{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T16:52:07Z","timestamp":1771519927173,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2020,11,24]],"date-time":"2020-11-24T00:00:00Z","timestamp":1606176000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Systems composed of multiple sensors for exteroceptive perception are becoming increasingly common, such as mobile robots or highly monitored spaces. However, to combine and fuse those sensors to create a larger and more robust representation of the perceived scene, the sensors need to be properly registered among them, that is, all relative geometric transformations must be known. This calibration procedure is challenging as, traditionally, human intervention is required in variate extents. This paper proposes a nearly automatic method where the best set of geometric transformations among any number of sensors is obtained by processing and combining the individual pairwise transformations obtained from an experimental method. Besides eliminating some experimental outliers with a standard criterion, the method exploits the possibility of obtaining better geometric transformations between all pairs of sensors by combining them within some restrictions to obtain a more precise transformation, and thus a better calibration. Although other data sources are possible, in this approach, 3D point clouds are obtained by each sensor, which correspond to the successive centers of a moving ball its field of view. The method can be applied to any sensors able to detect the ball and the 3D position of its center, namely, LIDARs, mono cameras (visual or infrared), stereo cameras, and TOF cameras. Results demonstrate that calibration is improved when compared to methods in previous works that do not address the outliers problem and, depending on the context, as explained in the results section, the multi-pairwise technique can be used in two different methodologies to reduce uncertainty in the calibration process.<\/jats:p>","DOI":"10.3390\/s20236717","type":"journal-article","created":{"date-parts":[[2020,11,24]],"date-time":"2020-11-24T09:06:28Z","timestamp":1606208788000},"page":"6717","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Multi-Sensor Extrinsic Calibration Using an Extended Set of Pairwise Geometric Transformations"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1283-7388","authenticated-orcid":false,"given":"Vitor","family":"Santos","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering(DEM), Institute of Electronics and Informatics Engineering of Aveiro (IEETA), University of Aveiro, 3810-193 Aveiro, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5229-4229","authenticated-orcid":false,"given":"Daniela","family":"Rato","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering(DEM), Institute of Electronics and Informatics Engineering of Aveiro (IEETA), University of Aveiro, 3810-193 Aveiro, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3754-2749","authenticated-orcid":false,"given":"Paulo","family":"Dias","sequence":"additional","affiliation":[{"name":"Departament of Electronics, Telecommunications and Informatics (DETI),\r\nInstitute of Electronics and Informatics Engineering of Aveiro (IEETA), University of Aveiro, 3810-193 Aveiro, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9288-5058","authenticated-orcid":false,"given":"Miguel","family":"Oliveira","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering(DEM), Institute of Electronics and Informatics Engineering of Aveiro (IEETA), University of Aveiro, 3810-193 Aveiro, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/j.robot.2016.05.010","article-title":"Self calibration of multiple LIDARs and cameras on autonomous vehicles","volume":"83","author":"Pereira","year":"2016","journal-title":"Robot. Auton. Syst."},{"key":"ref_2","unstructured":"Czyzewski, M.A. (2018). An Extremely Efficient Chess-board Detection for Non-trivial Photos. arXiv."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hu, D., DeTone, D., and Malisiewicz, T. (2019, January 15\u201320). Deep ChArUco: Dark ChArUco Marker Pose Estimation. Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR), Long Beach, CA, USA.","DOI":"10.1109\/CVPR.2019.00863"},{"key":"ref_4","unstructured":"Zhang, Q., and Pless, R. (October, January 28). Extrinsic calibration of a camera and laser range finder (improves camera calibration). Proceedings of the IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566), Sendai, Japan."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1016\/j.patcog.2015.09.023","article-title":"Generation of fiducial marker dictionaries using Mixed Integer Linear Programming","volume":"51","year":"2016","journal-title":"Pattern Recognit."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.imavis.2018.05.004","article-title":"Speeded up detection of squared fiducial markers","volume":"76","year":"2018","journal-title":"Image Vis. Comput."},{"key":"ref_7","unstructured":"Ruan, M., and Huber, D. (2014, January 8\u201311). Calibration of 3D Sensors Using a Spherical Target. Proceedings of the 2nd International Conference on 3D Vision, Tokyo, Japan."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Rato, D., and Santos, V. (2020, January 15\u201317). Automatic Registration of IR and RGB Cameras using a Target detected with Deep Learning. Proceedings of the IEEE International Conference on Autonomous Robot Systems and Competitions (ICARSC), Ponta Delgada, Portugal.","DOI":"10.1109\/ICARSC49921.2020.9096168"},{"key":"ref_9","unstructured":"Kwon, Y.C., Jang, J.W., and Choi, O. (2018, January 17\u201320). Automatic sphere detection for extrinsic calibration of multiple RGBD cameras. Proceedings of the 18th International Conference on Control, Automation and Systems (ICCAS), Daegwallyeong, Korea."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Almeida, M., Dias, P., Oliveira, M., and Santos, V. (2012, January 25\u201327). 3D-2D Laser Range Finder Calibration Using a Conic Based Geometry Shape. Proceedings of the Image Analysis and Recognition, Aveiro, Portugal.","DOI":"10.1007\/978-3-642-31295-3_37"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Mueller, G.R., and Wuensche, H. (2017, January 16\u201319). Continuous stereo camera calibration in urban scenarios. Proceedings of the IEEE 20th International Conference on Intelligent Transportation Systems (ITSC), Yokohama, Japan.","DOI":"10.1109\/ITSC.2017.8317675"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Wu, L., and Zhu, B. (2015, January 2\u20135). Binocular stereovision camera calibration. Proceedings of the IEEE International Conference on Mechatronics and Automation (ICMA), Beijing, China.","DOI":"10.1109\/ICMA.2015.7237903"},{"key":"ref_13","unstructured":"Su, R., Zhong, J., Li, Q., Qi, S., Zhang, H., and Wang, T. (2016, January 3\u20135). An automatic calibration system for binocular stereo imaging. Proceedings of the IEEE Advanced Information Management, Communicates, Electronic and Automation Control Conference (IMCEC), Xi\u2019an, China."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ling, Y., and Shen, S. (2016, January 9\u201314). High-precision online markerless stereo extrinsic calibration. Proceedings of the IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, Korea.","DOI":"10.1109\/IROS.2016.7759283"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1109\/TIP.2018.2870930","article-title":"Rectification Using Different Types of Cameras Attached to a Vehicle","volume":"28","author":"Dinh","year":"2019","journal-title":"IEEE Trans. Image Process."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Khan, A., Aragon-Camarasa, G., Sun, L., and Siebert, J.P. (2016, January 3\u20137). On the calibration of active binocular and RGBD vision systems for dual-arm robots. Proceedings of the IEEE International Conference on Robotics and Biomimetics (ROBIO), Qingdao, China.","DOI":"10.1109\/ROBIO.2016.7866616"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1109\/TRO.2018.2853742","article-title":"Robust Intrinsic and Extrinsic Calibration of RGB-D Cameras","volume":"34","author":"Basso","year":"2018","journal-title":"IEEE Trans. Robot."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Qiao, Y., Tang, B., Wang, Y., and Peng, L. (2013, January 26\u201328). A new approach to self-calibration of hand-eye vision systems. Proceedings of the International Conference on Computational Problem-Solving (ICCP), Jiuzhai, China.","DOI":"10.1109\/ICCPS.2013.6893596"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zhang, C., and Zhang, Z. (2011, January 11\u201315). Calibration between depth and color sensors for commodity depth cameras. Proceedings of the IEEE International Conference on Multimedia and Expo, Barcelona, Spain.","DOI":"10.1109\/ICME.2011.6012191"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2685","DOI":"10.1109\/JSEN.2018.2889805","article-title":"Accurate Intrinsic and Extrinsic Calibration of RGB-D Cameras with GP-Based Depth Correction","volume":"19","author":"Chen","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2097","DOI":"10.1109\/TPAMI.2012.18","article-title":"A Minimal Solution for the Extrinsic Calibration of a Camera and a Laser-Rangefinder","volume":"34","author":"Vasconcelos","year":"2012","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"H\u00e4selich, M., Bing, R., and Paulus, D. (2012, January 12\u201314). Calibration of multiple cameras to a 3D laser range finder. Proceedings of the IEEE International Conference on Emerging Signal Processing Applications, Las Vegas, NV, USA.","DOI":"10.1109\/ESPA.2012.6152437"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Chen, Z., Yang, X., Zhang, C., and Jiang, S. (2016, January 15\u201317). Extrinsic calibration of a laser range finder and a camera based on the automatic detection of line feature. Proceedings of the 9th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics (CISP-BMEI), Datong, China.","DOI":"10.1109\/CISP-BMEI.2016.7852753"},{"key":"ref_24","unstructured":"Velas, M., Spanel, M., Materna, Z., and Herout, A. (2014, January 2\u20135). Calibration of RGB camera with velodyne LiDAR. Proceedings of the 22nd International Conference in Central European Computer Graphics, Visualization and Computer Vision, Plzen, Czech Republic."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Guindel, C., Beltr\u00e1n, J., Mart\u00edn, D., and Garc\u00eda, F. (2017, January 16\u201319). Automatic extrinsic calibration for lidar-stereo vehicle sensor setups. Proceedings of the IEEE 20th International Conference on Intelligent Transportation Systems (ITSC), Yokohama, Japan.","DOI":"10.1109\/ITSC.2017.8317829"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Lee, G., Lee, J., and Park, S. (2017, January 16\u201318). Calibration of VLP-16 Lidar and multi-view cameras using a ball for 360 degree 3D color map acquisition. Proceedings of the IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI), Daegu, Korea.","DOI":"10.1109\/MFI.2017.8170408"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Levinson, J., and Thrun, S. (2013, January 24\u201328). Automatic Online Calibration of Cameras and Lasers. Proceedings of the Conference: Robotics: Science and Systems, Berlin, Germany.","DOI":"10.15607\/RSS.2013.IX.029"},{"key":"ref_28","unstructured":"Gao, D., Duan, J., Yang, X., and Zheng, B. (2010, January 7\u20139). A method of spatial calibration for camera and radar. Proceedings of the 8th World Congress on Intelligent Control and Automation, Jinan, China."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Santos, V., Almeida, J., \u00c1vila, E., Gameiro, D., Oliveira, M., Pascoal, R., Sabino, R., and Stein, P. (2010, January 19\u201322). ATLASCAR\u2014 Technologies for a computer assisted driving system, on board a common automobile. Proceedings of the 13th International IEEE Conference on Intelligent Transportation Systems, Funchal, Portugal.","DOI":"10.1109\/ITSC.2010.5625031"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Liao, Y., Li, G., Ju, Z., Liu, H., and Jiang, D. (2017, January 27\u201331). Joint kinect and multiple external cameras simultaneous calibration. Proceedings of the 2nd International Conference on Advanced Robotics and Mechatronics (ICARM), Hefei, China.","DOI":"10.1109\/ICARM.2017.8273179"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1109\/TRO.2016.2529645","article-title":"A General Approach to Spatiotemporal Calibration in Multisensor Systems","volume":"32","author":"Rehder","year":"2016","journal-title":"IEEE Trans. Robot."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Pradeep, V., Konolige, K., and Berger, E. (2014). Calibrating a Multi-arm Multi-sensor Robot: A Bundle Adjustment Approach. Experimental Robotics: The 12th International Symposium on Experimental Robotics, Springer.","DOI":"10.1007\/978-3-642-28572-1_15"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"103558","DOI":"10.1016\/j.robot.2020.103558","article-title":"A ROS framework for the extrinsic calibration of intelligent vehicles: A multi-sensor, multi-modal approach","volume":"131","author":"Oliveira","year":"2020","journal-title":"Robot. Auton. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1109\/TPAMI.1987.4767965","article-title":"Least-Squares Fitting of Two 3-D Point Sets","volume":"9","author":"Arun","year":"1987","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_35","unstructured":"Gusikhin, O., and Madani, K. (2020). ATLASCAR: A Sample of the Quests and Concerns for Autonomous Cars. Informatics in Control, Automation and Robotics, Springer International Publishing."},{"key":"ref_36","unstructured":"Wu, Y., Kirillov, A., Massa, F., Lo, W.Y., and Girshick, R. (2020, July 22). Detectron2. Available online: https:\/\/github.com\/facebookresearch\/detectron2."},{"key":"ref_37","unstructured":"Chauvenet, W. (1863). Method of Least Squares. Appendix to Manual of Spherical and Practical Astronomy, Vol. 2, Lippincott."},{"key":"ref_38","unstructured":"Barnett, V., and Lewis, T. (1994). Outliers in Statistical Data, John Wiley & Sons. [3rd ed.]."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/23\/6717\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:36:41Z","timestamp":1760179001000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/23\/6717"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,24]]},"references-count":38,"journal-issue":{"issue":"23","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["s20236717"],"URL":"https:\/\/doi.org\/10.3390\/s20236717","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,24]]}}}