From Rank Estimation to Rank Approximation: Rank Residual Constraint for Image Denoising

In this paper, we propose a novel approach for the rank minimization problem, termed rank residual constraint (RRC). Different from existing low-rank based approaches, such as the well-known weighted nuclear norm minimization (WNNM) and nuclear norm minimization (NNM), which aim to estimate the underlying low-rank matrix directly from the corrupted observation, we progressively approximate or approach the underlying low-rank matrix via minimizing the rank residual. By integrating the image nonlocal self-similarity (NSS) prior with the proposed RRC model, we develop an iterative algorithm for image denoising. To this end, we first present a recursive based nonlocal means method to obtain a good reference of the original image patch groups, and then the rank residual of the image patch groups between this reference and the noisy image is minimized to achieve a better estimate of the desired image. In this manner, both the reference and the estimated image in each iteration are improved gradually and jointly. Based on the group-based sparse representation model, we further provide a theoretical analysis on the feasibility of the proposed RRC model. Experimental results demonstrate that the proposed RRC model outperforms many state-of-the-art denoising methods in both the objective and perceptual qualities.