In this paper, we develop a new low-rank matrix recovery algorithm for image denoising. We incorporate the total variation (TV) norm and the pixel range constraint into the existing reweighted low-rank matrix analysis to achieve structural smoothness and to significantly improve quality in the recovered image. Our proposed mathematical formulation of the low-rank matrix recovery problem combines the nuclear norm, TV norm, and norm, thereby allowing us to exploit the low-rank property of natural images, enhance the structural smoothness, and detect and remove large sparse noise. Using the iterative alternating direction and fast gradient projection methods, we develop an algorithm to solve the proposed challenging non-convex optimization problem. We conduct extensive performance evaluations on single-image denoising, hyper-spectral image denoising, and video background modeling from corrupted images. Our experimental results demonstrate that the proposed method outperforms the state-of-the-art low-rank matrix recovery methods, particularly for large random noise. For example, when the density of random sparse noise is 30%, for single-image denoising, our proposed method is able to improve the quality of the restored image by up to 4.21 dB over existing methods.

Low-rank approximation has been successfully used for dimensionality reduction, image noise removal, and image restoration. In existing work, input images are often reshaped to a matrix of vectors before low-rank decomposition. It has been observed that this procedure will destroy the inherent two-dimensional correlation within images. To address this issue, the generalized low-rank approximation of matrices (GLRAM) method has been recently developed, which is able to perform low-rank decomposition of multiple matrices directly without the need for vector reshaping. In this paper, we propose a new robust generalized low-rank matrices decomposition method, which further extends the existing GLRAM method by incorporating rank minimization into the decomposition process. Specifically, our method aims to minimize the sum of nuclear norms and l1-norms. We develop a new optimization method, called alternating direction matrices tri-factorization method, to solve the minimization problem. We mathematically prove the convergence of the proposed algorithm. Our extensive experimental results demonstrate that our method significantly outperforms existing GLRAM methods.