{"id":"https://openalex.org/W4393056421","doi":"https://doi.org/10.3390/axioms13030209","title":"A Method for Solving Ill-Conditioned Nonlinear Least Squares Problems and Its Application in Image Distortion Correction Using Self-Calibration","display_name":"A Method for Solving Ill-Conditioned Nonlinear Least Squares Problems and Its Application in Image Distortion Correction Using Self-Calibration","publication_year":2024,"publication_date":"2024-03-21","ids":{"openalex":"https://openalex.org/W4393056421","doi":"https://doi.org/10.3390/axioms13030209"},"language":"en","primary_location":{"id":"doi:10.3390/axioms13030209","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/axioms13030209","pdf_url":"https://www.mdpi.com/2075-1680/13/3/209/pdf?version=1711093726","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/13/3/209/pdf?version=1711093726","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100385092","display_name":"Luyao Wang","orcid":"https://orcid.org/0000-0001-9275-1896"},"institutions":[{"id":"https://openalex.org/I80143920","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98","country_code":"CN","type":"education","lineage":["https://openalex.org/I80143920"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Luyao Wang","raw_affiliation_strings":["College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China"],"affiliations":[{"raw_affiliation_string":"College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China","institution_ids":["https://openalex.org/I80143920"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5090095535","display_name":"Guolin Liu","orcid":"https://orcid.org/0000-0001-5044-1750"},"institutions":[{"id":"https://openalex.org/I80143920","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98","country_code":"CN","type":"education","lineage":["https://openalex.org/I80143920"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Guolin Liu","raw_affiliation_strings":["College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China"],"affiliations":[{"raw_affiliation_string":"College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China","institution_ids":["https://openalex.org/I80143920"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5090095535","https://openalex.org/A5100385092"],"corresponding_institution_ids":["https://openalex.org/I80143920"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.2536,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.46839186,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"13","issue":"3","first_page":"209","last_page":"209"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10638","display_name":"Optical measurement and interference techniques","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10638","display_name":"Optical measurement and interference techniques","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T14158","display_name":"Optical Systems and Laser Technology","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13114","display_name":"Image Processing Techniques and Applications","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2214","display_name":"Media Technology"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.6585699319839478},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.6326751708984375},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6089097857475281},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6035788655281067},{"id":"https://openalex.org/keywords/distortion","display_name":"Distortion (music)","score":0.5223751664161682},{"id":"https://openalex.org/keywords/calibration","display_name":"Calibration","score":0.5023524761199951},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.4957607686519623},{"id":"https://openalex.org/keywords/quadratic-function","display_name":"Quadratic function","score":0.47566813230514526},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.47147300839424133},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.45977553725242615},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.44864165782928467},{"id":"https://openalex.org/keywords/condition-number","display_name":"Condition number","score":0.42888593673706055},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3834748864173889},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.34445661306381226},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.25705650448799133},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.1353287696838379},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12414640188217163}],"concepts":[{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.6585699319839478},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6326751708984375},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6089097857475281},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6035788655281067},{"id":"https://openalex.org/C126780896","wikidata":"https://www.wikidata.org/wiki/Q899871","display_name":"Distortion (music)","level":4,"score":0.5223751664161682},{"id":"https://openalex.org/C165838908","wikidata":"https://www.wikidata.org/wiki/Q736777","display_name":"Calibration","level":2,"score":0.5023524761199951},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4957607686519623},{"id":"https://openalex.org/C166437778","wikidata":"https://www.wikidata.org/wiki/Q50695","display_name":"Quadratic function","level":3,"score":0.47566813230514526},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.47147300839424133},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.45977553725242615},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.44864165782928467},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.42888593673706055},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3834748864173889},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.34445661306381226},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25705650448799133},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.1353287696838379},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12414640188217163},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C194257627","wikidata":"https://www.wikidata.org/wiki/Q211554","display_name":"Amplifier","level":3,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C2776257435","wikidata":"https://www.wikidata.org/wiki/Q1576430","display_name":"Bandwidth (computing)","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms13030209","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/axioms13030209","pdf_url":"https://www.mdpi.com/2075-1680/13/3/209/pdf?version=1711093726","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:bcd1b25fb66045e788741f39eb634443","is_oa":true,"landing_page_url":"https://doaj.org/article/bcd1b25fb66045e788741f39eb634443","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 13, Iss 3, p 209 (2024)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/13/3/209/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms13030209","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms13030209","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/axioms13030209","pdf_url":"https://www.mdpi.com/2075-1680/13/3/209/pdf?version=1711093726","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1231421488","display_name":null,"funder_award_id":"under","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3073698158","display_name":null,"funder_award_id":"ZR2020MD043","funder_id":"https://openalex.org/F4320324174","funder_display_name":"Natural Science Foundation of Shandong Province"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G391238517","display_name":null,"funder_award_id":", and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G591479136","display_name":null,"funder_award_id":"ZR2020","funder_id":"https://openalex.org/F4320324174","funder_display_name":"Natural Science Foundation of Shandong Province"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G922198256","display_name":null,"funder_award_id":"42074009","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320324174","display_name":"Natural Science Foundation of Shandong Province","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4393056421.pdf","grobid_xml":"https://content.openalex.org/works/W4393056421.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W2014725748","https://openalex.org/W2056176734","https://openalex.org/W2066043610","https://openalex.org/W2074177806","https://openalex.org/W2087070363","https://openalex.org/W2256578114","https://openalex.org/W2374042613","https://openalex.org/W2767160078","https://openalex.org/W2805921369","https://openalex.org/W2937340101","https://openalex.org/W2963638855","https://openalex.org/W2994855420","https://openalex.org/W2996829056","https://openalex.org/W3177470681","https://openalex.org/W3196008395","https://openalex.org/W4226060434","https://openalex.org/W4391631791"],"related_works":["https://openalex.org/W3180482864","https://openalex.org/W269925151","https://openalex.org/W2375378023","https://openalex.org/W3096788047","https://openalex.org/W3145558417","https://openalex.org/W2101513891","https://openalex.org/W4386731488","https://openalex.org/W2372562015","https://openalex.org/W2572738404","https://openalex.org/W2024156986"],"abstract_inverted_index":{"In":[0,126,194],"this":[1,47],"study,":[2],"the":[3,6,16,25,31,35,38,41,50,64,83,87,94,112,115,123,127,139,142,147,152,158,163,180,185,188,195,198,205,219,225,231,238,246,249,252,255],"ill-conditioning":[4,239],"of":[5,18,27,30,40,130,133,141,165,187,207,248,254],"iterative":[7,36],"method":[8],"for":[9,21,53,69],"nonlinear":[10,55],"models":[11,167],"is":[12,44,67,90,119,170,176,201,261],"discussed.":[13],"Due":[14],"to":[15,81,92,111,121,183,203,240],"effectiveness":[17],"ridge":[19,173],"estimation":[20],"ill-conditioned":[22,54],"problems":[23],"and":[24,86,106,114,151,162,168,234,244],"lack":[26],"a":[28,134,208,211,241],"combination":[29],"H-K":[32,65,88,159],"formula":[33,66,89,160],"with":[34],"method,":[37],"improvement":[39],"LM":[42,51,60,117,143,154,221,257],"algorithm":[43,52,61,118,144,155,190,200,222,258],"studied":[45],"in":[46,97,191,259],"paper.":[48],"Considering":[49],"least":[56],"squares,":[57],"an":[58],"improved":[59,116,153,189,199,220,256],"based":[62,145,156],"on":[63,146,157,179],"proposed":[68],"image":[70],"distortion":[71],"correction":[72],"using":[73,210],"self-calibration.":[74],"Three":[75],"finite":[76],"difference":[77],"methods":[78],"are":[79,109],"used":[80,91,120],"approximate":[82],"Jacobian":[84],"matrix,":[85],"calculate":[93],"damping":[95,181],"factor":[96,182],"each":[98],"iteration.":[99],"The":[100,214],"Brown":[101],"model,":[102],"quadratic":[103],"polynomial":[104],"model":[105,108,124],"Fourier":[107],"applied":[110,202],"self-calibration,":[113],"solve":[122],"parameters.":[125],"simulation":[128],"experiment":[129],"space":[131],"resection":[132],"single":[135,212],"image,":[136],"we":[137],"evaluate":[138],"performance":[140],"gain":[148],"ratio":[149],"(LMh)":[150],"(LMHK),":[161],"accuracy":[164,229],"different":[166],"algorithms":[169],"compared.":[171],"A":[172],"trace":[174],"analysis":[175],"carried":[177],"out":[178],"illustrate":[184],"effects":[186],"handling":[192],"ill-conditioning.":[193],"second":[196],"experiment,":[197],"measure":[204],"diameter":[206],"coin":[209],"camera.":[213],"experimental":[215],"results":[216],"show":[217],"that":[218],"can":[223,236],"reach":[224],"same":[226],"or":[227],"higher":[228],"as":[230],"LMh":[232],"algorithm,":[233],"it":[235],"weaken":[237],"certain":[242],"extent":[243],"enhance":[245],"stability":[247],"solution.":[250],"Meanwhile,":[251],"applicability":[253],"self-calibration":[260],"verified.":[262]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}