{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T02:03:13Z","timestamp":1776132193067,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T00:00:00Z","timestamp":1672963200000},"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":["12103075"],"award-info":[{"award-number":["12103075"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Three-dimensional (3D) shape measurement for specular surfaces is becoming increasingly important in various applications. A novel orthogonal dual-frequency fringe is proposed in the specular surface shape measurement to overcome the phase jumping and discontinuities in spatial phase unwrapping. The fringe recalibrated high-accuracy phase information from its high-frequency fringe component with low-ambiguity phase information from its low-frequency fringe component. An improved Fourier transform deflectometry method based on the orthogonal dual-frequency fringe is proposed to measure 3D specular surface shapes. Simulation results showed that the orthogonal dual-frequency Fourier transform deflectometry (ODD) method could precisely reconstruct flat surfaces with an error of 2.16 nm rms, and concave surfaces with an error of 1.86 \u03bcm rms. Experimental results showed that the reconstructed shapes of both the flat mirror and the concave mirror measured by the ODD measurement system were highly comparable to those obtained by the phase-measuring deflectometry (PMD) method. This new fringe provides a distinctive approach to structured pattern construction and reduces the phase unwrapping ambiguities in specular surface shape measurement. The ODD method can achieve accurate 3D shape measurement for specular surfaces by sampling only one fringe, providing a possible basis for future real-time measurement of specular surfaces.<\/jats:p>","DOI":"10.3390\/s23020674","type":"journal-article","created":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T06:38:27Z","timestamp":1673246307000},"page":"674","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Specular Surface Shape Measurement with Orthogonal Dual-Frequency Fourier Transform Deflectometry"],"prefix":"10.3390","volume":"23","author":[{"given":"Zhiming","family":"Li","sequence":"first","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China"},{"name":"School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dayi","family":"Yin","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuanyu","family":"Yang","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Quan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huixing","family":"Gong","sequence":"additional","affiliation":[{"name":"Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China"},{"name":"School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6457","DOI":"10.1364\/AO.56.006457","article-title":"Measurement of picometer-scale mirror dynamics","volume":"56","author":"Saif","year":"2017","journal-title":"Appl. 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