Rician compressed sensing for fast and stable signal reconstruction in diffusion MRI

Abstract

The advent of the theory of compressed sensing (CS) has revolutionized multiple areas of applied sciences, a particularly important instance of which is medical imaging. In particular, the theory provides a solution to the problem of long acquisition times, which is intrinsic in diffusion MRI (dMRI). As a specific instance of dMRI, this work focuses on high angular resolution diffusion imaging (HARDI), which is known to excel in delineating multiple diffusion flows through a given voxel within the brain. Specifically, to reduce the acquisition time, CS allows undersampling the HARDI data by employing fewer diffusion-encoding gradients than it is required by the classical sampling theory. Subsequently, the undersampled data is used to recover the original signals by means of non-linear decoding. In earlier reconstruction methods, such decoding has been carried out under a Gaussian model for measurement noises, instead of the Rician model which is known to prevail in MRI. Accordingly, the main contribution of the present work is twofold. First, we introduce a way to substantially improve the stability of the CS-based reconstruction of HARDI signals under the assumption of Gaussian noises. Second, we extend this approach to the case of Rician noise statistics. In addition to providing formal developments of the reconstruction algorithm based on Rician statistics, we also detail a computationally efficient numerical scheme which can be used to implement the above reconstruction. Finally, the methods based on the Gaussian and the Rician noise models are compared using both simulated and in-vivo MRI data under various measurement conditions.