{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T14:07:03Z","timestamp":1777039623763,"version":"3.51.4"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T00:00:00Z","timestamp":1770249600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T00:00:00Z","timestamp":1770249600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100007655","name":"Czech Technical University in Prague","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100007655","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int J Comput Vis"],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Homography refers to a specific type of transformation that relates two images of the same planar surface taken from different perspectives. Recovering motion parameters from a homography matrix is a classic problem in computer vision. It is important to derive a fast and stable solution to homography decomposition, since it forms a critical component of many vision systems,\n                    <jats:italic>e<\/jats:italic>\n                    .\n                    <jats:italic>g<\/jats:italic>\n                    .,\u00a0in Structure-from-Motion and visual localization. The current state-of-the-art solvers can be categorized into two types of methods, the numerical procedures based on singular value decomposition (SVD), and the closed-form solution. The SVD-based methods are stable but time-consuming, while the existing closed-form solution is faster but less stable. In this paper, we discuss the homography decomposition problem from a different viewpoint. In contrast to the existing methods which focus on the properties of the homography matrix, we propose a new method that uses three random point correspondences to obtain the motion parameters in closed form. The proposed method is conceptually simple, easy to understand and implement, and has a good geometrical interpretation. This solution can be seen as an alternative to the existing closed-form solution. We also discuss the configurations where the closed-form solutions might be unstable and present a framework for homography decomposition taking into account both the efficiency and stability.\n                  <\/jats:p>","DOI":"10.1007\/s11263-025-02680-4","type":"journal-article","created":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T07:24:48Z","timestamp":1770276288000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Homography Decomposition Revisited"],"prefix":"10.1007","volume":"134","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7448-6686","authenticated-orcid":false,"given":"Yaqing","family":"Ding","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zuzana","family":"Kukelova","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,5]]},"reference":[{"key":"2680_CR1","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"M Abramowitz","year":"1968","unstructured":"Abramowitz, M., & Stegun, I. A. (1968). Handbook of mathematical functions with formulas, graphs, and mathematical tables. US Government printing office."},{"key":"2680_CR2","doi-asserted-by":"crossref","unstructured":"Ding, Y., Chien, C. H., Larsson, V., \u00c5str\u00f6m, K. & Kimia, B. (2023). Minimal solutions to generalized three-view relative pose problem. In International Conference on Computer Vision (ICCV).","DOI":"10.1109\/ICCV51070.2023.00749"},{"issue":"03","key":"2680_CR3","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1142\/S0218001488000285","volume":"2","author":"OD Faugeras","year":"1988","unstructured":"Faugeras, O. D., & Lustman, F. (1988). Motion and structure from motion in a piecewise planar environment. International Journal of Pattern Recognition and Artificial Intelligence, 2(03), 485\u2013508.","journal-title":"International Journal of Pattern Recognition and Artificial Intelligence"},{"issue":"6","key":"2680_CR4","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1145\/358669.358692","volume":"24","author":"MA Fischler","year":"1981","unstructured":"Fischler, M. A., & Bolles, R. C. (1981). Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM, 24(6), 381\u2013395.","journal-title":"Communications of the ACM"},{"key":"2680_CR5","volume-title":"Multiple view geometry in computer vision","author":"R Hartley","year":"2003","unstructured":"Hartley, R., & Zisserman, A. (2003). Multiple view geometry in computer vision. Cambridge University Press."},{"key":"2680_CR6","doi-asserted-by":"crossref","unstructured":"Harville, D. A. (1998). Matrix algebra from a statistician\u2019s perspective.","DOI":"10.1007\/b98818"},{"issue":"2","key":"2680_CR7","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/s10851-009-0161-2","volume":"35","author":"DQ Huynh","year":"2009","unstructured":"Huynh, D. Q. (2009). Metrics for 3d rotations: Comparison and analysis. Journal of Mathematical Imaging and Vision, 35(2), 155\u2013164.","journal-title":"Journal of Mathematical Imaging and Vision"},{"key":"2680_CR8","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-21779-6","volume-title":"An invitation to 3-d vision: From images to geometric models","author":"Y Ma","year":"2004","unstructured":"Ma, Y., Soatto, S., Ko\u0161eck\u00e1, J., & Sastry, S. (2004). An invitation to 3-d vision: From images to geometric models (Vol. 26). New York: Springer."},{"key":"2680_CR9","volume-title":"Deeper understanding of the homography decomposition for vision-based control","author":"E Malis","year":"2007","unstructured":"Malis, E., & Vargas Villanueva, M. (2007). Deeper understanding of the homography decomposition for vision-based control. Inria: Doctoral dissertation."},{"issue":"5","key":"2680_CR10","doi-asserted-by":"publisher","first-page":"1255","DOI":"10.1109\/TRO.2017.2705103","volume":"33","author":"R Mur-Artal","year":"2017","unstructured":"Mur-Artal, R., & Tard\u00f3s, J. D. (2017). ORB-SLAM2: An open-source slam system for monocular, stereo, and rgb-d cameras. IEEE Transactions on Robotics, 33(5), 1255\u20131262.","journal-title":"IEEE Transactions on Robotics"},{"issue":"6","key":"2680_CR11","doi-asserted-by":"publisher","first-page":"756","DOI":"10.1109\/TPAMI.2004.17","volume":"26","author":"D Nist\u00e9r","year":"2004","unstructured":"Nist\u00e9r, D. (2004). An efficient solution to the five-point relative pose problem. Trans Pattern Analysis and Machine Intelligence (PAMI), 26(6), 756\u2013770.","journal-title":"Trans Pattern Analysis and Machine Intelligence (PAMI)"},{"key":"2680_CR12","doi-asserted-by":"crossref","unstructured":"Sarlin, P. E., Cadena, C., Siegwart, R., & Dymczyk, M. (2019). From coarse to fine: Robust hierarchical localization at large scale. In CVPR.","DOI":"10.1109\/CVPR.2019.01300"},{"issue":"9","key":"2680_CR13","doi-asserted-by":"publisher","first-page":"1744","DOI":"10.1109\/TPAMI.2016.2611662","volume":"39","author":"T Sattler","year":"2016","unstructured":"Sattler, T., Leibe, B., & Kobbelt, L. (2016). Efficient & effective prioritized matching for large-scale image-based localization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(9), 1744\u20131756.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"2680_CR14","doi-asserted-by":"crossref","unstructured":"Schonberger, J. L., & Frahm, J. M. (2016). Structure-from-motion revisited. In Computer Vision and Pattern Recognition (CVPR) (pp. 4104\u20134113).","DOI":"10.1109\/CVPR.2016.445"},{"issue":"1","key":"2680_CR15","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s10851-017-0741-5","volume":"60","author":"Z Yang","year":"2018","unstructured":"Yang, Z., Tang, L., & He, L. (2018). A new analytical method for relative camera pose estimation using unknown coplanar points. Journal of Mathematical Imaging and Vision, 60(1), 33\u201349.","journal-title":"Journal of Mathematical Imaging and Vision"},{"key":"2680_CR16","doi-asserted-by":"crossref","unstructured":"Zhang, Z., & Hanson, A. R. (1995). Scaled euclidean 3d reconstruction based on externally uncalibrated cameras. In Proceedings of International Symposium on Computer Vision-ISCV (IEEE, pp. 37\u201342).","DOI":"10.1109\/ISCV.1995.476974"},{"key":"2680_CR17","unstructured":"Zhang, Z., & Hanson, A. R. (1996). 3d reconstruction based on homography mapping. In Proc ARPA Image Understanding Workshop."},{"key":"2680_CR18","volume-title":"CRC standard mathematical tables and formulas","author":"D Zwillinger","year":"2018","unstructured":"Zwillinger, D. (2018). CRC standard mathematical tables and formulas. Chapman and hall\/CRC."}],"container-title":["International Journal of Computer Vision"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11263-025-02680-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11263-025-02680-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11263-025-02680-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T08:36:25Z","timestamp":1774600585000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11263-025-02680-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,5]]},"references-count":18,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["2680"],"URL":"https:\/\/doi.org\/10.1007\/s11263-025-02680-4","relation":{},"ISSN":["0920-5691","1573-1405"],"issn-type":[{"value":"0920-5691","type":"print"},{"value":"1573-1405","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,5]]},"assertion":[{"value":"17 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"102"}}