{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:35:43Z","timestamp":1787236543036,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1719538"],"award-info":[{"award-number":["DMS-1719538"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Algebra Geometry"],"published-print":{"date-parts":[[2022,3]]},"abstract":"<jats:p>A fundamental question in computer vision is whether a set of point pairs is the image of a scene that lies in front of two cameras. Such a scene and the cameras together are known as a chiral reconstruction of the point pairs. In this paper, we provide a complete classification of $k$ point pairs for which a chiral reconstruction exists. The existence of chiral reconstructions is equivalent to the nonemptiness of certain semialgebraic sets. We describe these sets and develop tools to certify their nonemptiness. For up to three point pairs, we prove that a chiral reconstruction always exists while the set of five or more point pairs that do not have a chiral reconstruction is Zariski-dense. We show that for five generic point pairs, the chiral region is bounded by line segments in a Schl\u00e4fli double six on a cubic surface with 27 real lines. Four point pairs have a chiral reconstruction unless they belong to two nongeneric combinatorial types, in which case they may or may not.<\/jats:p>","DOI":"10.1137\/20m1381848","type":"journal-article","created":{"date-parts":[[2022,3,1]],"date-time":"2022-03-01T12:01:06Z","timestamp":1646136066000},"page":"41-76","source":"Crossref","is-referenced-by-count":2,"title":["Existence of Two View Chiral Reconstructions"],"prefix":"10.1137","volume":"6","author":[{"given":"Andrew","family":"Pryhuber","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rainer","family":"Sinn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rekha R.","family":"Thomas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,3,1]]},"reference":[{"key":"atypb1","unstructured":"S. Agarwal, A. Pryhuber, R. Sinn, and R. R. Thomas,\n                      The Chiral Domain of a Camera Arrangement\n                      , preprint,https:\/\/arxiv.org\/abs\/2003.09265, 2020."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700023740"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-19.2.245"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s10711-007-9144-x"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"O. Chum, T. Werner, and T. Pajdla,\n                      Joint orientation of epipoles\n                      , in Proceedings of the British Machine Vision Conference, 2003.","DOI":"10.5244\/C.17.6"},{"key":"atypb6","first-page":"55","author":"Dolgachev I. V.","year":"2005","journal-title":"Milan"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"I. V. Dolgachev,\n                      Classical Algebraic Geometry: A Modern View\n                      , Cambridge University Press, Cambridge, UK, 2012.","DOI":"10.1017\/CBO9781139084437"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(88)90054-0"},{"key":"atypb9","unstructured":"A. V. Geramita,\n                      Lectures on the nonsingular cubic surface in ${P}^3$\n                      , in The Curves Seminar at Queen's, Vol. VI (Kingston, ON, 1989), Queen's Papers in Pure and Appl. Math. 83, Queen's University, Kingston, ON, Canada, 1989, Exp. no. A, 106 pp."},{"key":"atypb10","unstructured":"J. Harris,\n                      Algebraic Geometry: A First Course\n                      , Grad. Texts in Math. 133, Springer-Verlag, New York, 1995."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1023\/A:1007984508483"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"R. I. Hartley and A. Zisserman,\n                      Multiple View Geometry in Computer Vision\n                      , 2nd ed., Cambridge University Press, Cambridge, UK, 2003.","DOI":"10.1017\/CBO9780511811685"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"R. Hartshorne,\n                      Algebraic Geometry\n                      , Springer-Verlag, New York, Heidelberg, 1977.","DOI":"10.1007\/978-1-4757-3849-0"},{"key":"atypb14","unstructured":"A. Henderson,\n                      The Twenty-Seven Lines upon the Cubic Surface\n                      , Hafner, New York, 1960."},{"key":"atypb15","unstructured":"D. Hilbert and S. Cohn-Vossen,\n                      Geometry and the Imagination\n                      , Chelsea, New York, 1952."},{"key":"atypb16","first-page":"147","author":"Laveau S.","year":"1996","journal-title":"London"},{"key":"atypb17","unstructured":"H.L. Lee,\n                      On the Existence of a Projective Reconstruction\n                      , preprint,https:\/\/arxiv.org\/abs\/1608.05518v1, 2016."},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"S. Maybank,\n                      Theory of Reconstruction from Image Motion\n                      , Springer-Verlag, Berlin, 1993.","DOI":"10.1007\/978-3-642-77557-4"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"M. Reid,\n                      Undergraduate Algebraic Geometry\n                      , London Math. Soc. Stud. Texts 12, Cambridge University Press, Cambridge, UK, 1988.","DOI":"10.1017\/CBO9781139163699"},{"key":"atypb20","unstructured":"B. Segre,\n                      The Non-singular Cubic Surfaces\n                      , Oxford University Press, Oxford, UK, 1942."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV.2003.1238459"},{"key":"atypb22","unstructured":"T. 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Pajdla,\n                      Oriented matching constraints\n                      , in Proceedings of the British Machine Vision Conference, 2001.","DOI":"10.5244\/C.15.46"}],"container-title":["SIAM Journal on Applied Algebra and Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1381848","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:47:54Z","timestamp":1787233674000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1381848"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3]]},"references-count":24,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,3]]}},"alternative-id":["10.1137\/20M1381848"],"URL":"https:\/\/doi.org\/10.1137\/20m1381848","relation":{},"ISSN":["2470-6566"],"issn-type":[{"value":"2470-6566","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3]]}}}