Denoising 3-D magnitude magnetic resonance images based on weighted nuclear norm minimization

A new denoising algorithm based on low-rank matrix approximation (LRMA) with regularization of weighted nuclear norm minimization (WNNM) is proposed to remove Rician noise of magnetic resonance (MR) images. This technique simply groups similar non-local cubic blocks from noisy 3D MR data into a patch matrix with each block lexicographically vectorizing to be as a column, calculates the singular value decomposition (SVD) on this matrix, then the closed-form solution of LRMA is achieved by hard-thresholding different singular values with a different threshold. The denoised blocks are obtained from this estimate of the low-rank matrix, and the final estimate of the whole noise-free MR data is built up by aggregating all the denoised exemplar blocks that are overlapped each other. To further improve the denoising performance of the WNNM algorithm, we first realize the above denoising procedure in a two-iteration regularization framework, and then a simple non local means (NLM) filter based on single-pixel patch is utilized to reduce the intensity jumping at the homogeneous area. The proposed denoising algorithm was compared with related state-of-the-art methods and produced very competitive results over synthetic and real 3D MR data.