Noise correction for HARDI and HYDI data obtained with multi-channel coils and Sum of Squares reconstruction: An anisotropic extension of the LMMSE

Parallel magnetic resonance imaging (MRI) yields noisy magnitude data, described in most cases as following a noncentral χ distribution when the signals received by the coils are combined as the sum of their squares. One well-known case of this noncentral χ noise model is the Rician model, but it is only valid in the case of single-channel acquisition. Although the use of parallel MRI is increasingly common, most of the correction methods still perform Rician noise removal, yielding an erroneous result due to an incorrect noise model. Moreover, the existence of noise correlations in phased array systems renders noise nonstationary and further modifies the noise description in parallel MRI. However, the noncentral χ model has been demonstrated to work as a good approximation as long as effective voxelwise parameters are used. A good correction step, adapted to the right noise model, is of paramount importance, especially when working with diffusion-weighted MR data, whose signal-to-noise ratio is low. In this paper, we present a noise removal technique designed to be fast enough for integration into a real-time reconstruction system, thus offering the convenience of obtaining corrected data almost instantaneously during the MRI scan. Our method employs the noncentral χ noise model and uses a simplified method to account for noise correlations; this leads to an efficient and rapid correction. The method consists of an anisotropic extension of the Linear Minimum Mean Square Error estimator (LMMSE) that is a far better edge-preserving method than the traditional LMMSE and addresses noncentral χ distributions along with empirically computed global effective parameters. The results on simulated and real data demonstrate that this anisotropic extended LMMSE outperforms the original LMMSE on images corrupted by noncentral χ noise. Moreover, in comparison with the existing LMMSE technique incorporating the estimation of voxelwise effective parameters, our method yields improved results.