for simplicity. Second, logdet () can not well approximate the rank which is used as a surrogate function for the rank in NLR, because it is a fixed function and the optimization results by this function essentially deviate from the real solution of original minimization problem. In this paper, we propose a nonlocal low-rank tensor regularization (NLRT) approach toward exploiting the original structural information of image patches and structured sparsity of similar patches. We also exploit the use of Schatten p-norm as a nonconvex relaxation for the tensor rank. To further improve the computational efficiency of the proposed algorithm, we have developed a fast implementation utilizing the alternative direction multiplier method technique. Experimental results have demonstrated that the proposed NLRT approach significantly outperforms existing state-of-the-art CS algorithms for image recovery.