  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.
