Group-invariant moments under tomographic projections
2026-04-09 • Information Theory
Information Theory
AI summaryⓘ
The authors study a problem where you observe 2D projections of a rotated unknown object in higher dimensions. They show that by looking at certain statistical moments (averages of powers) of these projected observations, you can recover similar moments of the original object, regardless of the overall dimension. They also provide a step-by-step method to do this recovery. This means any prior ways to identify objects using these moments without projection also work when using projections, which is useful in fields like cryo-electron microscopy.
tomographic projectionHaar uniform distributionSO(n) groupmoment (statistics)group-invariant momentrotational invariancecryo-electron microscopy (cryo-EM)covariancedimensionality reductionalgorithmic recovery
Authors
Amnon Balanov, Tamir Bendory, Dan Edidin
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.