  Convolutional networks are successful, but they have recently been
outperformed by new neural networks that are equivariant under rotations and
translations. These new networks work better because they do not struggle with
learning each possible orientation of each image feature separately. So far,
they have been proposed for 2D and 3D data. Here we generalize them to 6D
diffusion MRI data, ensuring joint equivariance under 3D roto-translations in
image space and the matching 3D rotations in $q$-space, as dictated by the
image formation. Such equivariant deep learning is appropriate for diffusion
MRI, because microstructural and macrostructural features such as neural fibers
can appear at many different orientations, and because even
non-rotation-equivariant deep learning has so far been the best method for many
diffusion MRI tasks. We validate our equivariant method on multiple-sclerosis
lesion segmentation. Our proposed neural networks yield better results and
require fewer scans for training compared to non-rotation-equivariant deep
learning. They also inherit all the advantages of deep learning over classical
diffusion MRI methods. Our implementation is available at
https://github.com/philip-mueller/equivariant-deep-dmri and can be used off the
shelf without understanding the mathematical background.
