Joint 6D k-q Space Compressed Sensing for Accelerated High Angular Resolution Diffusion MRI

High Angular Resolution Diffusion Imaging (HARDI) avoids the Gaussian diffusion assumption that is inherent in Diffusion Tensor Imaging (DTI), and is capable of characterizing complex white matter micro-structure with greater precision. However, HARDI methods such as Diffusion Spectrum Imaging (DSI) typically require significantly more signal measurements than DTI, resulting in prohibitively long scanning times. One of the goals in HARDI research is therefore to improve estimation of quantities such as the Ensemble Average Propagator (EAP) and the Orientation Distribution Function (ODF) with a limited number of diffusion-weighted measurements. A popular approach to this problem, Compressed Sensing (CS), affords highly accurate signal reconstruction using significantly fewer (sub-Nyquist) data points than required traditionally. Existing approaches to CS diffusion MRI (CS-dMRI) mainly focus on applying CS in the \(\mathbf {q}\)-space of diffusion signal measurements and fail to take into consideration information redundancy in the \(\mathbf {k}\)-space. In this paper, we propose a framework, called 6-Dimensional Compressed Sensing diffusion MRI (6D-CS-dMRI), for reconstruction of the diffusion signal and the EAP from data sub-sampled in both 3D \(\mathbf {k}\)-space and 3D \(\mathbf {q}\)-space. To our knowledge, 6D-CS-dMRI is the first work that applies compressed sensing in the full 6D \(\mathbf {k}\)-\(\mathbf {q}\) space and reconstructs the diffusion signal in the full continuous \(\mathbf {q}\)-space and the EAP in continuous displacement space. Experimental results on synthetic and real data demonstrate that, compared with full DSI sampling in \(\mathbf {k}\)-\(\mathbf {q}\) space, 6D-CS-dMRI yields excellent diffusion signal and EAP reconstruction with low root-mean-square error (RMSE) using 11 times less samples (3-fold reduction in \(\mathbf {k}\)-space and 3.7-fold reduction in \(\mathbf {q}\)-space).