Convex optimization based low-rank matrix decomposition for image restoration

This paper addresses the problem of image denoising in the presence of significant corruption. Our method seeks an optimal set of image domain transformations such that the matrix of transformed images can be decomposed as the sum of a sparse matrix of errors and a low-rank matrix of recovered denoised images. We reduce this optimization problem to a sequence of convex programs minimizing the sum of the ℓ1-normℓ1-norm and the nuclear norm of the two component matrices, which can be solved efficiently using scalable convex optimization techniques. We verify the efficacy of the proposed image denoising algorithm through extensive experiments on both numerical simulations and different types of images, demonstrating its highly competent objective performance compared with several state-of-the-art methods for matrix decomposition and image denoising. Our subjective quality results compare favorably with those obtained by existing techniques, especially at high noise levels and with a large amount of missing data.