{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T16:04:18Z","timestamp":1740153858468,"version":"3.37.3"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,11,8]],"date-time":"2018-11-08T00:00:00Z","timestamp":1541635200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2018,11,8]],"date-time":"2018-11-08T00:00:00Z","timestamp":1541635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61703092"],"award-info":[{"award-number":["61703092"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61472075"],"award-info":[{"award-number":["61472075"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["IPSJ T Comput Vis Appl"],"published-print":{"date-parts":[[2018,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we propose a two-step method to give a unified explanation of camera calibration with two coplanar conics. Various kinds of conics-based patterns in which often two parameters are unknown have been studied in previous literatures. The key in such algorithms is to adopt different strategies to compute the world-to-image projective transformation (also called 2D homography). In the first step of our method, we show that two unknown parameters can always be computed in general cases by utilizing the underlying constraints on all parameters through the projective transformation (mathematically called projective invariants). The accompanied ambiguity problem is that the solutions of the unknown parameters are multiple. In the second step, the four intersection points (real or complex) of two totally known conics are utilized to compute the homography. The ambiguity in this step arises from the point correspondence problem. This results in multiple possibilities of correspondences followed by the ambiguous homographies. After analyzing the reasons of the two kinds of ambiguities, we apply the Centre Circle constraint to completely remove them. Finally, the experiments are shown to validate the proposed technique.<\/jats:p>","DOI":"10.1186\/s41074-018-0050-y","type":"journal-article","created":{"date-parts":[[2018,11,8]],"date-time":"2018-11-08T13:42:39Z","timestamp":1541684559000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Camera calibration with coplanar conics: a unified explanation and ambiguity analysis"],"prefix":"10.1186","volume":"10","author":[{"given":"Shen","family":"Cai","sequence":"first","affiliation":[]},{"given":"Zhanhao","family":"Wu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,11,8]]},"reference":[{"issue":"1","key":"50_CR1","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1007\/BF01440844","volume":"10","author":"S Ma","year":"1993","unstructured":"Ma S (1993) Conics-based stereo, motion estimation, and pose determination. Int J Comput Vis (IJCV) 10(1):7\u201325.","journal-title":"Int J Comput Vis (IJCV)"},{"key":"50_CR2","volume-title":"Geometric invariance in computer vision","author":"JL Mundy","year":"1992","unstructured":"Mundy JL, Zisserman A (1992) Geometric invariance in computer vision. MIT Press, Europe."},{"issue":"2","key":"50_CR3","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1109\/34.481540","volume":"18","author":"L Quan","year":"1996","unstructured":"Quan L (1996) Conic reconstruction and correspondence from two views. IEEE Trans Pattern Anal Mach Intell (PAMI) 18(2):151\u2013160.","journal-title":"IEEE Trans Pattern Anal Mach Intell (PAMI)"},{"issue":"10","key":"50_CR4","doi-asserted-by":"publisher","first-page":"971","DOI":"10.1109\/34.99233","volume":"13","author":"D Forsyth","year":"1991","unstructured":"Forsyth D, Mundy JL, Zisserman A, Coelho C, Heller A, Rothwell C (1991) Invariant descriptors for 3D object recognition and pose. IEEE Trans Pattern Anal Mach Intell (PAMI) 13(10):971\u2013991.","journal-title":"IEEE Trans Pattern Anal Mach Intell (PAMI)"},{"issue":"4","key":"50_CR5","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1016\/0262-8856(92)90056-9","volume":"10","author":"C Rothwell","year":"1992","unstructured":"Rothwell C, Zisserman A, Marinos C, Forsyth DA, Mundy JL (1992) Relative motion and pose from arbitrary plane curves. Image Vis Comput (IVC) 10(4):250\u2013262.","journal-title":"Image Vis Comput (IVC)"},{"issue":"4","key":"50_CR6","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/j.imavis.2005.11.008","volume":"24","author":"Y Wu","year":"2006","unstructured":"Wu Y, Li X, Wu F, Hu Z (2006) Coplanar circles, quasi-affine invariance and calibration. Image Vis Comput (IVC) 24(4):319\u2013326.","journal-title":"Image Vis Comput (IVC)"},{"key":"50_CR7","unstructured":"Ying X, Zha H (2007) Camera calibration using principal-axes aligned conics. Asisan Conference on Computer Vision (ACCV)."},{"issue":"4","key":"50_CR8","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1109\/TPAMI.2005.80","volume":"27","author":"JS Kim","year":"2005","unstructured":"Kim JS, Gurdjos P, Kweon IS (2005) Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration. IEEE Trans Pattern Anal Mach Intell (PAMI) 27(4):637\u2013642.","journal-title":"IEEE Trans Pattern Anal Mach Intell (PAMI)"},{"key":"50_CR9","unstructured":"Gurdjos P, Kim JS, Kweon IS (2006) Euclidean structure from confocal conics: theory and application to camera calibration. in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, New York."},{"key":"50_CR10","doi-asserted-by":"crossref","unstructured":"Gurdjos P, Sturm P, Wu Y (2006) Euclidean structure from n\u22652 parallel circles: theory and algorithms. European Conference on Computer Vision (ECCV). Springer, Graz.","DOI":"10.1007\/11744023_19"},{"key":"50_CR11","unstructured":"Zhao Z (2011) Conics with a common axis of symmetry: properties and applications to camera calibration:1079\u20131084. Twenty-Second International Joint Conference on Artificial Intelligence (IJCAI). IJCAI\/AAAI, Barcelona, Catalonia."},{"issue":"3","key":"50_CR12","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1007\/s00138-012-0446-0","volume":"24","author":"S Cai","year":"2013","unstructured":"Cai S, Zhao Z, Huang L, Liu Y (2013) Camera calibration with enclosing ellipses by an extended application of generalized eigenvalue decomposition. Mach Vis Appl 24(3):513\u2013520.","journal-title":"Mach Vis Appl"},{"issue":"1","key":"50_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s12204-013-1361-x","volume":"18","author":"S Cai","year":"2013","unstructured":"Cai S, Wang C, Yan Y, Liu Y (2013) Analysis of the pencil of conics with double complex contact and its application to camera calibration. J Shanghai Jiaotong Univ (Science) 18(1):1\u20136.","journal-title":"J Shanghai Jiaotong Univ (Science)"},{"key":"50_CR14","first-page":"4010","volume-title":"International Conference on Pattern Recognition","author":"Z Zhao","year":"2014","unstructured":"Zhao Z (2014) Euclidean structure from conic feature correspondences In: International Conference on Pattern Recognition, 4010\u20134014.. IEEE, Stockholm."},{"key":"50_CR15","first-page":"314","volume-title":"IEEE International Conference on Robotics and Biomimetics","author":"W Wang","year":"2018","unstructured":"Wang W, Zhang X, Qi Y, Cheng W (2018) An algorithm of pose estimation based on conic correspondences In: IEEE International Conference on Robotics and Biomimetics, 314\u2013319.. IEEE, Macau."},{"key":"50_CR16","doi-asserted-by":"publisher","first-page":"252","DOI":"10.1007\/3-540-47979-1_17","volume-title":"Computer Vision \u2014 ECCV 2002","author":"Pierre Gurdjos","year":"2002","unstructured":"Gurdjos P, Crouzil A, Payrissat R (2002) Another way of looking at plane-based calibration: the centre circle constraint. Eur Conf Comput Vis (ECCV). Springer, Copenhagen.pp 252\u2013266."},{"key":"50_CR17","volume-title":"Applications D\u00a1 Analyse et de G\u00a1\u00e4eometrie - Trait\u00a1\u00e4e des Propri\u00a1\u00e4et\u00a1\u00e4es Projectives des Figures","author":"JV Poncelet","year":"1862","unstructured":"Poncelet JV (1862) Applications D\u00a1 Analyse et de G\u00a1\u00e4eometrie - Trait\u00a1\u00e4e des Propri\u00a1\u00e4et\u00a1\u00e4es Projectives des Figures. Tome I. Imprimerie de Mallet-Bachelier, Paris."},{"issue":"22","key":"50_CR18","doi-asserted-by":"publisher","first-page":"5369","DOI":"10.1364\/AO.51.005369","volume":"51","author":"S Cai","year":"2012","unstructured":"Cai S, Huang L, Liu Y (2012) Automatically obtaining the correspondences of 4 coplanar points for an un-calibrated camera. Appl Opt 51(22):5369\u20135376.","journal-title":"Appl Opt"},{"key":"50_CR19","volume-title":"Multiple view geometry in computer vision, Second Ed","author":"R Hartley","year":"2003","unstructured":"Hartley R, Zisserman A (2003) Multiple view geometry in computer vision, Second Ed. Cambridge Univ. Press, New York."},{"issue":"11","key":"50_CR20","doi-asserted-by":"publisher","first-page":"1330","DOI":"10.1109\/34.888718","volume":"22","author":"Z Zhang","year":"2000","unstructured":"Zhang Z (2000) A flexible new technique for camera calibration. IEEE Trans Pattern Anal Mach Intell (PAMI) 22(11):1330\u20131334.","journal-title":"IEEE Trans Pattern Anal Mach Intell (PAMI)"},{"key":"50_CR21","volume-title":"IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR)","author":"PF Sturm","year":"1999","unstructured":"Sturm PF, Maybank SJ (1999) On plane-based camera calibration: a general algorithm, singularities, applications In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR).. IEEE, Fort Collins."},{"key":"50_CR22","unstructured":"Forsyth D, Mundy JL, Zisserman A, Brown CM (1990) Invariance - a new framework for vision. International Conference on Computer Vision (ICCV). IEEE, Osaka."},{"key":"50_CR23","unstructured":"Gros P, Quan L (1992) Technique report: projective invariants for vision."},{"key":"50_CR24","volume-title":"Algebraic projective geometry","author":"J Semple","year":"1952","unstructured":"Semple J, Kneebone G (1952) Algebraic projective geometry. Oxford Univ. Press, Oxford."},{"key":"50_CR25","unstructured":"Bouguet J (2011) Camera Calibration Toolbox for Matlab. http:\/\/www.vision.caltech.edu\/bouguetj\/calib_doc\/index.html."},{"issue":"1","key":"50_CR26","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1023\/A:1007984508483","volume":"26","author":"R Hartley","year":"1998","unstructured":"Hartley R (1998) Chirality. Int J Comput Vis (IJCV) 26(1):41\u201361.","journal-title":"Int J Comput Vis (IJCV)"},{"issue":"5","key":"50_CR27","doi-asserted-by":"publisher","first-page":"476","DOI":"10.1109\/34.765658","volume":"21","author":"AW Fitzgibbon","year":"1999","unstructured":"Fitzgibbon AW, Pilu M, Fisher RB (1999) Direct least-squares fitting of ellipses. IEEE Trans Pattern Anal Mach Intell (PAMI) 21(5):476\u2013480.","journal-title":"IEEE Trans Pattern Anal Mach Intell (PAMI)"},{"key":"50_CR28","first-page":"766","volume-title":"European Conference on Computer Vision","author":"S Placht","year":"2014","unstructured":"Placht S, F\u00fcrsattel P, Mengue EA, Hofmann H, Schaller C, Balda M, Angelopoulou E (2014) Rochade: robust checkerboard advanced detection for camera calibration In: European Conference on Computer Vision, 766\u2013779.. Springer, Cham."}],"container-title":["IPSJ Transactions on Computer Vision and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s41074-018-0050-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s41074-018-0050-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s41074-018-0050-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T12:10:29Z","timestamp":1627647029000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1186\/s41074-018-0050-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,8]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,12]]}},"alternative-id":["50"],"URL":"https:\/\/doi.org\/10.1186\/s41074-018-0050-y","relation":{},"ISSN":["1882-6695"],"issn-type":[{"type":"electronic","value":"1882-6695"}],"subject":[],"published":{"date-parts":[[2018,11,8]]},"assertion":[{"value":"13 June 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 October 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Publisher\u2019s Note"}}],"article-number":"14"}}