{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T07:37:20Z","timestamp":1780472240056,"version":"3.54.1"},"reference-count":56,"publisher":"MDPI AG","issue":"18","license":[{"start":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T00:00:00Z","timestamp":1694736000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Madrid Government (Comunidad de Madrid Spain)","award":["APBI-CM-UC3M"],"award-info":[{"award-number":["APBI-CM-UC3M"]}]},{"name":"Madrid Government (Comunidad de Madrid Spain)","award":["PID2022-136468OB-I00"],"award-info":[{"award-number":["PID2022-136468OB-I00"]}]},{"name":"Madrid Government (Comunidad de Madrid Spain)","award":["PID2022-142015OB-I00"],"award-info":[{"award-number":["PID2022-142015OB-I00"]}]},{"name":"Research and Technological Innovation Regional Programme and by the FEDER\/Ministry of Science and Innovation\u2014Agencia Estatal de Investigacion (AEI)","award":["APBI-CM-UC3M"],"award-info":[{"award-number":["APBI-CM-UC3M"]}]},{"name":"Research and Technological Innovation Regional Programme and by the FEDER\/Ministry of Science and Innovation\u2014Agencia Estatal de Investigacion (AEI)","award":["PID2022-136468OB-I00"],"award-info":[{"award-number":["PID2022-136468OB-I00"]}]},{"name":"Research and Technological Innovation Regional Programme and by the FEDER\/Ministry of Science and Innovation\u2014Agencia Estatal de Investigacion (AEI)","award":["PID2022-142015OB-I00"],"award-info":[{"award-number":["PID2022-142015OB-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Camera calibration is necessary for many machine vision applications. The calibration methods are based on linear or non-linear optimization techniques that aim to find the best estimate of the camera parameters. One of the most commonly used methods in computer vision for the calibration of intrinsic camera parameters and lens distortion (interior orientation) is Zhang\u2019s method. Additionally, the uncertainty of the camera parameters is normally estimated by assuming that their variability can be explained by the images of the different poses of a checkerboard. However, the degree of reliability for both the best parameter values and their associated uncertainties has not yet been verified. Inaccurate estimates of intrinsic and extrinsic parameters during camera calibration may introduce additional biases in post-processing. This is why we propose a novel Bayesian inference-based approach that has allowed us to evaluate the degree of certainty of Zhang\u2019s camera calibration procedure. For this purpose, the a prioriprobability was assumed to be the one estimated by Zhang, and the intrinsic parameters were recalibrated by Bayesian inversion. The uncertainty of the intrinsic parameters was found to differ from the ones estimated with Zhang\u2019s method. However, the major source of inaccuracy is caused by the procedure for calculating the extrinsic parameters. The procedure used in the novel Bayesian inference-based approach significantly improves the reliability of the predictions of the image points, as it optimizes the extrinsic parameters.<\/jats:p>","DOI":"10.3390\/s23187903","type":"journal-article","created":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T04:49:27Z","timestamp":1694753367000},"page":"7903","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Novel Bayesian Inference-Based Approach for the Uncertainty Characterization of Zhang\u2019s Camera Calibration Method"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5817-9718","authenticated-orcid":false,"given":"Ram\u00f3n","family":"Guti\u00e9rrez-Moizant","sequence":"first","affiliation":[{"name":"Mechanical Engineering Department, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan\u00e9s, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5377-0023","authenticated-orcid":false,"given":"Mar\u00eda Jes\u00fas L.","family":"Boada","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Department, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan\u00e9s, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1929-0872","authenticated-orcid":false,"given":"Mar\u00eda","family":"Ram\u00edrez-Berasategui","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Department, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan\u00e9s, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0212-6400","authenticated-orcid":false,"given":"Abdulla","family":"Al-Kaff","sequence":"additional","affiliation":[{"name":"Systems Engineering and Automation, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan\u00e9s, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Aber, J.S., Marzolff, I., and Ries, J.B. (2010). Small-Format Aerial Photography, Elsevier.","DOI":"10.1016\/B978-0-444-53260-2.10008-0"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Champion, E. (2021). Virtual Heritage: A Guide, Ubiquity Press.","DOI":"10.5334\/bck"},{"key":"ref_3","unstructured":"J\u00e4hne, B., and Hau\u00dfecker, H. (2000). Computer Vision and Applications, Elsevier."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1016\/j.cell.2011.11.001","article-title":"Computer vision in cell biology","volume":"147","author":"Danuser","year":"2011","journal-title":"Cell"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Klette, R. (2014). Concise Computer Vision, Springer.","DOI":"10.1007\/978-1-4471-6320-6"},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.isprsjprs.2015.10.006","article-title":"Sensor modelling and camera calibration for close-range photogrammetry","volume":"115","author":"Luhmann","year":"2016","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/S0924-2716(97)00005-1","article-title":"Digital camera self-calibration","volume":"52","author":"Fraser","year":"1997","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Geiger, A., Moosmann, F., Car, \u00d6., and Schuster, B. (2012, January 14\u201318). Automatic camera and range sensor calibration using a single shot. Proceedings of the 2012 IEEE International Conference on Robotics and Automation, St Paul, MN, USA.","DOI":"10.1109\/ICRA.2012.6224570"},{"key":"ref_10","unstructured":"Nyimbili, P.H., Demirel, H., Seker, D., and Erden, T. (2016, January 27\u201330). Structure from motion (sfm)-approaches and applications. Proceedings of the International Scientific Conference on Applied Sciences, Antalya, Turkey."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2684","DOI":"10.1109\/TII.2022.3190366","article-title":"Computer vision enabled building digital twin using building information model","volume":"19","author":"Zhou","year":"2022","journal-title":"IEEE Trans. Ind. Inf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"19954","DOI":"10.1109\/TITS.2022.3182410","article-title":"A review of vision-based traffic semantic understanding in ITSs","volume":"23","author":"Chen","year":"2022","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_13","unstructured":"Orghidan, R., Salvi, J., Gordan, M., and Orza, B. (2012, January 9\u201312). Camera calibration using two or three vanishing points. Proceedings of the 2012 Federated Conference on Computer science and information systems (FedCSIS), Wroclaw, Poland."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Mei, C., and Rives, P. (2007, January 10\u201314). Single view point omnidirectional camera calibration from planar grids. Proceedings of the 2007 IEEE International Conference on Robotics and Automation, Roma, Italy.","DOI":"10.1109\/ROBOT.2007.364084"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1109\/JRA.1987.1087109","article-title":"A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses","volume":"3","author":"Tsai","year":"1987","journal-title":"IEEE J. Robot. Autom."},{"key":"ref_16","unstructured":"Heikkila, J., and Silv\u00e9n, O. (1997, January 17\u201319). A four-step camera calibration procedure with implicit image correction. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Juan, Puerto Rico."},{"key":"ref_17","unstructured":"Sundareswara, R., and Schrater, P.R. (2005, January 13\u201316). Bayesian modelling of camera calibration and reconstruction. Proceedings of the Fifth International Conference on 3-D Digital Imaging and Modeling (3DIM\u201905), Ottawa, ON, Canada."},{"key":"ref_18","unstructured":"Jorio, A., and Dresselhaus, M. (2016). Reference Module in Materials Science and Materials Engineering, Elsevier."},{"key":"ref_19","unstructured":"Rasmussen, K., Kondrup, J.B., Allard, A., Demeyer, S., Fischer, N., Barton, E., Partridge, D., Wright, L., B\u00e4r, M., and Fiebach, H. (2015). Novel Mathematical and Statistical Approaches to Uncertainty Evaluation: Best Practice Guide to Uncertainty Evaluation for Computationally Expensive Models, Euramet."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Wang, C., Qiang, X., Xu, M., and Wu, T. (2022). Recent advances in surrogate modeling methods for uncertainty quantification and propagation. Symmetry, 14.","DOI":"10.3390\/sym14061219"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"109619","DOI":"10.1016\/j.ymssp.2022.109619","article-title":"A state-of-the-art review on uncertainty analysis of rotor systems","volume":"183","author":"Fu","year":"2023","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.mfglet.2022.02.001","article-title":"Bayesian optimization for inverse calibration of expensive computer models: A case study for Johnson-Cook model in machining","volume":"32","author":"Karandikar","year":"2022","journal-title":"Manuf. Lett."},{"key":"ref_23","unstructured":"Zhang, C., and Sun, L. (2022). Bayesian Calibration of the Intelligent Driver Model. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"232502","DOI":"10.1103\/PhysRevLett.122.232502","article-title":"Direct comparison between Bayesian and frequentist uncertainty quantification for nuclear reactions","volume":"122","author":"King","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.patrec.2015.12.008","article-title":"Automated checkerboard detection and indexing using circular boundaries","volume":"71","author":"Bok","year":"2016","journal-title":"Pattern Recognit. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Feng, X., Cao, M., Wang, H., and Collier, M. (2008, January 1\u20133). The comparison of camera calibration methods based on structured-light measurement. Proceedings of the 2008 Congress on Image and Signal Processing, Octeville, France.","DOI":"10.1109\/CISP.2008.163"},{"key":"ref_27","unstructured":"Burger, W. (2016). Zhang\u2019s Camera Calibration Algorithm: In-Depth Tutorial and Implementation, University of Applied Sciences Upper Austria. HGB16-05."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Rojtberg, P., and Kuijper, A. (2018, January 16\u201320). Efficient pose selection for interactive camera calibration. Proceedings of the 2018 IEEE International Symposium on Mixed and Augmented Reality (ISMAR), Munich, Germany.","DOI":"10.1109\/ISMAR.2018.00026"},{"key":"ref_29","unstructured":"Peng, S., and Sturm, P. (November, January 27). Calibration wizard: A guidance system for camera calibration based on modelling geometric and corner uncertainty. Proceedings of the IEEE\/CVF International Conference on Computer Vision, Seoul, Republic of Korea."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Richardson, A., Strom, J., and Olson, E. (2013, January 3\u20137). AprilCal: Assisted and repeatable camera calibration. Proceedings of the 2013 IEEE\/RSJ International Conference on Intelligent Robots and Systems, Tokyo, Japan.","DOI":"10.1109\/IROS.2013.6696595"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1016\/j.procir.2015.12.108","article-title":"Analysis of camera calibration with respect to measurement accuracy","volume":"41","author":"Semeniuta","year":"2016","journal-title":"Procedia Cirp"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s11263-021-01528-x","article-title":"Inferring bias and uncertainty in camera calibration","volume":"130","author":"Hagemann","year":"2022","journal-title":"Int. J. Comput. Vis."},{"key":"ref_33","unstructured":"MATLAB (2022). Version 9.12.0.2039608 (R2022a) Update 5 (R2022a), The MathWorks Inc."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Wang, Y., Li, Y., and Zheng, J. (2010, January 23\u201325). A camera calibration technique based on OpenCV. Proceedings of the 3rd International Conference on Information Sciences and Interaction Sciences, Chengdu, China.","DOI":"10.1109\/ICICIS.2010.5534797"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40068-015-0031-4","article-title":"Evaluation of parameter uncertainties in nonlinear regression using Microsoft Excel Spreadsheet","volume":"4","author":"Hu","year":"2015","journal-title":"Environ. Syst. Res."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Garc\u00eda-Alfonso, H., and C\u00f3rdova-Esparza, D.M. (2018, January 1\u20133). Comparison of uncertainty analysis of the Montecarlo and Latin Hypercube algorithms in a camera calibration model. Proceedings of the 2018 IEEE 2nd Colombian Conference on Robotics and Automation (CCRA), Barranquilla, Colombia.","DOI":"10.1109\/CCRA.2018.8588138"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"012010","DOI":"10.1088\/1757-899X\/1048\/1\/012010","article-title":"Stability analysis of intrinsic camera calibration using probability distributions","volume":"Volume 1048","author":"Zhan","year":"2021","journal-title":"IOP Conference Series: Materials Science and Engineering"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Kainz, O., Jakab, F., Fecil\u2019ak, P., V\u00e1pen\u00edk, R., De\u00e1k, A., and Cymbal\u00e1k, D. (2016, January 24\u201325). Estimation of camera intrinsic matrix parameters and its utilization in the extraction of dimensional units. Proceedings of the 2016 International Conference on Emerging eLearning Technologies and Applications (ICETA), Stary Smokovec, Slovakia.","DOI":"10.1109\/ICETA.2016.7802057"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"509","DOI":"10.5194\/isprsarchives-XL-5-509-2014","article-title":"Method for measuring lens distortion by using pinhole lens","volume":"40","author":"Reznicek","year":"2014","journal-title":"Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci."},{"key":"ref_40","first-page":"51","article-title":"Lens distortion for close-range photogrammetry","volume":"52","author":"Fryer","year":"1986","journal-title":"Photogramm. Eng. Remote Sens."},{"key":"ref_41","first-page":"855","article-title":"Close-range camera calibration","volume":"37","author":"Duane","year":"1971","journal-title":"Photogramm. Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/j.patcog.2007.06.012","article-title":"A new calibration model of camera lens distortion","volume":"41","author":"Wang","year":"2008","journal-title":"Pattern Recognit."},{"key":"ref_43","unstructured":"Sudhanshu, K. (2022, December 01). CarND-Camera-Calibration. Available online: https:\/\/github.com\/udacity\/CarND-Camera-Calibration\/tree\/master."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1111\/rssb.12182","article-title":"A frequentist approach to computer model calibration","volume":"79","author":"Wong","year":"2017","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.compfluid.2017.11.007","article-title":"An efficient Bayesian uncertainty quantification approach with application to k-\u03c9-\u03b3 transition modeling","volume":"161","author":"Zhang","year":"2018","journal-title":"Comput. Fluids"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"105414","DOI":"10.1016\/j.rinp.2022.105414","article-title":"Bayesian inverse uncertainty quantification of the physical model parameters for the spallation neutron source first target station","volume":"36","author":"Radaideh","year":"2022","journal-title":"Results Phys."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Marelli, S., and Sudret, B. (2014, January 13\u201316). UQLab: A framework for uncertainty quantification in Matlab. Proceedings of the 2nd International Conference on Vulnerability and Risk Analysis and Management (ICVRAM2014), Liverpool, UK.","DOI":"10.1061\/9780784413609.257"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"143","DOI":"10.3758\/s13423-016-1015-8","article-title":"A simple introduction to Markov Chain Monte\u2013Carlo sampling","volume":"25","author":"Cassey","year":"2018","journal-title":"Psychon. Bull. Rev."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/j.nucengdes.2018.06.004","article-title":"Inverse uncertainty quantification using the modular Bayesian approach based on Gaussian process, Part 1: Theory","volume":"335","author":"Wu","year":"2018","journal-title":"Nucl. Eng. Des."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Kruschke, J. (2014). Doing Bayesian Data Analysis: A Ttutorial with R, JAGS, and Stan, Academic Press.","DOI":"10.1016\/B978-0-12-405888-0.00008-8"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1198\/jcgs.2009.06134","article-title":"Examples of adaptive MCMC","volume":"18","author":"Roberts","year":"2009","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1186\/s40623-022-01645-y","article-title":"Comparison between the Hamiltonian Monte Carlo method and the Metropolis\u2013Hastings method for coseismic fault model estimation","volume":"74","author":"Yamada","year":"2022","journal-title":"Earth Planets Space"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1548","DOI":"10.1080\/15732479.2019.1628077","article-title":"Adaptive Markov chain Monte Carlo algorithms for Bayesian inference: Recent advances and comparative study","volume":"15","author":"Jin","year":"2019","journal-title":"Struct. Infrastruct. Eng."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1076","DOI":"10.1016\/j.ijrobp.2021.12.011","article-title":"Understanding the differences between Bayesian and frequentist statistics","volume":"112","author":"Mistry","year":"2022","journal-title":"Int. J. Radiat. Oncol. Biol. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1177\/0361198120910149","article-title":"Uncertainty quantification of simplified viscoelastic continuum damage fatigue model using the bayesian inference-based markov chain monte carlo method","volume":"2674","author":"Ding","year":"2020","journal-title":"Transp. Res. Rec."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1088\/0026-1394\/40\/5\/305","article-title":"On use of Bayesian statistics to make the Guide to the Expression of Uncertainty in Measurement consistent","volume":"40","author":"Kacker","year":"2003","journal-title":"Metrologia"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/18\/7903\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:51:33Z","timestamp":1760129493000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/18\/7903"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,15]]},"references-count":56,"journal-issue":{"issue":"18","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["s23187903"],"URL":"https:\/\/doi.org\/10.3390\/s23187903","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,15]]}}}