Group-invariant moments under tomographic projections

2026-04-09

AI Summary

This study addresses the recovery of high-dimensional group-invariant moments of an object from low-dimensional tomographic projections acquired under random rotations, enabling structural identification. Under a model combining random rotations with linear projection observations, the authors prove that when the moment order does not exceed the projection dimension, the d-th order moments of the projected data uniquely determine the d-th order Haar-orbit invariant moments of the original object, independent of the ambient dimension. Leveraging group representation theory, Haar measure, and tensor moment analysis, the paper establishes an identifiability theory for tomography and provides an explicit reconstruction algorithm. Notably, in the 3D-to-2D projection setting, second-order moments alone reproduce the classical covariance-based identification results from cryo-electron microscopy, successfully extending existing identifiability theory for group-invariant moments—previously limited to non-projective models—to the tomographic projection scenario.

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.

Problem

• group-invariant moments
• tomographic projections
• Haar-orbit moments
• cryo-EM
• moment identifiability

Innovation

• group-invariant moments
• tomographic projections
• Haar-orbit moments
• moment recovery
• rotational invariance