Group-invariant moments under tomographic projections

TL;DR

This paper demonstrates that moments of projected data under random rotations uniquely determine the invariant moments of the original object, enabling reconstruction in tomographic imaging with explicit algorithms.

Contribution

It establishes a theoretical link between projected moments and full object moments under rotations, with an explicit recovery procedure applicable to tomographic data.

Findings

Projected moments determine full Haar-orbit moments of the object.

Explicit algorithms are provided for recovering invariant moments from projections.

Results extend identifiability from unprojected to tomographic models at the same moment order.

Abstract

Let $f:\mathbb{R}^n\to\mathbb{R}$ be an unknown object, and suppose the observations are tomographic projections of randomly rotated copies of $f$ of the form $Y = P(R\cdot f)$, where $R$ is Haar-uniform in $\mathrm{SO}(n)$ and $P$ is the projection onto an $m$-dimensional subspace, so that $Y:\mathbb{R}^m\to\mathbb{R}$. We prove that, whenever $d\le m$, the $d$-th order moment of the projected data determines the full $d$-th order Haar-orbit moment of $f$, independently of the ambient dimension $n$. 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 $d$-th order group-invariant moment extends directly to the tomographic setting at the same moment order. In particular, for $n=3$, $m=2$, and $d=2$, our result recovers a classical result in the cryo-EM literature: the covariance of the 2D projection images determines the second order rotationally invariant moment of the underlying 3D object.