Group-Invariant Moments under Tomographic Projections

Abstract.

Let [Formula: see text] be an unknown object, and suppose that the observations are tomographic projections of randomly rotated copies of [Formula: see text] of the form [Formula: see text], where [Formula: see text] is Haar-uniform in [Formula: see text] and [Formula: see text] is the projection onto an [Formula: see text]-dimensional subspace so that [Formula: see text]. We prove that whenever [Formula: see text], the [Formula: see text]th-order moment of the projected data determines the full [Formula: see text]th-order Haar-orbit moment of [Formula: see text], independently of the ambient dimension [Formula: see text]. We further provide an explicit algorithmic procedure for recovering the latter from the former. As a consequence, any identifiability result for the unprojected model based on the [Formula: see text]th-order group-invariant moment extends directly to the tomographic setting at the same moment order. In particular, for [Formula: see text], [Formula: see text], and [Formula: see text], our result recovers a classical result in the cryogenic electron microscopy literature: The covariance of the two-dimensional projection images determines the second-order rotationally invariant moment of the underlying three-dimensional object.