Group-invariant moments under tomographic projections
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915927452286976 |
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| author | Balanov, Amnon Bendory, Tamir Edidin, Dan |
| author_facet | Balanov, Amnon Bendory, Tamir Edidin, Dan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_08330 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Group-invariant moments under tomographic projections Balanov, Amnon Bendory, Tamir Edidin, Dan Signal Processing Information Theory 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. |
| title | Group-invariant moments under tomographic projections |
| topic | Signal Processing Information Theory |
| url | https://arxiv.org/abs/2604.08330 |