{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T21:10:52Z","timestamp":1792789852564,"version":"4.5.0"},"reference-count":26,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2016,4,28]],"date-time":"2016-04-28T00:00:00Z","timestamp":1461801600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shanghai High Technology Research Program","award":["15JC1402500"],"award-info":[{"award-number":["15JC1402500"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In order to acquire an accurate three-dimensional (3D) measurement, the traditional fringe projection technique applies complex and laborious procedures to compensate for the errors that exist in the vision system. However, the error sources in the vision system are very complex, such as lens distortion, lens defocus, and fringe pattern nonsinusoidality. Some errors cannot even be explained or rendered with clear expressions and are difficult to compensate directly as a result. In this paper, an approach is proposed that avoids the complex and laborious compensation procedure for error sources but still promises accurate 3D measurement. It is realized by the mathematical model extension technique. The parameters of the extended mathematical model for the \u2019phase to 3D coordinates transformation\u2019 are derived using the least-squares parameter estimation algorithm. In addition, a phase-coding method based on a frequency analysis is proposed for the absolute phase map retrieval to spatially isolated objects. The results demonstrate the validity and the accuracy of the proposed flexible fringe projection vision system on spatially continuous and discontinuous objects for 3D measurement.<\/jats:p>","DOI":"10.3390\/s16050612","type":"journal-article","created":{"date-parts":[[2016,4,28]],"date-time":"2016-04-28T10:25:55Z","timestamp":1461839155000},"page":"612","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A Flexible Fringe Projection Vision System with Extended Mathematical Model for Accurate Three-Dimensional Measurement"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4060-1607","authenticated-orcid":false,"given":"Suzhi","family":"Xiao","sequence":"first","affiliation":[{"name":"School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road Minhang District, Shanghai 200240, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei","family":"Tao","sequence":"additional","affiliation":[{"name":"School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road Minhang District, Shanghai 200240, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hui","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road Minhang District, Shanghai 200240, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2016,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.optlaseng.2009.03.008","article-title":"Recent progresses on real-time 3D shape measurement using digital fringe projection techniques","volume":"48","author":"Zhang","year":"2010","journal-title":"Opt. 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