{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T07:52:44Z","timestamp":1771573964300,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2017,7,7]],"date-time":"2017-07-07T00:00:00Z","timestamp":1499385600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2018,8]]},"DOI":"10.1007\/s10208-017-9361-0","type":"journal-article","created":{"date-parts":[[2017,7,7]],"date-time":"2017-07-07T18:46:33Z","timestamp":1499453193000},"page":"1043-1071","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Distortion Varieties"],"prefix":"10.1007","volume":"18","author":[{"given":"Joe","family":"Kileel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zuzana","family":"Kukelova","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tomas","family":"Pajdla","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bernd","family":"Sturmfels","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2017,7,7]]},"reference":[{"key":"9361_CR1","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/S0747-7171(02)00140-2","volume":"35","author":"E Arnold","year":"2003","unstructured":"E.\u00a0Arnold, Modular algorithms for computing Gr\u00f6bner bases, J. Symbolic Comput. 35 (2003) 403\u2013419.","journal-title":"J. Symbolic Comput."},{"key":"9361_CR2","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1016\/j.jalgebra.2015.10.008","volume":"448","author":"C Bocci","year":"2016","unstructured":"C.\u00a0Bocci, E.\u00a0Carlini and J.\u00a0Kileel, Hadamard products of linear spaces, J. Algebra 448 (2016) 595\u2013617.","journal-title":"J. Algebra"},{"key":"9361_CR3","unstructured":"M.\u00a0Bujnak, Algebraic solutions to absolute pose problems, Doctoral Thesis, Czech Technical University in Prague, 2012."},{"key":"9361_CR4","doi-asserted-by":"crossref","unstructured":"M.\u00a0Bujnak, Z.\u00a0Kukelova, and T.\u00a0Pajdla, 3D reconstruction from image collections with a single known focal length, in Proceedings of the 2009 IEEE International Conference on Computer Vision (ICCV 2009), pp.\u00a0351\u2013358.","DOI":"10.1109\/ICCV.2009.5459402"},{"key":"9361_CR5","doi-asserted-by":"crossref","unstructured":"M.\u00a0Bujnak, Z.\u00a0Kukelova and T.\u00a0Pajdla, Making minimal solvers fast, in Proceedings of the 2012 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2012), pp.\u00a01506\u20131513.","DOI":"10.1109\/CVPR.2012.6247840"},{"key":"9361_CR6","doi-asserted-by":"crossref","unstructured":"M.\u00a0Byrod, Z.\u00a0Kukelova, K.\u00a0Josephson, T.\u00a0Pajdla and K.\u00a0\u00c5str\u00f6m, Fast and robust numerical solutions to minimal problems for cameras with radial distortion, in Proceedings of the 2008 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2008), pp.\u00a01\u20138.","DOI":"10.1109\/CVPR.2008.4587674"},{"key":"9361_CR7","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1007\/s00209-012-1095-8","volume":"274","author":"E Cattani","year":"2013","unstructured":"E.\u00a0Cattani, M.\u00a0A.\u00a0Cueto, A.\u00a0Dickenstein, S.\u00a0Di Rocco and B.\u00a0Sturmfels, Mixed discriminants, Math. Z. 274 (2013) 761\u2013778.","journal-title":"Math. Z."},{"key":"9361_CR8","volume-title":"Toric varieties, Graduate Studies in Mathematics 124","author":"D Cox","year":"2011","unstructured":"D.\u00a0Cox, J.\u00a0Little and H.\u00a0Schenck, Toric varieties, Graduate Studies in Mathematics 124, American Mathematical Society, Providence, 2011."},{"key":"9361_CR9","doi-asserted-by":"crossref","unstructured":"J.\u00a0Dalbec and B.\u00a0Sturmfels, Introduction to Chow forms, in Invariant methods in discrete and computational geometry (N.\u00a0White, ed.), Springer, New York, 1995, pp.\u00a037\u201358.","DOI":"10.1007\/978-94-015-8402-9_2"},{"key":"9361_CR10","volume-title":"Sur deux probl\u00e8mes de reconstruction, Technical Report 882","author":"M D\u00e9mazure","year":"1988","unstructured":"M.\u00a0D\u00e9mazure, Sur deux probl\u00e8mes de reconstruction, Technical Report 882, INRIA, Rocquencourt, 1988."},{"key":"9361_CR11","doi-asserted-by":"crossref","unstructured":"S.\u00a0Di Rocco, Linear toric fibrations, in Combinatorial algebraic geometry, Lecture Notes in Mathematics 2108, Springer, Cham, 2014, pp.\u00a0119\u2013147.","DOI":"10.1007\/978-3-319-04870-3_4"},{"key":"9361_CR12","doi-asserted-by":"crossref","unstructured":"D.\u00a0Eisenbud and J.\u00a0Harris, On varieties of minimal degree (a centennial account), in Algebraic geometry, Bowdoin 1985, Part 1, Proceedings of Symposia in Pure Mathematics 46 (S.J.\u00a0Bloch, ed.), American Mathematics Society, Providence, 1987, pp.\u00a03\u201313.","DOI":"10.1090\/pspum\/046.1\/927946"},{"key":"9361_CR13","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/S0022-4049(99)00005-5","volume":"139","author":"J-C Faug\u00e8re","year":"1999","unstructured":"J.-C.\u00a0Faug\u00e8re, A new efficient algorithm for computing Gr\u00f6bner bases (F4), J. Pure Appl. Algebra 139 (1999) 61\u201388.","journal-title":"J. Pure Appl. Algebra"},{"key":"9361_CR14","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1145\/358669.358692","volume":"24","author":"M Fischler","year":"1981","unstructured":"M.\u00a0Fischler and R.\u00a0Bolles, Random sample consensus:\u00a0a paradigm for model fitting with applications to image analysis and automated cartography, Commun. ACM 24 (1981) 381\u2013395.","journal-title":"Commun. ACM"},{"key":"9361_CR15","doi-asserted-by":"crossref","unstructured":"A.\u00a0Fitzgibbon, Simultaneous linear estimation of multiple view geometry and lens distortion, in Proceedings of the 2001 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2001), pp.\u00a0125\u2013132.","DOI":"10.1109\/CVPR.2001.990465"},{"key":"9361_CR16","unstructured":"G.\u00a0Fl\u00f8ystad, J.\u00a0Kileel and G.\u00a0Ottaviani, The Chow form of the essential variety in computer vision, J. Symbolic Comput., to appear."},{"key":"9361_CR17","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-8176-4771-1","volume-title":"Discriminants, resultants and multidimensional determinants","author":"IM Gel\u2019fand","year":"1994","unstructured":"I.M.\u00a0Gel\u2019fand, M.M.\u00a0Kapranov and A.V.\u00a0Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkh\u00e4user, Boston, 1994."},{"key":"9361_CR18","unstructured":"D.\u00a0Grayson and M.\u00a0Stillman, Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu\/Macaulay2\/ ."},{"key":"9361_CR19","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2189-8","volume-title":"Algebraic geometry: a first course, Graduate Texts in Mathematics 133","author":"J Harris","year":"1992","unstructured":"J.\u00a0Harris, Algebraic geometry: a first course, Graduate Texts in Mathematics 133, Springer-Verlag, New York, 1992."},{"key":"9361_CR20","doi-asserted-by":"crossref","unstructured":"R.\u00a0Hartley and A.\u00a0Zisserman, Multiple view geometry in computer vision, Cambridge University Press, 2nd ed., Cambridge, 2003.","DOI":"10.1017\/CBO9780511811685"},{"key":"9361_CR21","unstructured":"A.\u00a0Jensen, Gfan, a software system for Gr\u00f6bner fans and tropical varieties, Available at http:\/\/home.imf.au.dk\/jensen\/software\/gfan\/gfan.html ."},{"key":"9361_CR22","doi-asserted-by":"crossref","unstructured":"F.\u00a0Jiang, Y.\u00a0Kuang, J.E.\u00a0Solem and K.\u00a0\u00c5str\u00f6m, A minimal solution to relative pose with unknown focal length and radial distortion, in Proceedings of the 2014 Asian Conference on Computer Vision (ACCV 2014), pp.\u00a0443\u2013456.","DOI":"10.1007\/978-3-319-16808-1_30"},{"key":"9361_CR23","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1215\/S0012-7094-92-06707-X","volume":"67","author":"M Kapranov","year":"1992","unstructured":"M.\u00a0Kapranov, B.\u00a0Sturmfels and A.\u00a0Zelevinski, Chow polytopes and general resultants, Duke Math. J. 67 (1992) 189\u2013218.","journal-title":"Duke Math. J."},{"key":"9361_CR24","doi-asserted-by":"crossref","unstructured":"Y.\u00a0Kuang, J.E.\u00a0Solem, F.\u00a0Kahl and K.\u00a0\u00c5str\u00f6m, Minimal solvers for relative pose with a single unknown radial distortion, in Proceedings of 2014 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2014), pp.\u00a033\u201340.","DOI":"10.1109\/CVPR.2014.12"},{"key":"9361_CR25","unstructured":"Z.\u00a0Kukelova, Algebraic Methods in Computer Vision, Doctoral Thesis, Czech Technical University in Prague, 2013."},{"key":"9361_CR26","doi-asserted-by":"crossref","unstructured":"Z.\u00a0Kukelova, M.\u00a0Bujnak and T.\u00a0Pajdla, Automatic Generator of Minimal Problem Solvers, in Proceedings of the 2008 European Conference on Computer Vision (ECCV 2008), Lecture Notes in Computer Science 5304, Springer 2008, pp.\u00a0302\u2013315.","DOI":"10.1007\/978-3-540-88690-7_23"},{"key":"9361_CR27","doi-asserted-by":"crossref","unstructured":"D.\u00a0Maclagan and B.\u00a0Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, 2015.","DOI":"10.1090\/gsm\/161"},{"key":"9361_CR28","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1109\/TPAMI.2006.151","volume":"28","author":"B Micusik","year":"2006","unstructured":"B.\u00a0Micusik and T.\u00a0Pajdla, Structure from motion with wide circular field of view cameras, IEEE T. Pattern Anal. 28 (2006) 1135\u20131149.","journal-title":"IEEE T. Pattern Anal."},{"key":"9361_CR29","unstructured":"G.\u00a0Newsam, D.\u00a0Q.\u00a0Huynh, M.\u00a0Brooks and H.\u00a0P.\u00a0Pan, Recovering unknown focal lengths in self-calibration: an essentially linear algorithm and degenerate configurations. ISPRS J. Photogramm., vol.\u00a0XXXI-B3 (1996), 575\u2013580."},{"key":"9361_CR30","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1109\/TPAMI.2004.17","volume":"26","author":"D Nist\u00e9r","year":"2004","unstructured":"D.\u00a0Nist\u00e9r, An efficient solution to the five-point relative pose problem, IEEE T. Pattern Anal. 26 (2004) 756\u2013770.","journal-title":"IEEE T. Pattern Anal."},{"key":"9361_CR31","doi-asserted-by":"crossref","first-page":"1211","DOI":"10.4310\/MRL.2008.v15.n6.a11","volume":"15","author":"S Petrovi\u0107","year":"2008","unstructured":"S.\u00a0Petrovi\u0107, On the universal Gr\u00f6bner bases of varieties of minimal degree, Math. Res. Lett. 15 (2008) 1211\u20131221.","journal-title":"Math. Res. Lett."},{"key":"9361_CR32","doi-asserted-by":"crossref","unstructured":"H.\u00a0Stewenius, D.\u00a0Nist\u00e9r, F.\u00a0Kahl and F.\u00a0Schaffalitzky, A minimal solution for relative pose with unknown focal length, in Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2005), pp.\u00a0789\u2013794.","DOI":"10.1109\/CVPR.2005.36"},{"key":"9361_CR33","doi-asserted-by":"crossref","unstructured":"P.\u00a0Sturm, On focal length calibration from two views, in Proceedings of the 2001 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2001), pp.\u00a0145\u2013150.","DOI":"10.1109\/CVPR.2001.990940"},{"key":"9361_CR34","doi-asserted-by":"crossref","unstructured":"B.\u00a0Sturmfels, Gr\u00f6bner bases and convex polytopes, American Mathematical Society, University Lectures Series 8, Providence, 1996.","DOI":"10.1090\/ulect\/008"},{"key":"9361_CR35","doi-asserted-by":"crossref","unstructured":"B.\u00a0Sturmfels, Solving systems of polynomial equations, American Mathematical Society, CBMS Regional Conferences Series 97, Providence, 2002.","DOI":"10.1090\/cbms\/097"},{"key":"9361_CR36","doi-asserted-by":"crossref","unstructured":"C.\u00a0Traverso, Gr\u00f6bner trace algorithms, in Proceedings of the 1988 International Symposium of Symbolic and Algebraic Computation (ISSAC 1988), Lecture Notes in Computer Science 358 (P.\u00a0Gianni, ed.), Springer-Verlag, Berlin Heidelberg, 1989, pp.\u00a0125\u2013138.","DOI":"10.1007\/3-540-51084-2_12"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-017-9361-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-017-9361-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-017-9361-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T15:30:17Z","timestamp":1750519817000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-017-9361-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,7]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2018,8]]}},"alternative-id":["9361"],"URL":"https:\/\/doi.org\/10.1007\/s10208-017-9361-0","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,7,7]]}}}