{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T15:51:29Z","timestamp":1782834689996,"version":"3.54.5"},"reference-count":33,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,6,8]],"date-time":"2023-06-08T00:00:00Z","timestamp":1686182400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004488","name":"Croatian Science Foundation","doi-asserted-by":"publisher","award":["HRZZ-IP-2019-04-9157"],"award-info":[{"award-number":["HRZZ-IP-2019-04-9157"]}],"id":[{"id":"10.13039\/501100004488","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004488","name":"Croatian Science Foundation","doi-asserted-by":"publisher","award":["DOK-2021-02-9474"],"award-info":[{"award-number":["DOK-2021-02-9474"]}],"id":[{"id":"10.13039\/501100004488","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The development of a robust 3D imaging system for underwater applications is a crucial process in underwater imaging where the physical properties of the underwater environment make the implementation of such systems challenging. Calibration is an essential step in the application of such imaging systems and is performed to acquire the parameters of the image formation model and to enable 3D reconstruction. We present a novel calibration method for an underwater 3D imaging system comprising a pair of cameras, of a projector, and of a single glass interface that is shared between cameras and projector(s). The image formation model is based on the axial camera model. The proposed calibration uses a numerical optimization of a 3D cost function to determine all system parameters, thus avoiding the minimization of re-projection errors which require numerically solving a 12th order polynomial equation multiple times for each observed point. We also propose a novel stable approach to estimate the axis of the axial camera model. The proposed calibration was experimentally evaluated on four different glass interfaces, wherein several quantitative results were reported, including the re-projection error. The achieved mean angular error of the system\u2019s axis was under 6\u2218, and the mean absolute errors for the reconstruction of a flat surface were 1.38 mm for normal glass interfaces and 2.82 mm for the laminated glass interface, which is more than sufficient for application.<\/jats:p>","DOI":"10.3390\/s23125444","type":"journal-article","created":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T02:03:18Z","timestamp":1686276198000},"page":"5444","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Calibration of a Structured Light Imaging System in Two-Layer Flat Refractive Geometry for Underwater Imaging"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4633-8755","authenticated-orcid":false,"given":"Domagoj","family":"Zoraja","sequence":"first","affiliation":[{"name":"University of Zagreb Faculty of Electrical Engineering and Computing, Unska 3, HR-10000 Zagreb, Croatia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3054-002X","authenticated-orcid":false,"given":"Tomislav","family":"Petkovi\u0107","sequence":"additional","affiliation":[{"name":"University of Zagreb Faculty of Electrical Engineering and Computing, Unska 3, HR-10000 Zagreb, Croatia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4868-5884","authenticated-orcid":false,"given":"Josep","family":"Forest","sequence":"additional","affiliation":[{"name":"Computer Vision and Robotics Research Institute (VICOROB), University of Girona, 17003 Girona, Catalunya, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5415-3630","authenticated-orcid":false,"given":"Tomislav","family":"Pribani\u0107","sequence":"additional","affiliation":[{"name":"University of Zagreb Faculty of Electrical Engineering and Computing, Unska 3, HR-10000 Zagreb, Croatia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Roman, C., Inglis, G., and Rutter, J. (2010, January 4\u201327). Application of structured light imaging for high resolution mapping of underwater archaeological sites. Proceedings of the OCEANS\u201910 IEEE SYDNEY, Sydney, Australia.","DOI":"10.1109\/OCEANSSYD.2010.5603672"},{"key":"ref_2","first-page":"87910G","article-title":"A review of techniques for the identification and measurement of fish in underwater stereo-video image sequences","volume":"Volume 8791","author":"Shortis","year":"2013","journal-title":"Videometrics, Range Imaging, and Applications XII and Automated Visual Inspection"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lopes, F., Silva, H., Almeida, J., Martins, A., and Silva, E. (2015, January 18\u201321). Structured light system for underwater inspection operations. Proceedings of the OCEANS 2015\u2013Genova, Genova, Italy.","DOI":"10.1109\/OCEANS-Genova.2015.7271564"},{"key":"ref_4","first-page":"1","article-title":"Underwater 3D Scanner to Counteract Refraction: Calibration and Experimental Results","volume":"27","author":"Forest","year":"2022","journal-title":"IEEE\/ASME Trans. Mechatronics"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"50231","DOI":"10.1109\/ACCESS.2021.3069189","article-title":"Underwater 3D Scanner Model Using a Biaxial MEMS Mirror","volume":"9","author":"Palomer","year":"2021","journal-title":"IEEE Access"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIM.2023.3280499","article-title":"Development, Calibration, and Image Processing of Underwater Structured Light Vision System: A Survey","volume":"72","author":"Fan","year":"2023","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"McCarthy, J.K., Benjamin, J., Winton, T., and van Duivenvoorde, W. (2019). 3D Recording and Interpretation for Maritime Archaeology, Springer International Publishing.","DOI":"10.1007\/978-3-030-03635-5"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"18140","DOI":"10.3390\/s150818140","article-title":"On the Accuracy Potential in Underwater\/Multimedia Photogrammetry","volume":"15","author":"Maas","year":"2015","journal-title":"Sensors"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zoraja, D., Petkovi\u0107, T., Pribani\u0107, T., and Forest, J. (2022, January 23\u201327). Projector Calibration in a Two-Layer Flat Refractive Geometry for Underwater Imaging. Proceedings of the 2022 45th Jubilee International Convention on Information, Communication and Electronic Technology (MIPRO), Opatija, Croatia.","DOI":"10.23919\/MIPRO55190.2022.9803454"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hartley, R.I., and Zisserman, A. (2004). Multiple View Geometry in Computer Vision, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511811685"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1364\/AOP.3.000128","article-title":"Structured-light 3D surface imaging: A tutorial","volume":"3","author":"Geng","year":"2011","journal-title":"Adv. Opt. Photon."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2666","DOI":"10.1016\/j.patcog.2010.03.004","article-title":"A state of the art in structured light patterns for surface profilometry","volume":"43","author":"Salvi","year":"2010","journal-title":"Pattern Recognit."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1016\/j.isprsjprs.2011.02.009","article-title":"Experimentation of structured light and stereo vision for underwater 3D reconstruction","volume":"66","author":"Bruno","year":"2011","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.isprsjprs.2016.01.014","article-title":"A structured light method for underwater surface reconstruction","volume":"114","author":"Sarafraz","year":"2016","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Narasimhan, S., Nayar, S., Sun, B., and Koppal, S. (2005, January 17\u201321). Structured light in scattering media. Proceedings of the Tenth IEEE International Conference on Computer Vision (ICCV\u201905), Beijing, China.","DOI":"10.1109\/ICCV.2005.232"},{"key":"ref_16","first-page":"55","article-title":"Underwater 3D measurements with advanced camera modelling","volume":"90","author":"Munkelt","year":"2022","journal-title":"PFG\u2014J. Photogramm. Remote Sens. Geoinf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1119\/1.2342908","article-title":"The pinhole camera: Imaging without lenses or mirrors","volume":"27","author":"Young","year":"1989","journal-title":"Phys. Teach."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Dellaert, F., Frahm, J.M., Pollefeys, M., Leal-Taix\u00e9, L., and Rosenhahn, B. (2012). Outdoor and Large-Scale Real-World Scene Analysis, Proceedings of the 15th International Conference on Theoretical Foundations of Computer Vision: Outdoor and Large-Scale Real-World Scene Analysis, Dagstuhl Castle, Germany, 26 June\u20131 July 2011, Springer.","DOI":"10.1007\/978-3-642-34091-8"},{"key":"ref_19","unstructured":"Agrawal, A., Ramalingam, S., Taguchi, Y., and Chari, V. (2022, November 28). A Theory of Multi-Layer Flat Refractive Geometry. Available online: http:\/\/www.amitkagrawal.com\/cvpr12\/FlatRefraction.html."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ramalingam, S., Sturm, P., and Lodha, S. (2006, January 13\u201316). Theory and Calibration for Axial Cameras. Proceedings of the Computer Vision\u2014ACCV 2006: 7th Asian Conference on Computer Vision, Hyderabad, India.","DOI":"10.1007\/11612032_71"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Petkovi\u0107, T., Gasparini, S., and Pribani\u0107, T. (October, January 28). A Note on Geometric Calibration of Multiple Cameras and Projectors. Proceedings of the 2020 43rd International Convention on Information, Communication and Electronic Technology (MIPRO), Opatija, Croatia.","DOI":"10.23919\/MIPRO48935.2020.9245316"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.oceaneng.2017.01.029","article-title":"The Pinax-model for accurate and efficient refraction correction of underwater cameras in flat-pane housings","volume":"133","author":"Pfingsthorn","year":"2017","journal-title":"Ocean. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"083601","DOI":"10.1117\/1.2336196","article-title":"Novel method for structured light system calibration","volume":"45","author":"Zhang","year":"2006","journal-title":"Opt. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1330","DOI":"10.1109\/34.888718","article-title":"A flexible new technique for camera calibration","volume":"22","author":"Zhang","year":"2000","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kawahara, R., Nobuhara, S., and Matsuyama, T. (2013, January 1\u20138). A Pixel-Wise Varifocal Camera Model for Efficient Forward Projection and Linear Extrinsic Calibration of Underwater Cameras with Flat Housings. Proceedings of the 2013 IEEE International Conference on Computer Vision Workshops, Sydney, Australia.","DOI":"10.1109\/ICCVW.2013.112"},{"key":"ref_26","unstructured":"Bouguet, J.Y. (2022, November 29). Camera Calibration Toolbox for Matlab. Available online: http:\/\/robots.stanford.edu\/cs223b04\/JeanYvesCalib\/."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1112","DOI":"10.1016\/j.jbiomech.2012.01.004","article-title":"Comparison of different camera calibration approaches for underwater applications","volume":"45","author":"Silvatti","year":"2012","journal-title":"J. Biomech."},{"key":"ref_28","unstructured":"Demazure, M. (1988). Sur Deux Problemes de Reconstruction. [Ph.D. Thesis, National Institute for Research in Digital Science and Technology]."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1109\/TPAMI.2004.17","article-title":"An efficient solution to the five-point relative pose problem","volume":"26","author":"Nister","year":"2004","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3415","DOI":"10.1364\/AO.53.003415","article-title":"Novel calibration method for structured-light system with an out-of-focus projector","volume":"53","author":"Li","year":"2014","journal-title":"Appl. Opt."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"90321","DOI":"10.1109\/ACCESS.2022.3200388","article-title":"Multiprojector Multicamera Structured Light Surface Scanner","volume":"10","year":"2022","journal-title":"IEEE Access"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.optlaseng.2016.09.006","article-title":"Temporal phase unwrapping using orthographic projection","volume":"90","year":"2017","journal-title":"Opt. Lasers Eng."},{"key":"ref_33","unstructured":"Mathworks (2022, December 02). Optimization Toolbox. Available online: https:\/\/www.mathworks.com\/help\/optim\/ug\/lsqnonlin.html."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/12\/5444\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:50:56Z","timestamp":1760125856000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/12\/5444"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,8]]},"references-count":33,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["s23125444"],"URL":"https:\/\/doi.org\/10.3390\/s23125444","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,8]]}}}