  Diffusion MRI (dMRI) provides the ability to reconstruct neuronal fibers in
the brain, $\textit{in vivo}$, by measuring water diffusion along angular
gradient directions in q-space. High angular resolution diffusion imaging
(HARDI) can produce better estimates of fiber orientation than the popularly
used diffusion tensor imaging, but the high number of samples needed to
estimate diffusivity requires longer patient scan times. To accelerate dMRI,
compressed sensing (CS) has been utilized by exploiting a sparse dictionary
representation of the data, discovered through sparse coding. The sparser the
representation, the fewer samples are needed to reconstruct a high resolution
signal with limited information loss, and so an important area of research has
focused on finding the sparsest possible representation of dMRI. Current
reconstruction methods however, rely on an angular representation $\textit{per
voxel}$ with added spatial regularization, and so, for non-zero signals, one is
required to have at least one non-zero coefficient per voxel. This means that
the global level of sparsity must be greater than the number of voxels. In
contrast, we propose a joint spatial-angular representation of dMRI that will
allow us to achieve levels of global sparsity that are below the number of
voxels. A major challenge, however, is the computational complexity of solving
a global sparse coding problem over large-scale dMRI. In this work, we present
novel adaptations of popular sparse coding algorithms that become better suited
for solving large-scale problems by exploiting spatial-angular separability.
Our experiments show that our method achieves significantly sparser
representations of HARDI than is possible by the state of the art.
