{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T07:40:05Z","timestamp":1790840405246,"version":"4.1.0"},"reference-count":40,"publisher":"Optica Publishing Group","issue":"10","license":[{"start":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T00:00:00Z","timestamp":1789948800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/doi.org\/10.1364\/OA_License_v2#VOR"},{"start":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T00:00:00Z","timestamp":1789948800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/opg.optica.org\/policies\/opg-tdm-policy.json"}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"short-container-title":["J. Opt. Soc. Am. A"],"published-print":{"date-parts":[[2026,10,1]]},"abstract":"<jats:p>\n                    Color formation, the mapping from spectral reflectances through an illuminant and observer to color responses, is a bounded linear operator from a Hilbert space of spectral functions to the space\n                    <jats:italic>R<\/jats:italic>\n                    <jats:sup>\n                      <jats:italic>C<\/jats:italic>\n                    <\/jats:sup>\n                    of color responses, where\n                    <jats:italic>C<\/jats:italic>\n                    is the number of sensor classes of the observer (\n                    <jats:italic>C<\/jats:italic>\n                    =3 for a trichromat). This paper develops the algebraic ecosystem of this framework and uses it to establish two results on the limits of metamerism. Metameric equivalence classes are cosets of the kernel, making color space the quotient\n                    <jats:italic>L<\/jats:italic>\n                    <jats:sup>2<\/jats:sup>\n                    \/ker(W); the pseudoinverse is the canonical section selecting minimum-norm representatives; physical constraints cut each coset to a compact convex body whose image is the object-color gamut (a zonotope); and two complementary types of domain restriction, linear-model bases and convex-combination \u201cnaturalness\u201d domains, transfer the algebraic structure to a finite-dimensional effective operator. Within this framework we prove:\n                    <jats:italic toggle=\"yes\">(i)<\/jats:italic>\n                    non-trivial absolute metamers, reflectance pairs that match under every possible illuminant, cannot exist as long as every wavelength excites at least one photoreceptor class;\n                    <jats:italic toggle=\"yes\">(ii)<\/jats:italic>\n                    a two-sided dimensional threshold for metamers stable across a restricted family of illuminants: such metamers exist whenever the illuminant family spans fewer than \u2308\n                    <jats:italic>S<\/jats:italic>\n                    \/\n                    <jats:italic>C<\/jats:italic>\n                    \u2309 dimensions, where\n                    <jats:italic>S<\/jats:italic>\n                    is the number of sampled wavelengths, and cannot exist once the stacked operator attains full rank\u2014e.g., for a trichromatic observer (\n                    <jats:italic>C<\/jats:italic>\n                    =3) and the CIE daylight family, whose span has dimension\n                    <jats:italic>d<\/jats:italic>\n                    =2 or 3, in an\n                    <jats:italic>S<\/jats:italic>\n                    =31-dimensional spectral representation this threshold is far from being reached. The existence half of (ii) was given in discrete form by Burns, Cohen, and Kuznetsov and in dual form by Brainard, Wandell, and Cowan, both in 1989. What is added here is the impossibility direction, the two-sided threshold, and its transport to a restricted domain. A worked example with the CIE 1931 observer under D65, the CIE daylight basis, and a 10-dimensional PCA basis of the SOCS dataset illustrates the theory and constructs daylight-absolute metamer pairs by linear programming. A companion paper [J. Opt. Soc. Am. A (to be published)] extends the framework to two operators, treating metamer mismatching and chromatic adaptation.\n                  <\/jats:p>","DOI":"10.1364\/josaa.606555","type":"journal-article","created":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T18:00:10Z","timestamp":1786384810000},"page":"1614","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Color formation as a Hilbert space operator: quotient structures and limits of metamerism"],"prefix":"10.1364","volume":"43","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7759-413X","authenticated-orcid":true,"given":"Peter","family":"Morovic","sequence":"first","affiliation":[{"name":"HP Inc."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9983-8211","authenticated-orcid":true,"given":"J\u00e1n","family":"Morovic","sequence":"additional","affiliation":[{"name":"HP Inc."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"285","published-online":{"date-parts":[[2026,9,21]]},"reference":[{"key":"josaa-43-10-1614-R1","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1002\/andp.19203682103","type":"journal-article","volume":"63","author":"Schr\u00f6dinger","year":"1920","journal-title":"Ann. Phys."},{"key":"josaa-43-10-1614-R2","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1364\/JOSA.48.000451","type":"journal-article","volume":"48","author":"Wyszecki","year":"1958","journal-title":"J. Opt. Soc. Am."},{"key":"josaa-43-10-1614-R3","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1098\/rspl.1859.0074","type":"journal-article","volume":"150","author":"Maxwell","year":"1860","journal-title":"Philos. Trans. R. Soc. London"},{"key":"josaa-43-10-1614-R4","doi-asserted-by":"publisher","first-page":"537","DOI":"10.2307\/1422186","type":"journal-article","volume":"95","author":"Cohen","year":"1982","journal-title":"Am. J. Psychol."},{"key":"josaa-43-10-1614-R5","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1002\/col.5080130106","type":"journal-article","volume":"13","author":"Cohen","year":"1988","journal-title":"Color Res. Appl."},{"key":"josaa-43-10-1614-R6","doi-asserted-by":"publisher","first-page":"1905","DOI":"10.1364\/JOSAA.9.001905","type":"journal-article","volume":"9","author":"Marimont","year":"1992","journal-title":"J. Opt. Soc. Am. A"},{"key":"josaa-43-10-1614-R8","doi-asserted-by":"publisher","first-page":"810","DOI":"10.1364\/JOSAA.22.000810","type":"journal-article","volume":"22","author":"Finlayson","year":"2005","journal-title":"J. Opt. Soc. Am. A"},{"key":"josaa-43-10-1614-R9","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1167\/9.11.5","type":"journal-article","volume":"9","author":"Logvinenko","year":"2009","journal-title":"J. Vis."},{"key":"josaa-43-10-1614-R10","first-page":"111","article-title":"Real world metamer sets: or how we came to love noise","volume-title":"29th Color and Imaging Conference","author":"Morovi\u010d","year":"2021"},{"key":"josaa-43-10-1614-R11","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1002\/col.5080140106","type":"journal-article","volume":"14","author":"Burns","year":"1989","journal-title":"Color Res. Appl."},{"key":"josaa-43-10-1614-R12","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1109\/10.16459","type":"journal-article","volume":"36","author":"Brainard","year":"1989","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"josaa-43-10-1614-R13","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/BF00275798","type":"journal-article","volume":"1","author":"Resnikoff","year":"1974","journal-title":"J. Math. Biol."},{"key":"josaa-43-10-1614-R14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S1076-5670(05)38001-3","type":"journal-article","volume":"138","author":"Lenz","year":"2005","journal-title":"Adv. Imaging Electron. Phys."},{"key":"josaa-43-10-1614-R15","doi-asserted-by":"publisher","author":"Davis","year":"2018","DOI":"10.48550\/arXiv.1811.00990","type":"preprint"},{"key":"josaa-43-10-1614-R16","doi-asserted-by":"publisher","first-page":"e70028","DOI":"10.1002\/col.70028","type":"journal-article","volume":"51","author":"Wold","year":"2026","journal-title":"Color Res. Appl."},{"key":"josaa-43-10-1614-R17","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1002\/col.20713","type":"journal-article","volume":"38","author":"Centore","year":"2013","journal-title":"Color Res. Appl."},{"key":"josaa-43-10-1614-R18","doi-asserted-by":"publisher","first-page":"1673","DOI":"10.1364\/JOSAA.3.001673","type":"journal-article","volume":"3","author":"Maloney","year":"1986","journal-title":"J. Opt. Soc. Am. A"},{"key":"josaa-43-10-1614-R19","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1364\/JOSA.54.001031","type":"journal-article","volume":"54","author":"Judd","year":"1964","journal-title":"J. Opt. Soc. Am."},{"key":"josaa-43-10-1614-R20","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1002\/col.20086","type":"journal-article","volume":"30","author":"Tzeng","year":"2005","journal-title":"Color Res. Appl."},{"key":"josaa-43-10-1614-R23","series-title":"","volume-title":"Methods of Modern Mathematical Physics","volume":"I","author":"Reed","year":"1980"},{"key":"josaa-43-10-1614-R24","volume-title":"A Course in Functional Analysis","author":"Conway","year":"1990"},{"key":"josaa-43-10-1614-R25","doi-asserted-by":"publisher","first-page":"1814","DOI":"10.1364\/JOSAA.23.001814","type":"journal-article","volume":"23","author":"Morovi\u010d","year":"2006","journal-title":"J. Opt. Soc. Am. A"},{"key":"josaa-43-10-1614-R26","doi-asserted-by":"crossref","DOI":"10.2352\/CIC.2023.31.1.27","article-title":"An optimality property of matrix-R theorem, its extension, and the application to hyperspectral PAN-sharpening","volume-title":"31st Color and Imaging Conference","author":"Lin","year":"2023"},{"key":"josaa-43-10-1614-R27","doi-asserted-by":"publisher","first-page":"536","DOI":"10.1287\/moor.12.3.536","type":"journal-article","volume":"12","author":"Samet","year":"1987","journal-title":"Math. Oper. Res."},{"key":"josaa-43-10-1614-R28","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1098\/rspb.1983.0090","type":"journal-article","volume":"220","author":"Buchsbaum","year":"1983","journal-title":"Proc. R. Soc. London B"},{"key":"josaa-43-10-1614-R30","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1364\/JOSA.25.000249","type":"journal-article","volume":"25","author":"MacAdam","year":"1935","journal-title":"J. Opt. Soc. Am."},{"key":"josaa-43-10-1614-R31","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1167\/jov.25.2.2","type":"journal-article","volume":"25","author":"Logvinenko","year":"2025","journal-title":"J. Vis."},{"key":"josaa-43-10-1614-R32","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1364\/JOSAA.13.000436","type":"journal-article","volume":"13","author":"Speigle","year":"1996","journal-title":"J. Opt. Soc. Am. A"},{"key":"josaa-43-10-1614-R33","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1002\/andp.18531650505","type":"journal-article","volume":"165","author":"Grassmann","year":"1853","journal-title":"Ann. Phys."},{"key":"josaa-43-10-1614-R34","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1109\/TIP.2013.2283148","type":"journal-article","volume":"23","author":"Logvinenko","year":"2014","journal-title":"IEEE Trans. Image Process."},{"key":"josaa-43-10-1614-R35","volume-title":"Color Science: Concepts and Methods, Quantitative Data and Formulae","author":"Wyszecki","year":"1982"},{"key":"josaa-43-10-1614-R36","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1090\/S0002-9947-1969-0256265-X","type":"journal-article","volume":"145","author":"Bolker","year":"1969","journal-title":"Trans. Am. Math. Soc."},{"key":"josaa-43-10-1614-R37","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/BF00276901","type":"journal-article","volume":"11","author":"Brill","year":"1981","journal-title":"J. Math. Biol."},{"key":"josaa-43-10-1614-R38","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1002\/col.5080160108","type":"journal-article","volume":"16","author":"Trussell","year":"1991","journal-title":"Color Res. Appl."},{"key":"josaa-43-10-1614-R39","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1167\/16.15.8","type":"journal-article","volume":"16","author":"Witzel","year":"2016","journal-title":"J. Vis."},{"key":"josaa-43-10-1614-R40","doi-asserted-by":"publisher","first-page":"369","DOI":"10.3758\/BF03342963","type":"journal-article","volume":"1","author":"Cohen","year":"1964","journal-title":"Psychon. Sci."},{"key":"josaa-43-10-1614-R41","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1364\/JOSAA.6.000318","type":"journal-article","volume":"6","author":"Parkkinen","year":"1989","journal-title":"J. Opt. Soc. Am. A"},{"key":"josaa-43-10-1614-R42","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1111\/j.1520-6378.1994.tb00053.x","type":"journal-article","volume":"19","author":"Vrhel","year":"1994","journal-title":"Color Res. Appl."},{"key":"josaa-43-10-1614-R43","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1109\/TPAMI.1987.4767868","type":"journal-article","volume":"9","author":"Wandell","year":"1987","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"josaa-43-10-1614-R44","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.1364\/JOSAA.20.001261","type":"journal-article","volume":"20","author":"DiCarlo","year":"2003","journal-title":"J. Opt. Soc. Am. A"}],"container-title":["Journal of the Optical Society of America A"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/opg.optica.org\/viewmedia.cfm?URI=josaa-43-10-1614&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T19:16:33Z","timestamp":1790018193000},"score":1,"resource":{"primary":{"URL":"https:\/\/opg.optica.org\/abstract.cfm?URI=josaa-43-10-1614"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,21]]},"references-count":40,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2026]]},"published-print":{"date-parts":[[2026]]}},"URL":"https:\/\/doi.org\/10.1364\/josaa.606555","relation":{},"ISSN":["1084-7529","1520-8532"],"issn-type":[{"value":"1084-7529","type":"print"},{"value":"1520-8532","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,21]]},"assertion":[{"value":"Optica Publishing Group","name":"publisher","label":"This article is maintained by"},{"value":"https:\/\/doi.org\/10.1364\/JOSAA.606555","name":"articlelink","label":"Crossref DOI link to publisher maintained version"},{"value":"research-article","name":"content_type","label":"Article type"},{"value":"Screened by Similarity Check","name":"cross_check","label":"Similarity check"},{"value":"Yes","order":0,"name":"peer_reviewed","label":"Peer reviewed","group":{"name":"peer_review","label":"Peer review"}},{"value":"Single blind","order":1,"name":"review_process","label":"Review process","group":{"name":"peer_review","label":"Peer review"}},{"value":"28 May 2026","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"8 August 2026","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"21 September 2026","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}},{"value":"\u00a9 2026 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI) training, and similar technologies, are reserved.","name":"copyright","label":"Copyright"}]}}