{"paper":{"title":"Group-invariant moments under tomographic projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The d-th moments of random tomographic projections determine the full d-th order rotation-invariant moments of the original object whenever d is at most the projection dimension.","cross_cats":["cs.IT","math.IT"],"primary_cat":"eess.SP","authors_text":"Amnon Balanov, Dan Edidin, Tamir Bendory","submitted_at":"2026-04-09T15:03:34Z","abstract_excerpt":"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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that, whenever d≤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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The observations are of the form Y = P(R·f) where R is Haar-uniform in SO(n) and P projects onto an m-dimensional subspace; the determination holds only for d ≤ m and may fail if rotations are not uniform or if d exceeds m.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"d-th order moments of m-dimensional projections determine the d-th order Haar-invariant moments of the n-dimensional object for d ≤ m, independently of n.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The d-th moments of random tomographic projections determine the full d-th order rotation-invariant moments of the original object whenever d is at most the projection dimension.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"49cdd2289719825d42ae0207787c1a6f01f787f2209a75bc8d8a05ce18826a4b"},"source":{"id":"2604.08330","kind":"arxiv","version":2},"verdict":{"id":"efd1e788-f088-4b38-9bdd-5b0502dd7f02","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T17:27:10.315368Z","strongest_claim":"We prove that, whenever d≤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.","one_line_summary":"d-th order moments of m-dimensional projections determine the d-th order Haar-invariant moments of the n-dimensional object for d ≤ m, independently of n.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The observations are of the form Y = P(R·f) where R is Haar-uniform in SO(n) and P projects onto an m-dimensional subspace; the determination holds only for d ≤ m and may fail if rotations are not uniform or if d exceeds m.","pith_extraction_headline":"The d-th moments of random tomographic projections determine the full d-th order rotation-invariant moments of the original object whenever d is at most the projection dimension."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.08330/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}