Compactification of the Rigid Motions Group in Image Processing

Image processing problems in general, and in particular in the field of single-particle cryo-electron microscopy, often require considering images up to their rotations and translations. Such problems were tackled successfully when considering images up to rotations only, using quantities which are invariant to the action of rotations on images. Extending these methods to cases where translations are involved is more complicated. Here we present a computationally feasible and theoretically sound approximate invariant to the action of rotations and translations on images. It allows one to approximately reduce image processing problems to similar problems over the sphere, a compact domain acted on by the group of 3D rotations, a compact group. We show that this invariant is induced by a family of mappings deforming, and thereby compactifying, the group structure of rotations and translations of the plane, i.e., the group of rigid motions, into the group of 3D rotations. Furthermore, we demonstrate its viability in two image processing tasks: multi-reference alignment and classification. To our knowledge, this is the first instance of a quantity that is either exactly or approximately invariant to rotations and translations of images that both rests on a sound theoretical foundation and also applicable in practice.