{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:52:04Z","timestamp":1760241124378,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2019,11,27]],"date-time":"2019-11-27T00:00:00Z","timestamp":1574812800000},"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":["51705485"],"award-info":[{"award-number":["51705485"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002851","name":"China Academy of Engineering Physics","doi-asserted-by":"publisher","award":["K1204,K1156"],"award-info":[{"award-number":["K1204,K1156"]}],"id":[{"id":"10.13039\/501100002851","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Traditional filtering methods only focused on improving the peak signal-to-noise ratio of the single fringe pattern, which ignore the filtering effect on phase extraction. Fringe phase-shifting field based fuzzy quotient space-oriented partial differential equations filtering method is proposed to reduce the phase error caused by Gaussian noise while filtering. First, the phase error distribution that is caused by Gaussian noise is analyzed. Furthermore, by introducing the fringe phase-shifting field and the theory of fuzzy quotient space, the modified filtering direction can be adaptively obtained, which transforms the traditional single image filtering into multi-image filtering. Finally, the improved fourth-order oriented partial differential equations with fidelity item filtering method is established. Experiments demonstrated that the proposed method achieves a higher signal-to-noise ratio and lower phase error caused by noise, while also retaining more edge details.<\/jats:p>","DOI":"10.3390\/s19235202","type":"journal-article","created":{"date-parts":[[2019,11,27]],"date-time":"2019-11-27T11:07:00Z","timestamp":1574852820000},"page":"5202","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Fringe Phase-Shifting Field Based Fuzzy Quotient Space-Oriented Partial Differential Equations Filtering Method for Gaussian Noise-Induced Phase Error"],"prefix":"10.3390","volume":"19","author":[{"given":"Changzhi","family":"Yu","sequence":"first","affiliation":[{"name":"Institute of Mechanical Manufacturing Technology, China Academy of Engineering Physics, Mianyang 621999, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fang","family":"Ji","sequence":"additional","affiliation":[{"name":"Institute of Mechanical Manufacturing Technology, China Academy of Engineering Physics, Mianyang 621999, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3398-8708","authenticated-orcid":false,"given":"Junpeng","family":"Xue","sequence":"additional","affiliation":[{"name":"School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yajun","family":"Wang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"18421","DOI":"10.1364\/OE.27.018421","article-title":"High-accuracy 3D shape measurement of translucent objects by fringe projection profilometry","volume":"27","author":"Xu","year":"2019","journal-title":"Opt. 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