{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T12:41:55Z","timestamp":1785415315646,"version":"3.56.0"},"reference-count":30,"publisher":"Optica Publishing Group","issue":"7","license":[{"start":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T00:00:00Z","timestamp":1781568000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/doi.org\/10.1364\/OA_License_v2#VOR"},{"start":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T00:00:00Z","timestamp":1781568000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/opg.optica.org\/policies\/opg-tdm-policy.json"}],"funder":[{"DOI":"10.13039\/501100004731","name":"Zhejiang Provincial Natural Science Foundation","doi-asserted-by":"crossref","award":["LR25F050002"],"award-info":[{"award-number":["LR25F050002"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"crossref"},{"id":"https:\/\/ror.org\/04aqat463","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Leading Innovative and Entrepreneur Team Introduction Program of Zhejiang","award":["2024R01001"],"award-info":[{"award-number":["2024R01001"]}]}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"short-container-title":["J. Opt. Soc. Am. A"],"published-print":{"date-parts":[[2026,7,1]]},"abstract":"<jats:p>Retrieving standard circular Zernike coefficients from wavefronts measured over non-standard circular apertures is crucial for modern optical engineering. While direct least-squares fitting and the orthogonal transformation method are commonly used, a systematic comparison of their numerical stability, computational costs, and noise robustness remains lacking. In this paper, we present a comparative study of five established algorithms [least-squares, Gram\u2013Schmidt, Cholesky factorization, QR factorization, and singular value decomposition (SVD)], introduce two short-recurrence variants to investigate the computational speed limits of recursive orthogonalization, and rigorously prove their mathematical equivalence under exact arithmetic and characterize how floating-point errors cause this equivalence to break down differently across methods. However, simulations on annular and eccentric apertures reveal that although all methods are reliable under well-conditioned masks, least-squares and Cholesky suffer precision loss and numerical collapse under ill-conditioned apertures. Furthermore, non-regularized methods suffer from noise amplification, making truncated SVD the most reliable method among those evaluated for retrieving physically meaningful coefficients. Based on these findings, we provide a practical guide for algorithm selection in optical engineering, helping researchers choose appropriate methods based on their specific measurement conditions.<\/jats:p>","DOI":"10.1364\/josaa.601210","type":"journal-article","created":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T19:00:10Z","timestamp":1779908410000},"page":"1053","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Numerical stability and efficiency in Zernike fitting for non-standard circular apertures"],"prefix":"10.1364","volume":"43","author":[{"given":"Churui","family":"Li","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00a2xv884","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gongjian","family":"Guo","sequence":"additional","affiliation":[{"name":"Tengtai Energy Technology Co., Ltd."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3931-6884","authenticated-orcid":true,"given":"Xiang","family":"Hao","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00a2xv884","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"285","published-online":{"date-parts":[[2026,6,16]]},"reference":[{"key":"josaa-43-7-1053-R2","doi-asserted-by":"publisher","first-page":"822","DOI":"10.1117\/12.924271","type":"journal-article","volume":"8442","author":"Feinberg","year":"2012","journal-title":"Proc. SPIE"},{"key":"josaa-43-7-1053-R3","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1117\/12.2572878","type":"journal-article","volume":"11517","author":"de Winter","year":"2020","journal-title":"Proc. SPIE"},{"key":"josaa-43-7-1053-R4","doi-asserted-by":"publisher","first-page":"023801","DOI":"10.1117\/1.JMM.21.2.023801","type":"journal-article","volume":"21","author":"de Winter","year":"2022","journal-title":"J. Micro\/Nanopatterning Mater. Metrol."},{"key":"josaa-43-7-1053-R5","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1117\/12.354219","type":"journal-article","volume":"3492","author":"Peng","year":"1999","journal-title":"Proc. SPIE"},{"key":"josaa-43-7-1053-R6","doi-asserted-by":"publisher","first-page":"2841","DOI":"10.1117\/1.1814767","type":"journal-article","volume":"43","author":"Miller","year":"2004","journal-title":"Opt. Eng."},{"key":"josaa-43-7-1053-R7","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1364\/JOSAA.24.000569","type":"journal-article","volume":"24","author":"Lundstr\u00f6m","year":"2007","journal-title":"J. Opt. Soc. Am. A"},{"key":"josaa-43-7-1053-R8","doi-asserted-by":"publisher","first-page":"2192","DOI":"10.1364\/JOSAA.26.002192","type":"journal-article","volume":"26","author":"Lundstr\u00f6m","year":"2009","journal-title":"J. Opt. Soc. Am. A"},{"key":"josaa-43-7-1053-R9","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1117\/12.928387","type":"journal-article","volume":"8486","author":"Zhang","year":"2012","journal-title":"Proc. SPIE"},{"key":"josaa-43-7-1053-R10","doi-asserted-by":"publisher","first-page":"C88","DOI":"10.1364\/AO.52.000C88","type":"journal-article","volume":"52","author":"Wang","year":"2013","journal-title":"Appl. Opt."},{"key":"josaa-43-7-1053-R11","doi-asserted-by":"publisher","first-page":"8121","DOI":"10.1364\/AO.33.008121","type":"journal-article","volume":"33","author":"Mahajan","year":"1994","journal-title":"Appl. Opt."},{"key":"josaa-43-7-1053-R12","doi-asserted-by":"publisher","first-page":"1083","DOI":"10.1364\/JOSA.64.001083","type":"journal-article","volume":"64","author":"Tatian","year":"1974","journal-title":"J. Opt. Soc. Am."},{"key":"josaa-43-7-1053-R13","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1364\/JOSA.71.000075","type":"journal-article","volume":"71","author":"Mahajan","year":"1981","journal-title":"J. Opt. Soc. Am."},{"key":"josaa-43-7-1053-R14","doi-asserted-by":"publisher","first-page":"2462","DOI":"10.1364\/OL.31.002462","type":"journal-article","volume":"31","author":"Mahajan","year":"2006","journal-title":"Opt. Lett."},{"key":"josaa-43-7-1053-R15","doi-asserted-by":"publisher","first-page":"2994","DOI":"10.1364\/JOSAA.24.002994","type":"journal-article","volume":"24","author":"Mahajan","year":"2007","journal-title":"J. Opt. Soc. Am. A"},{"key":"josaa-43-7-1053-R16","volume-title":"Optical Imaging and Aberrations, Part III: Wavefront Analysis","author":"Mahajan","year":"2013"},{"key":"josaa-43-7-1053-R17","doi-asserted-by":"crossref","DOI":"10.1002\/9780470135976","volume-title":"Optical Shop Testing","author":"Malacara","year":"2007"},{"key":"josaa-43-7-1053-R18","doi-asserted-by":"crossref","DOI":"10.1520\/STP33971S","article-title":"A computer program for analysis of interferometric data","volume-title":"Optical Interferograms: Reduction and Interpretation, ASTM Special Technical Publication 666","author":"Loomis","year":"1978"},{"key":"josaa-43-7-1053-R19","doi-asserted-by":"publisher","first-page":"1832","DOI":"10.1364\/AO.33.001832","type":"journal-article","volume":"33","author":"Swantner","year":"1994","journal-title":"Appl. Opt."},{"key":"josaa-43-7-1053-R20","doi-asserted-by":"publisher","first-page":"1510","DOI":"10.1364\/AO.19.001510","type":"journal-article","volume":"19","author":"Wang","year":"1980","journal-title":"Appl. Opt."},{"key":"josaa-43-7-1053-R21","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1364\/OL.32.000074","type":"journal-article","volume":"32","author":"Dai","year":"2007","journal-title":"Opt. Lett."},{"key":"josaa-43-7-1053-R22","doi-asserted-by":"publisher","first-page":"054112","DOI":"10.1117\/1.OE.63.5.054112","type":"journal-article","volume":"63","author":"Chai","year":"2024","journal-title":"Opt. Eng."},{"key":"josaa-43-7-1053-R23","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1117\/12.677984","type":"journal-article","volume":"6150","author":"Chang","year":"2006","journal-title":"Proc. SPIE"},{"key":"josaa-43-7-1053-R24","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719574","volume-title":"Numerical Linear Algebra","author":"Trefethen","year":"1997"},{"key":"josaa-43-7-1053-R25","volume-title":"Matrix Computations","author":"Golub","year":"2013"},{"key":"josaa-43-7-1053-R26","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1017\/S0305004100030401","type":"journal-article","volume":"51","author":"Penrose","year":"1955","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"josaa-43-7-1053-R27","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1364\/JOSA.66.000207","type":"journal-article","volume":"66","author":"Noll","year":"1976","journal-title":"J. Opt. Soc. Am."},{"key":"josaa-43-7-1053-R28","volume-title":"Solutions of Ill-Posed Problems","author":"Tikhonov","year":"1977"},{"key":"josaa-43-7-1053-R29","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/BF00330404","type":"journal-article","volume":"21","author":"Hewitt","year":"1979","journal-title":"Arch. for Hist. Exact Sci."},{"key":"josaa-43-7-1053-R30","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1364\/AO.37.000140","type":"journal-article","volume":"37","author":"Chanan","year":"1998","journal-title":"Appl. Opt."},{"key":"josaa-43-7-1053-R31","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1364\/OSAC.1.000715","type":"journal-article","volume":"1","author":"Janin-Potiron","year":"2018","journal-title":"OSA Continuum"}],"container-title":["Journal of the Optical Society of America A"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/opg.optica.org\/viewmedia.cfm?URI=josaa-43-7-1053&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T16:17:06Z","timestamp":1781626626000},"score":1,"resource":{"primary":{"URL":"https:\/\/opg.optica.org\/abstract.cfm?URI=josaa-43-7-1053"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,16]]},"references-count":30,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2026]]},"published-print":{"date-parts":[[2026]]}},"URL":"https:\/\/doi.org\/10.1364\/josaa.601210","relation":{},"ISSN":["1084-7529","1520-8532"],"issn-type":[{"value":"1084-7529","type":"print"},{"value":"1520-8532","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,16]]},"assertion":[{"value":"Optica Publishing Group","name":"publisher","label":"This article is maintained by"},{"value":"https:\/\/doi.org\/10.1364\/JOSAA.601210","name":"articlelink","label":"Crossref DOI link to publisher maintained version"},{"value":"research-article","name":"content_type","label":"Article type"},{"value":"Screened by Similarity Check","name":"cross_check","label":"Similarity check"},{"value":"Yes","order":0,"name":"peer_reviewed","label":"Peer reviewed","group":{"name":"peer_review","label":"Peer review"}},{"value":"Single blind","order":1,"name":"review_process","label":"Review process","group":{"name":"peer_review","label":"Peer review"}},{"value":"8 April 2026","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"27 May 2026","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"16 June 2026","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}},{"value":"\u00a9 2026 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI) training, and similar technologies, are reserved.","name":"copyright","label":"Copyright"}]}}