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.

翻译：设$f:\mathbb{R}^n\to\mathbb{R}$为一个未知物体，观测数据为随机旋转后的$f$的层析投影，形式为$Y = P(R\cdot f)$，其中$R$在$\mathrm{SO}(n)$上服从Haar均匀分布，$P$为到$m$维子空间的投影算子，从而$Y:\mathbb{R}^m\to\mathbb{R}$。我们证明：只要$d\le m$，投影数据的$d$阶矩完全决定了$f$的完整$d$阶Haar轨道矩，且该结果与空间维度$n$无关。我们进一步给出从前者恢复后者的显式算法流程。作为推论，任何基于$d$阶群等变矩的未投影模型的可辨识性结果，可直接以相同矩阶数推广至层析成像场景。特别地，当$n=3$、$m=2$、$d=2$时，我们的结果恢复了低温电子显微镜文献中的经典结论：二维投影图像的协方差矩阵决定了三维物体的二阶旋转不变矩。