{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T18:16:11Z","timestamp":1772043371720,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2021,9,28]],"date-time":"2021-09-28T00:00:00Z","timestamp":1632787200000},"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":["62075143"],"award-info":[{"award-number":["62075143"]}],"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":["U20A20215"],"award-info":[{"award-number":["U20A20215"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Key Scientific Apparatus Development Project of China","award":["2013YQ490879"],"award-info":[{"award-number":["2013YQ490879"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Crossed-grating phase-shifting profilometry (CGPSP) has great utility in three-dimensional shape measurement due to its ability to acquire horizontal and vertical phase maps in a single measurement. However, CGPSP is extremely sensitive to the non-linearity effect of a digital fringe projection system, which is not studied in depth yet. In this paper, a mathematical model is established to analyze the phase error caused by the non-linearity effect. Subsequently, two methods used to eliminate the non-linearity error are discussed in detail. To be specific, a double five-step algorithm based on the mathematical model is proposed to passively suppress the second non-linearity. Furthermore, a precoding gamma correction method based on probability distribution function is introduced to actively attenuate the non-linearity of the captured crossed fringe. The comparison results show that the active gamma correction method requires less fringe patterns and can more effectively reduce the non-linearity error compared with the passive method. Finally, employing CGPSP with gamma correction, a faster and reliable inverse pattern projection is realized with less fringe patterns.<\/jats:p>","DOI":"10.3390\/s21196475","type":"journal-article","created":{"date-parts":[[2021,9,28]],"date-time":"2021-09-28T21:39:29Z","timestamp":1632865169000},"page":"6475","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Phase Error Analysis and Correction for Crossed-Grating Phase-Shifting Profilometry"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6725-9955","authenticated-orcid":false,"given":"Fuqian","family":"Li","sequence":"first","affiliation":[{"name":"Department of Opto-Electronics, Sichuan University, Chengdu 610065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenjing","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Opto-Electronics, Sichuan University, Chengdu 610065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.optlaseng.2018.04.019","article-title":"Phase shifting algorithms for fringe projection profilometry: A review","volume":"109","author":"Zuo","year":"2018","journal-title":"Opt. 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