  Diffusion-weighted MRI measures the direction and scale of the local
diffusion process in every voxel through its spectrum in q-space, typically
acquired in one or more shells. Recent developments in microstructure imaging
and multi-tissue decomposition have sparked renewed attention in the radial
b-value dependence of the signal. Applications in motion correction and outlier
rejection therefore require a compact linear signal representation that extends
over the radial as well as angular domain. Here, we introduce SHARD, a
data-driven representation of the q-space signal based on spherical harmonics
and a radial decomposition into orthonormal components. This representation
provides a complete, orthogonal signal basis, tailored to the spherical
geometry of q-space and calibrated to the data at hand. We demonstrate that the
rank-reduced decomposition outperforms model-based alternatives in human brain
data, whilst faithfully capturing the micro- and meso-structural information in
the signal. Furthermore, we validate the potential of joint radial-spherical as
compared to single-shell representations. As such, SHARD is optimally suited
for applications that require low-rank signal predictions, such as motion
correction and outlier rejection. Finally, we illustrate its application for
the latter using outlier robust regression.
