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We present Roto-Translation Equivariant Spherical Deconvolution (RT-ESD), an$E(3)\times SO(3)$ equivariant framework for sparse deconvolution of volumeswhere each voxel contains a spherical signal. Such 6D data naturally arises indiffusion MRI (dMRI), a medical imaging modality widely used to measuremicrostructure and structural connectivity. As each dMRI voxel is typically amixture of various overlapping structures, there is a need for blinddeconvolution to recover crossing anatomical structures such as white mattertracts. Existing dMRI work takes either an iterative or deep learning approachto sparse spherical deconvolution, yet it typically does not account forrelationships between neighboring measurements. This work constructsequivariant deep learning layers which respect to symmetries of spatialrotations, reflections, and translations, alongside the symmetries of voxelwisespherical rotations. As a result, RT-ESD improves on previous work acrossseveral 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 tractogramson the Tractometer dataset. Our implementation is available athttps://github.com/AxelElaldi/e3so3_conv

$E(3) \times SO(3)$-Equivariant Networks for Spherical Deconvolution in Diffusion MRI