{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:W6A5PCGVLJMNEOQP6THDTATZ3O","short_pith_number":"pith:W6A5PCGV","canonical_record":{"source":{"id":"2310.01233","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-10-02T14:23:12Z","cross_cats_sorted":[],"title_canon_sha256":"e377bc12b369d492e654231d010a0378191cdcf448b8e747982f4dfd8dfff163","abstract_canon_sha256":"21f35d14e4e5104b2740314fa0050a18a21e03cb60ebe57e21685427d731f6b8"},"schema_version":"1.0"},"canonical_sha256":"b781d788d55a58d23a0ff4ce398279db857c471a08e288c312281c8235ef0fe7","source":{"kind":"arxiv","id":"2310.01233","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2310.01233","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"arxiv_version","alias_value":"2310.01233v2","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2310.01233","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"pith_short_12","alias_value":"W6A5PCGVLJMN","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"pith_short_16","alias_value":"W6A5PCGVLJMNEOQP","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"pith_short_8","alias_value":"W6A5PCGV","created_at":"2026-07-05T08:39:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:W6A5PCGVLJMNEOQP6THDTATZ3O","target":"record","payload":{"canonical_record":{"source":{"id":"2310.01233","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-10-02T14:23:12Z","cross_cats_sorted":[],"title_canon_sha256":"e377bc12b369d492e654231d010a0378191cdcf448b8e747982f4dfd8dfff163","abstract_canon_sha256":"21f35d14e4e5104b2740314fa0050a18a21e03cb60ebe57e21685427d731f6b8"},"schema_version":"1.0"},"canonical_sha256":"b781d788d55a58d23a0ff4ce398279db857c471a08e288c312281c8235ef0fe7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:39:28.386224Z","signature_b64":"Ws4bxbmWd9RlAWn9Fx0iygYGxnnVJZ+U9Fpn0/qYswuexUScUufbjDA1Cu+0fByD8/OzOhPgC3KI/IdQIV2VBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b781d788d55a58d23a0ff4ce398279db857c471a08e288c312281c8235ef0fe7","last_reissued_at":"2026-07-05T08:39:28.385779Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:39:28.385779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2310.01233","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:39:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5SpWVX6Xf6WGng5GhOzAPpLboqyDY4RrTRZqXVpXTLHYXRGkRHNWCu1DcA0kIB6qd6+FmZtpbFNedzV+NEJ2BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T12:12:31.966111Z"},"content_sha256":"779089bc9943e6cdf0abe14d65b7df8c9321d588d6885170e717d0454fac6a37","schema_version":"1.0","event_id":"sha256:779089bc9943e6cdf0abe14d65b7df8c9321d588d6885170e717d0454fac6a37"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:W6A5PCGVLJMNEOQP6THDTATZ3O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Distributional Extension and Invertibility of the $k$-Plane Transform and Its Dual","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Michael Unser, Rahul Parhi","submitted_at":"2023-10-02T14:23:12Z","abstract_excerpt":"We investigate the distributional extension of the $k$-plane transform in $\\mathbb{R}^d$ and of related operators. We parameterize the $k$-plane domain as the Cartesian product of the Stiefel manifold of orthonormal $k$-frames in $\\mathbb{R}^d$ with $\\mathbb{R}^{d-k}$. This parameterization imposes an isotropy condition on the range of the $k$-plane transform which is analogous to the even condition on the range of the Radon transform. We use our distributional formalism to investigate the invertibility of the dual $k$-plane transform (the \"backprojection\" operator). We provide a systematic co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.01233","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2310.01233/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:39:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"prOrv9hM+ijWRtbL44xibjZD9R6Nf78UmORNhXChjxbDukVrZs1fkkrvmVYVsGdoZHhfC448VAEVCoT7plwdBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T12:12:31.967096Z"},"content_sha256":"f0e880124cc2650119675594dda7ba423bcf95f8cbf4656846d2f267b92f4299","schema_version":"1.0","event_id":"sha256:f0e880124cc2650119675594dda7ba423bcf95f8cbf4656846d2f267b92f4299"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W6A5PCGVLJMNEOQP6THDTATZ3O/bundle.json","state_url":"https://pith.science/pith/W6A5PCGVLJMNEOQP6THDTATZ3O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W6A5PCGVLJMNEOQP6THDTATZ3O/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-08T12:12:31Z","links":{"resolver":"https://pith.science/pith/W6A5PCGVLJMNEOQP6THDTATZ3O","bundle":"https://pith.science/pith/W6A5PCGVLJMNEOQP6THDTATZ3O/bundle.json","state":"https://pith.science/pith/W6A5PCGVLJMNEOQP6THDTATZ3O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W6A5PCGVLJMNEOQP6THDTATZ3O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:W6A5PCGVLJMNEOQP6THDTATZ3O","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"21f35d14e4e5104b2740314fa0050a18a21e03cb60ebe57e21685427d731f6b8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-10-02T14:23:12Z","title_canon_sha256":"e377bc12b369d492e654231d010a0378191cdcf448b8e747982f4dfd8dfff163"},"schema_version":"1.0","source":{"id":"2310.01233","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2310.01233","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"arxiv_version","alias_value":"2310.01233v2","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2310.01233","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"pith_short_12","alias_value":"W6A5PCGVLJMN","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"pith_short_16","alias_value":"W6A5PCGVLJMNEOQP","created_at":"2026-07-05T08:39:28Z"},{"alias_kind":"pith_short_8","alias_value":"W6A5PCGV","created_at":"2026-07-05T08:39:28Z"}],"graph_snapshots":[{"event_id":"sha256:f0e880124cc2650119675594dda7ba423bcf95f8cbf4656846d2f267b92f4299","target":"graph","created_at":"2026-07-05T08:39:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2310.01233/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate the distributional extension of the $k$-plane transform in $\\mathbb{R}^d$ and of related operators. We parameterize the $k$-plane domain as the Cartesian product of the Stiefel manifold of orthonormal $k$-frames in $\\mathbb{R}^d$ with $\\mathbb{R}^{d-k}$. This parameterization imposes an isotropy condition on the range of the $k$-plane transform which is analogous to the even condition on the range of the Radon transform. We use our distributional formalism to investigate the invertibility of the dual $k$-plane transform (the \"backprojection\" operator). We provide a systematic co","authors_text":"Michael Unser, Rahul Parhi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-10-02T14:23:12Z","title":"Distributional Extension and Invertibility of the $k$-Plane Transform and Its Dual"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.01233","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:779089bc9943e6cdf0abe14d65b7df8c9321d588d6885170e717d0454fac6a37","target":"record","created_at":"2026-07-05T08:39:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"21f35d14e4e5104b2740314fa0050a18a21e03cb60ebe57e21685427d731f6b8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2023-10-02T14:23:12Z","title_canon_sha256":"e377bc12b369d492e654231d010a0378191cdcf448b8e747982f4dfd8dfff163"},"schema_version":"1.0","source":{"id":"2310.01233","kind":"arxiv","version":2}},"canonical_sha256":"b781d788d55a58d23a0ff4ce398279db857c471a08e288c312281c8235ef0fe7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b781d788d55a58d23a0ff4ce398279db857c471a08e288c312281c8235ef0fe7","first_computed_at":"2026-07-05T08:39:28.385779Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:39:28.385779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ws4bxbmWd9RlAWn9Fx0iygYGxnnVJZ+U9Fpn0/qYswuexUScUufbjDA1Cu+0fByD8/OzOhPgC3KI/IdQIV2VBA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:39:28.386224Z","signed_message":"canonical_sha256_bytes"},"source_id":"2310.01233","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:779089bc9943e6cdf0abe14d65b7df8c9321d588d6885170e717d0454fac6a37","sha256:f0e880124cc2650119675594dda7ba423bcf95f8cbf4656846d2f267b92f4299"],"state_sha256":"3b17090a7c8fc1b1d8c7098feb9332700c0724c514fabc77abc0ae8f95889bc4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w38H4ddyY51s+/v58NE+1m9NVVYlndlLBvkMwSvm3FONsbKUdzEQanrf+mk4iltGRnaNzakghT4SUjdBPtk2Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T12:12:31.974164Z","bundle_sha256":"a1b4ad0acfe24c6234cd2bc5401582310a745909130ac295ea7e56e675f9081a"}}