{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T15:39:26Z","timestamp":1772206766127,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T00:00:00Z","timestamp":1710979200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42074009"],"award-info":[{"award-number":["42074009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["ZR2020MD043"],"award-info":[{"award-number":["ZR2020MD043"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["42074009"],"award-info":[{"award-number":["42074009"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2020MD043"],"award-info":[{"award-number":["ZR2020MD043"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this study, the ill-conditioning of the iterative method for nonlinear models is discussed. Due to the effectiveness of ridge estimation for ill-conditioned problems and the lack of a combination of the H-K formula with the iterative method, the improvement of the LM algorithm is studied in this paper. Considering the LM algorithm for ill-conditioned nonlinear least squares, an improved LM algorithm based on the H-K formula is proposed for image distortion correction using self-calibration. Three finite difference methods are used to approximate the Jacobian matrix, and the H-K formula is used to calculate the damping factor in each iteration. The Brown model, quadratic polynomial model and Fourier model are applied to the self-calibration, and the improved LM algorithm is used to solve the model parameters. In the simulation experiment of space resection of a single image, we evaluate the performance of the LM algorithm based on the gain ratio (LMh) and the improved LM algorithm based on the H-K formula (LMHK), and the accuracy of different models and algorithms is compared. A ridge trace analysis is carried out on the damping factor to illustrate the effects of the improved algorithm in handling ill-conditioning. In the second experiment, the improved algorithm is applied to measure the diameter of a coin using a single camera. The experimental results show that the improved LM algorithm can reach the same or higher accuracy as the LMh algorithm, and it can weaken the ill-conditioning to a certain extent and enhance the stability of the solution. Meanwhile, the applicability of the improved LM algorithm in self-calibration is verified.<\/jats:p>","DOI":"10.3390\/axioms13030209","type":"journal-article","created":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T11:37:22Z","timestamp":1711021042000},"page":"209","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Method for Solving Ill-Conditioned Nonlinear Least Squares Problems and Its Application in Image Distortion Correction Using Self-Calibration"],"prefix":"10.3390","volume":"13","author":[{"given":"Luyao","family":"Wang","sequence":"first","affiliation":[{"name":"College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"given":"Guolin","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,21]]},"reference":[{"key":"ref_1","first-page":"197","article-title":"GNSS Constrained Self-Calibration for Long Corridor UAV Image","volume":"49","author":"Huang","year":"2024","journal-title":"Geomat. Inf. Sci. Wuhan Univ."},{"key":"ref_2","first-page":"855","article-title":"Close-range camera calibration","volume":"37","author":"Brown","year":"1971","journal-title":"Photogramm. Eng."},{"key":"ref_3","unstructured":"Sun, P. (2019). Research on Key Techniques of Large Scale Dynamic Photogrammetry, Beijing University of Posts and Telecommunications."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2532","DOI":"10.3788\/OPE.20172509.2532","article-title":"Self-calibration based on Simplified Brown Nonlinear Camera Model and Modified BFGS Algorithm","volume":"25","author":"Gao","year":"2017","journal-title":"Opt. Precison Eng."},{"key":"ref_5","first-page":"215","article-title":"Comparison of pre- and self-calibrated camera calibration models for UAS-derived nadir imagery for a SfM application","volume":"43","author":"David","year":"2018","journal-title":"Prog. Phys. Geogr. Earth Environ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.isprsjprs.2012.05.004","article-title":"New Rigorous and Flexible Fourier Self-calibration Models for Airborne Camera Calibration","volume":"71","author":"Tang","year":"2012","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_7","first-page":"110","article-title":"Performance Analysis of a Generic Photogrammetric Distortion Model","volume":"41","author":"Sun","year":"2020","journal-title":"Spacecr. Recovery Remote Sens."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bian, Y., Wang, M., Chu, Y., Liu, Z., Chen, J., Xia, Z., and Fang, S. (2021). A Cost Function for the Uncertainty of Matching Point Distribution on Image Registration. Int. J. Geo-Inf., 10.","DOI":"10.3390\/ijgi10070438"},{"key":"ref_9","unstructured":"Tikhonov, A.N., and Arsenin, V.Y. (1977). Solutions of Ill-Posed Problems, John Wiley & Sons."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1007\/BF01937276","article-title":"The Truncated SVD as a Method for Regularization","volume":"27","author":"Hansen","year":"1987","journal-title":"BIT Numer. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.cam.2018.04.049","article-title":"Parameter determination for Tikhonov regularization problems in general form","volume":"343","author":"Park","year":"2018","journal-title":"J. Comput. Appl. Math"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2131","DOI":"10.1109\/TAC.2017.2754474","article-title":"Efficient quadratic penalization through the partial minimization technique","volume":"63","author":"Aravkin","year":"2018","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Guo, H., Liu, G., and Wang, L. (2021). An Improved Tikhonov-Regularized Variable Projection Algorithm for Separable Nonlinear Least Squares. Axioms, 10.","DOI":"10.3390\/axioms10030196"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1080\/00401706.1970.10488635","article-title":"Ridge Regression: Applications to Nonorthogonal Problems","volume":"12","author":"Hoerl","year":"1970","journal-title":"Technometrics"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/s00190-010-0403-5","article-title":"Fast GNSS Ambiguity Resolution as an Ill-posed Problem","volume":"84","author":"Li","year":"2010","journal-title":"J. Geod."},{"key":"ref_16","first-page":"1787","article-title":"Determination of Truncation Parameter based on the Differences of TSVD Parameter Estimates for Ill-posed Problems in Geodesy","volume":"51","author":"Lin","year":"2022","journal-title":"Acta Geod. Cartogr. Sin."},{"key":"ref_17","first-page":"465","article-title":"The Simulation Research on the Tikhonov Regularization Applied in Gravity Field Determination of GOCE Satellite Mission","volume":"39","author":"Xu","year":"2010","journal-title":"Acta Geod. Cartogr. Sin."},{"key":"ref_18","first-page":"883","article-title":"Construction Method of Regularization by Singular Value Decomposition of Design Matrix","volume":"45","author":"Lin","year":"2016","journal-title":"Acta Geod. Cartogr. Sin."},{"key":"ref_19","first-page":"403","article-title":"Ridge estimation algorithm to ill-posed uncertainty adjustment model","volume":"48","author":"Lu","year":"2019","journal-title":"Acta Geod. Cartogr. Sin."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1090\/qam\/10666","article-title":"A method for the solution of certain nonlinear problems in least squares","volume":"2","author":"Levenberg","year":"1944","journal-title":"Q. J. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/0111030","article-title":"An algorithm for the least-squares estimation of nonlinear parameters","volume":"11","author":"Marquardt","year":"1963","journal-title":"J. Soc. Ind. Appl. Math."},{"key":"ref_22","unstructured":"Madsen, K., Nielsen, H.B., and Tingleff, O. (2004). Methods for Non-Linear Least Squares Problems, Informatics and Mathematical Modelling, Technical University of Denmark. [2nd ed.]."},{"key":"ref_23","first-page":"18","article-title":"High-precision indoor positioning based on robust LM visual inertial odometer and pseudosatellite","volume":"51","author":"Yang","year":"2022","journal-title":"Acta Geod. Cartogr. Sin."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/209\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:17:33Z","timestamp":1760105853000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/209"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,21]]},"references-count":23,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["axioms13030209"],"URL":"https:\/\/doi.org\/10.3390\/axioms13030209","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,21]]}}}