Group-invariant moments under tomographic projections
This provides a theoretical foundation for tomographic reconstruction in fields like cryo-EM, enabling identifiability results to extend directly from unprojected to projected models, though it is incremental as it builds on existing moment-based methods.
The paper tackles the problem of recovering group-invariant moments of an unknown object from tomographic projections under random rotations, proving that the d-th order moment of projected data determines the full d-th order Haar-orbit moment of the object for d ≤ m, with an explicit recovery algorithm. It generalizes a classical result in cryo-EM, showing that for n=3, m=2, and d=2, the covariance of 2D projections determines the second order rotationally invariant moment of the 3D object.
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.