{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T14:10:18Z","timestamp":1782483018762,"version":"3.54.5"},"reference-count":57,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2021,11,13]],"date-time":"2021-11-13T00:00:00Z","timestamp":1636761600000},"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>The sphere target played a vital role in terrestrial LiDAR applications, and solving its geometrical center based on point cloud was a widely concerned problem. In this study, we proposed a newly finite random search algorithm for sphere target fitting. Based on the point cloud data and the geometric characteristics of the sphere target, the algorithm realized the target sphere fitting from the perspective of probability and statistics with the help of parameter estimation. Firstly, an initial constraint space was constructed, and the initial center and radius were determined by finite random search. Then, the optimal spherical center and radius were determined gradually through continuous iterative optimization. We tested the algorithm with the simulated and realistic point cloud. Experimental results showed that the proposed algorithm could be effectively applied to all kinds of point cloud fitting. When the coverage rate was bigger than 30%, the fitting accuracy could reach within 0.01 mm for all kinds of point clouds. When the coverage rate was less than 20%, the fitting accuracy can reach \u00b11 mm, although it was reduced to a certain extent.<\/jats:p>","DOI":"10.3390\/s21227546","type":"journal-article","created":{"date-parts":[[2021,11,14]],"date-time":"2021-11-14T20:51:53Z","timestamp":1636923113000},"page":"7546","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["An Algorithm for Fitting Sphere Target of Terrestrial LiDAR"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3483-8955","authenticated-orcid":false,"given":"Yintao","family":"Shi","sequence":"first","affiliation":[{"name":"School of Geomatics Science and Technology, Nanjing Tech University, Nanjing 211816, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gang","family":"Zhao","sequence":"additional","affiliation":[{"name":"Jiangsu Hydraulic Research Institute, Nanjing 210017, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maomei","family":"Wang","sequence":"additional","affiliation":[{"name":"Jiangsu Hydraulic Research Institute, Nanjing 210017, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Xu","sequence":"additional","affiliation":[{"name":"Jiangsu Hydraulic Research Institute, Nanjing 210017, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dadong","family":"Zhu","sequence":"additional","affiliation":[{"name":"Jiangsu Hydraulic Research Institute, Nanjing 210017, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1080\/17452750802047917","article-title":"From a 3D point cloud to an engineering CAD model: A knowledge-product-based approach for reverse engineering","volume":"3","author":"Durupt","year":"2008","journal-title":"Virtual Phys. Prototyp."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1144\/0016-76492007-100","article-title":"Terrestrial laser scanning in geology; data acquisition, processing and accuracy considerations","volume":"165","author":"Buckley","year":"2008","journal-title":"J. Geol Soc. Lond."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"e2210","DOI":"10.1002\/stc.2210","article-title":"Surface damage quantification of postearthquake building based on terrestrial laser scan data","volume":"25","author":"Dai","year":"2018","journal-title":"Struct. Control. Health Monit."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.aei.2019.02.007","article-title":"Applications of 3D point cloud data in the construction industry: A fifteen-year review from 2004 to 2018","volume":"39","author":"Wang","year":"2019","journal-title":"Adv. Eng. Inform."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Jo, Y., and Hong, S. (2019). Three-dimensional digital documentation of cultural heritage site based on the convergence of terrestrial laser scanning and unmanned aerial vehicle photogrammetry. ISPRS Int. J. Geo-Inf., 8.","DOI":"10.3390\/ijgi8020053"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.isprsjprs.2018.06.014","article-title":"Spherical target-based calibration of terrestrial laser scanner intensity. Application to colour information computation","volume":"144","author":"Bretagne","year":"2018","journal-title":"ISPRS J. Photogramm."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"T\u00f3th, T., Pusztai, Z., and Hajder, L. (August, January 31). Automatic LiDAR-camera calibration of extrinsic parameters using a spherical target. Proceedings of the 2020 IEEE International Conference on Robotics and Automation (ICRA), Paris, France.","DOI":"10.1109\/ICRA40945.2020.9197316"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"8816063","DOI":"10.1155\/2021\/8816063","article-title":"Accurate calibration of a self-developed vehicle-borne LiDAR scanning system","volume":"2021","author":"Guo","year":"2021","journal-title":"J. Sens."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"072001","DOI":"10.1088\/1361-6501\/abdae3","article-title":"Performance evaluation of terrestrial laser scanners\u2014A review","volume":"32","author":"Muralikrishnan","year":"2021","journal-title":"Meas. Sci. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1007\/s001700050039","article-title":"Registration and data merging for multiple sets of scan data","volume":"15","author":"Lai","year":"1999","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"083588","DOI":"10.1117\/1.JRS.8.083588","article-title":"Automatic registration of laser point cloud using precisely located sphere targets","volume":"8","author":"Wang","year":"2014","journal-title":"J. Appl. Remote Sens."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1109\/TPAMI.2007.45","article-title":"Camera calibration from images of spheres","volume":"29","author":"Zhang","year":"2007","journal-title":"IEEE Trans. Pattern Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3544","DOI":"10.1109\/TIM.2009.2018011","article-title":"Fitting spheres to range data from 3-D imaging systems","volume":"58","author":"Franaszek","year":"2009","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1109\/TIP.2010.2063035","article-title":"A stratified approach for camera calibration using spheres","volume":"20","author":"Wong","year":"2011","journal-title":"IEEE Trans. Image Process"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1109\/LRA.2017.2773668","article-title":"Quadcopter tracks quadcopter via real-time shape fitting","volume":"3","author":"Epstein","year":"2018","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1109\/TAES.2013.120107","article-title":"Extended object tracking with random hypersurface models","volume":"50","author":"Baum","year":"2014","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Epstein, D., and Feldman, D. (2020). Sphere fitting with applications to machine tracking. Algorithms, 13.","DOI":"10.3390\/a13080177"},{"key":"ref_18","unstructured":"Forbes, A.B. (1989). Robust Circle and Sphere Fitting by Least Squares, National Physical Laboratory Teddington."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0010-4655(94)90082-5","article-title":"Computing circles and spheres of arithmitic least squares","volume":"81","author":"Nievergelt","year":"1994","journal-title":"Comput. Phys. Commun."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1023\/A:1022675403441","article-title":"Least-square fitting with spheres","volume":"96","year":"1998","journal-title":"J. Optim. Theory Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"633","DOI":"10.6028\/jres.103.043","article-title":"Least-squares fitting algorithms of the NIST algorithm testing system","volume":"103","author":"Shakarji","year":"1998","journal-title":"J. Res. Natl. Inst. Stand. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2283","DOI":"10.1016\/S0031-3203(00)00152-7","article-title":"Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola","volume":"34","author":"Ahn","year":"2001","journal-title":"Pattern Recognit."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.autcon.2008.08.003","article-title":"Fast automatic registration of range images from 3D imaging systems using sphere targets","volume":"18","author":"Franaszek","year":"2009","journal-title":"Autom. Constr."},{"key":"ref_24","unstructured":"Clouse, D., and Padgett, C. (2002, January 9\u201316). Estimating the position of a sphere from range images. Proceedings of the IEEE Aerospace Conference, Big Sky, MT, USA."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Witzgall, C., Cheok, G.S., and Kearsley, A.J. (2006). Recovering circles and spheres from point data. Perspectives in Operations Research, Springer US.","DOI":"10.1007\/978-0-387-39934-8_22"},{"key":"ref_26","unstructured":"Sumith, Y. (2015). Fast geometric fit algorithm for sphere using exact solution. arXiv."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1016\/j.measurement.2019.02.025","article-title":"Novel method for sphere target detection and center estimation from mobile terrestrial laser scanner data","volume":"137","author":"Liu","year":"2019","journal-title":"Measurement"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"015012","DOI":"10.1088\/1361-6501\/abae3c","article-title":"A new high-precision sphere-fitting method with small segment angles","volume":"32","author":"Fei","year":"2020","journal-title":"Meas. Sci. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1109\/LSP.2021.3051851","article-title":"Hypersphere fitting from noisy data using an EM algorithm","volume":"28","author":"Lesouple","year":"2021","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1007\/s11200-014-0267-x","article-title":"On total least squares for quadratic form estimation","volume":"59","author":"Fang","year":"2015","journal-title":"Stud. Geophys. Geod."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"113","DOI":"10.2478\/v10156-011-0036-5","article-title":"Weighted total least squares formulated by standard least squares theory","volume":"2","author":"Jazaeri","year":"2012","journal-title":"J. Geod. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2283","DOI":"10.1016\/j.sigpro.2007.04.004","article-title":"Overview of total least-squares methods","volume":"87","author":"Markovsky","year":"2007","journal-title":"Signal Process."},{"key":"ref_33","first-page":"103","article-title":"Total least squares fitting with quadrics","volume":"11","author":"Spath","year":"2004","journal-title":"Pure Appl. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/j.procir.2016.02.104","article-title":"Theory and algorithm for planar datum establishment using constrained total least-squares","volume":"43","author":"Shakarji","year":"2016","journal-title":"Procedia CIRP"},{"key":"ref_35","unstructured":"Hopp, T.H., and Reeve, C.P. (1996). An Algorithm for Computing the Minimum Covering Sphere in Any Dimension, NIST."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/S0262-8856(96)01112-2","article-title":"Parameter estimation techniques: A tutorial with application to conic fitting","volume":"15","author":"Zhang","year":"1997","journal-title":"Image Vis. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1007\/s00211-009-0267-x","article-title":"Median spheres: Theory, algorithms, applications","volume":"114","author":"Nievergelt","year":"2010","journal-title":"Numer. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1007\/s00371-015-1157-0","article-title":"eSphere: Extracting spheres from unorganized point clouds","volume":"32","author":"Denis","year":"2016","journal-title":"Vis. Comput."},{"key":"ref_39","first-page":"W2","article-title":"Terrestrial laser scanning\u2013new perspectives in 3D surveying","volume":"36","author":"Mettenleiter","year":"2004","journal-title":"Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci."},{"key":"ref_40","first-page":"2273","article-title":"Research and implementation for scattered point cloud data denoising method","volume":"9","author":"Shi","year":"2016","journal-title":"Int. J. Earth Sci. Eng."},{"key":"ref_41","unstructured":"Milton, J.S., and Arnold, J.C. (1990). Introduction to Probability and Statistics, McGraw-Hill."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Dekking, F.M., Kraaikamp, C., Lopuha\u00e4, H.P., and Meester, L.E. (2005). A Modern Introduction to Probability and Statistics: Understanding Why and How, Springer Science & Business Media.","DOI":"10.1007\/1-84628-168-7"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"105001","DOI":"10.1088\/1361-6501\/aa8011","article-title":"Methods and considerations to determine sphere center from terrestrial laser scanner point cloud data","volume":"28","author":"Rachakonda","year":"2017","journal-title":"Meas. Sci. Technol."},{"key":"ref_44","unstructured":"Scheaffer, R.L., Mulekar, M.S., and McClave, J.T. (2010). Probability and Statistics for Engineers, Cengage Learning."},{"key":"ref_45","unstructured":"Montgomery, D.C., and Runger, G.C. (2018). Applied Statistics and Probability for Engineers, Wiley Hoboken."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Uren, J., and Price, W.F. (2010). Surveying for Engineers, Macmillan International Higher Education.","DOI":"10.1057\/978-1-137-05279-7"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Ghilani, C.D. (2017). Adjustment Computations: Spatial Data Analysis, John Wiley & Sons.","DOI":"10.1002\/9781119390664"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1002\/9781444311952.ch2","article-title":"Principles of 3D laser scanning","volume":"1","author":"Heritage","year":"2009","journal-title":"Laser Scanning Environ. Sci."},{"key":"ref_49","unstructured":"Kong, L., and Lv, F. (2021). Application of 3D Laser Scanning Technology in Engineering Field, EDP Sciences."},{"key":"ref_50","unstructured":"Marsh, D. (2005). Applied Geometry for Computer Graphics and CAD, Springer Science & Business Media."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Elman, R.S., Karpenko, N., and Merkurjev, A. (2008). The Algebraic and Geometric Theory of Quadratic Forms, American Mathematical Society.","DOI":"10.1090\/coll\/056"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"563","DOI":"10.4007\/annals.2021.193.2.4","article-title":"Integer multiplication in time O (n log n)","volume":"193","author":"Harvey","year":"2021","journal-title":"Ann. Math."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s00190-007-0190-9","article-title":"On weighted total least-squares adjustment for linear regression","volume":"82","author":"Schaffrin","year":"2008","journal-title":"J. Geodesy"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1007\/s00190-013-0643-2","article-title":"Weighted total least squares: Necessary and sufficient conditions, fixed and random parameters","volume":"87","author":"Fang","year":"2013","journal-title":"J. Geodesy"},{"key":"ref_55","unstructured":"Van Huffel, S., and Lemmerling, P. (2013). Total Least Squares and Errors-In-Variables Modeling: Analysis, Algorithms and Applications, Springer Science & Business Media."},{"key":"ref_56","unstructured":"Lu, W., and Wang, H. (2021). Approximation and quadrature by weighted least squares polynomials on the sphere. arXiv."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Mahboub, V., and Ebrahimzadeh, S. (2021). Non-linear block least-squares adjustment for a large number of observations. Surv. Rev., 1\u201311.","DOI":"10.1080\/00396265.2021.1970916"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/22\/7546\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:29:41Z","timestamp":1760167781000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/22\/7546"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,13]]},"references-count":57,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["s21227546"],"URL":"https:\/\/doi.org\/10.3390\/s21227546","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,13]]}}}