A new nonlocal low-rank regularization method with applications to magnetic resonance image denoising

Magnetic resonance (MR) images are frequently corrupted by Rician noise during image acquisition and transmission. And it is very challenging to restore MR data because Rician noise is signal-dependent. By exploring the nonlocal self-similarity of natural images and further using the low-rank prior of the matrices formed by nonlocal similar patches for 2D data or cubes for 3D data, we propose in this paper a new nonlocal low-rank regularization (NLRR) method including an optimization model and an efficient iterative algorithm to remove Rician noise. The proposed mathematical model consists of a data fidelity term derived from a maximum a posteriori estimation and a NLRR term using the log-det function. The resulting model in terms of approximated patch/cube matrices is non-convex and non-smooth. To solve this model, we propose an alternating reweighted minimization (ARM) algorithm using the Lipschitz-continuity of the gradient of the fidelity term and the concavity of the logarithmic function in the log-det function. The subproblems of the ARM algorithm have closed-form solutions and its limit points are first-order critical points of the problem. The ARM algorithm is further integrated with a two-stage scheme to enhance the denoising performance of the proposed NLRR method. Experimental results tested on 2D and 3D MR data, including simulated and real data, show that the NLRR method outperforms existing state-of-the-art methods for removing Rician noise.