Correlation Regularized Image Denoising via Low Rank Matrix Approximation

In this paper, we propose a new residual correlation based regularization method for image denoising and formulate the problem in the framework of Low Rank Matrix Approximation. For this purpose we propose an objective function which leads naturally to the weighted nuclear norm minimization. The objective function consists of a fidelity term and a regularization term which are the Frobenious norms of residual noise and the deviation of residual noise correlations from the true correlations. Convexity of the objective function is established and the closed form global minimum is obtained. A simple closed form formula for the residual noise singular values is derived. The formula involves the right singular vectors of the noisy observation matrix and the noise correlation matrix. Our solution explicitly shows how each singular value of the noisy observation matrix is thresholded to obtain the clean signal singular values. Hard and soft thresholding as well as the weighted thresholding of singular values come out as solutions of the proposed optimization problem. Simulation results in image denoising show that the proposed method performs as good as the weighted nuclear norm minimization method with better performance for most standard test images especially at high noise levels.