Joint k-q Space Compressed Sensing for Accelerated Multi-Shell Acquisition and Reconstruction of the diffusion signal and Ensemble Average Propagator

Introduction. High Angular Resolution Diffusion Imaging (HARDI) has been proposed to avoid the limitations of the conventional Diffusion Tensor Imaging (DTI) and to better explore white matter micro-structure non-invasively. However, HARDI methods normally require many more samples than DTI. For example, Diffusion Spectrum Imaging (DSI), estimates the diffusion Ensemble Average Propagator (EAP) from typically 515 diffusion-weighted images, taking nearly one hour. Therefore, one important area of research in HARDI is to improve estimation of quantities such the diffusion propagator, the orientation distribution function, and diffusion-weighted signals from ra educed number of diffusion-weighted measurements. A solution to this problem is compressed sensing (CS). Existing approaches to compressed sensing diffusion MRI (CS-DMRI) mainly focus on applying CS in the q-space of diffusion signal measurements. For example, it has been shown in [2] that it is possible to accurately estimate the diffusion signal, with low root-mean-square-error (RMSE), from about 100 diffusion-weighted measurements in multiple shells (multiple b-values). However, this line of work fails to harness information redundancy in the k-space. By sub-sampling in both k-space and q-space, the scanning time can be significantly reduced, while keeping good estimation quality. To our knowledge, [3,4] are the only works using joint k-q space CS schemes to estimate the fiber ODF from single-shell data. The utility of this joint CS approach in multi-shell acquisition has not been sufficiently studied. In the current work, we propose a general framework, called k-q compressed sensing diffusion MRI (kq-CS-DMRI), to estimate the diffusion signal and the diffusion propagator from data sub-sampled in both k-space and q-space. Theory: Naive CS-dMRI in k-q space. In the last several years, CS has been widely used to recover MR images from data sub-sampled in the k-space . Typically, each diffusion-weighted image is first independently reconstructed using k-space CS . All estimated DW images are then used in for CS estimation of signal in the q-space [4]. This approach however ignores the correlation between DW images. Our approach, described below, unifies CS in both k-space and q-space. Theory: Unified kqs-dMRI. Denote Qv as the partial Fourier sample vector of v-th volume Ev. Denote Ei as the vector of the diffusion weighted signals at voxel i. Assume diffusion signal vector Ei can be sparsely represented by a continuous basis set B, and ci is its representation coefficient vector. Then the estimation of the DW images via {ci} can be reconstructed by their partial Fourier samples by minimizing: { } 2