  Rigid motion tracking is paramount in many medical imaging applications where
movements need to be detected, corrected, or accounted for. Modern strategies
rely on convolutional neural networks (CNN) and pose this problem as rigid
registration. Yet, CNNs do not exploit natural symmetries in this task, as they
are equivariant to translations (their outputs shift with their inputs) but not
to rotations. Here we propose EquiTrack, the first method that uses recent
steerable SE(3)-equivariant CNNs (E-CNN) for motion tracking. While steerable
E-CNNs can extract corresponding features across different poses, testing them
on noisy medical images reveals that they do not have enough learning capacity
to learn noise invariance. Thus, we introduce a hybrid architecture that pairs
a denoiser with an E-CNN to decouple the processing of anatomically irrelevant
intensity features from the extraction of equivariant spatial features. Rigid
transforms are then estimated in closed-form. EquiTrack outperforms
state-of-the-art learning and optimisation methods for motion tracking in adult
brain MRI and fetal MRI time series. Our code is available at
https://github.com/BBillot/EquiTrack.
