The presence of noise results in quality deterioration of magnetic resonance (MR) images and thus limits the visual inspection and influence the quantitative measurements from the data. In this work, an efficient two stage linear minimum mean square error (LMMSE) method is proposed for the enhancement of magnitude MR images in which data in the presence of noise follows a Rician distribution. The conventional Rician LMMSE estimator determines a closed-form analytical solution to the aforementioned inverse problem. Even-though computationally efficient, this approach fails to take advantage of data redundancy in the 3D MR data and hence leads to a suboptimal filtering performance. Motivated by this observation, we put forward the concept of nonlocal implementation with LMMSE estimation method. To select appropriate samples for the nonlocal version of the LMMSE estimation, the similarity weights are computed using Euclidean distance between either the gray level values in the spatial domain or the coefficients in the transformed domain. Assuming that the signal dependent component of the noise is optimally suppressed by this filtering and the rest is a white and uncorrelated noise with the image, we adopt a second stage LMMSE filtering in the principal component analysis (PCA) domain to further enhance the image and the noise variance is adaptively adjusted. Experiments on both simulated and real data show that the proposed filters have excellent filtering performance over other state-of-the-art methods. 
Nonlocal linear minimum mean square error methods for denoising MRI