  We present Roto-Translation Equivariant Spherical Deconvolution (RT-ESD), an
$E(3)\times SO(3)$ equivariant framework for sparse deconvolution of volumes
where each voxel contains a spherical signal. Such 6D data naturally arises in
diffusion MRI (dMRI), a medical imaging modality widely used to measure
microstructure and structural connectivity. As each dMRI voxel is typically a
mixture of various overlapping structures, there is a need for blind
deconvolution to recover crossing anatomical structures such as white matter
tracts. Existing dMRI work takes either an iterative or deep learning approach
to sparse spherical deconvolution, yet it typically does not account for
relationships between neighboring measurements. This work constructs
equivariant deep learning layers which respect to symmetries of spatial
rotations, reflections, and translations, alongside the symmetries of voxelwise
spherical rotations. As a result, RT-ESD improves on previous work across
several tasks including fiber recovery on the DiSCo dataset,
deconvolution-derived partial volume estimation on real-world \textit{in vivo}
human brain dMRI, and improved downstream reconstruction of fiber tractograms
on the Tractometer dataset. Our implementation is available at
https://github.com/AxelElaldi/e3so3_conv
