{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T19:17:40Z","timestamp":1776280660037,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"21","license":[{"start":{"date-parts":[[2019,10,28]],"date-time":"2019-10-28T00:00:00Z","timestamp":1572220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In a multifrequency phase-shifting (MFPS) algorithm, the temporal phase unwrapping algorithm can extend the unambiguous phase range by transforming the measurement range from a short fringe pitch into an extended synthetic pitch of two different frequencies. However, this undesirably amplifies the uncertainty of measurement, with each single-frequency phase map retaining its measurement uncertainty, which is carried over to the final unwrapped phase maps in fringe-order calculations. This article analyzes possible causes and proposes a new absolute depth measurement algorithm to minimize the propagation of measurement uncertainty. Developed from normalized cross-correlation (NCC), the proposed algorithm can minimize wrong fringe-order calculations in the MFPS algorithm. The experimental results demonstrated that the proposed measurement method could effectively calibrate the wrong fringe order. Moreover, some extremely low signal-to-noise ratio (SNR) regions of a captured image could be correctly reconstructed (for surface profiles). The present findings confirmed measurement precision at one standard deviation below 5.4 \u00b5m, with an absolute distance measurement of 16 mm. The measurement accuracy of the absolute depth could be significantly improved from an unacceptable level of measured errors down to 0.5% of the overall measuring range. Additionally, the proposed algorithm was capable of extracting the absolute phase map in other optical measurement applications, such as distance measurements using interferometry.<\/jats:p>","DOI":"10.3390\/s19214683","type":"journal-article","created":{"date-parts":[[2019,10,28]],"date-time":"2019-10-28T11:26:13Z","timestamp":1572261973000},"page":"4683","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Absolute Depth Measurement Using Multiphase Normalized Cross-Correlation for Precise Optical Profilometry"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8972-128X","authenticated-orcid":false,"given":"Duc-Hieu","family":"Duong","sequence":"first","affiliation":[{"name":"Graduate Institute of Automation Technology, College of Mechanical &amp; Electrical Engineering, No. 1, Section 3, National Taipei University of Technology, Zhong-Xiao E. Rd, Da\u2019an District, Taipei City 10608, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7060-1344","authenticated-orcid":false,"given":"Chin-Sheng","family":"Chen","sequence":"additional","affiliation":[{"name":"Graduate Institute of Automation Technology, College of Mechanical &amp; Electrical Engineering, No. 1, Section 3, National Taipei University of Technology, Zhong-Xiao E. Rd, Da\u2019an District, Taipei City 10608, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9613-9936","authenticated-orcid":false,"given":"Liang-Chia","family":"Chen","sequence":"additional","affiliation":[{"name":"Graduate Institute of Automation Technology, College of Mechanical &amp; Electrical Engineering, No. 1, Section 3, National Taipei University of Technology, Zhong-Xiao E. Rd, Da\u2019an District, Taipei City 10608, Taiwan"},{"name":"Department of Mechanical Engineering, National Taiwan University, No. 1, Section 4, Roosevelt Rd, Da\u2019an District, Taipei City 10617, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,28]]},"reference":[{"key":"ref_1","unstructured":"Gasvik, K.J. (2003). Optical Metrology, John Wiley& Sons Ltd.\/Wiley Online Library. [3rd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3977","DOI":"10.1364\/AO.22.003977","article-title":"Fourier transform profilometry for the automatic measurement of 3-D object shapes","volume":"22","author":"Takeda","year":"1983","journal-title":"Appl. Opt."},{"key":"ref_3","unstructured":"Ghiglia, D.C., and Pritt, M.D. (1998). Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software, John Wiley & Son\/Wiley Online Library."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3268","DOI":"10.1364\/AO.28.003268","article-title":"Noise-immune phase unwrapping algorithm","volume":"28","author":"Huntley","year":"1989","journal-title":"Appl. Opt."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1029\/RS023i004p00713","article-title":"Satellite radar interferometry: Two-dimensional phase unwrapping","volume":"23","author":"Goldstein","year":"1988","journal-title":"Radio Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"720","DOI":"10.1109\/36.499751","article-title":"Interferometric SAR phase unwrapping using Green\u2019s formulation","volume":"34","author":"Fornaro","year":"1996","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2702","DOI":"10.1364\/JOSAA.14.002702","article-title":"Global and local phase-unwrapping techniques: A comparison","volume":"14","author":"Fornaro","year":"1997","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4802","DOI":"10.1364\/AO.39.004802","article-title":"Phase unwrapping with the branch-cut method: role of phase-field direction","volume":"39","author":"Gutmann","year":"2000","journal-title":"Appl. Opt."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1109\/TIP.2006.888351","article-title":"Phase unwrapping via graph cuts","volume":"16","author":"Valadao","year":"2007","journal-title":"IEEE Trans. Image Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7437","DOI":"10.1364\/AO.41.007437","article-title":"Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path","volume":"41","author":"Herraez","year":"2002","journal-title":"Appl. Opt."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"101907","DOI":"10.1117\/1.OE.52.10.101907","article-title":"Statistically guided improvements in speckle phase discontinuity predictions by machine learning systems","volume":"52","author":"Sawaf","year":"2013","journal-title":"Opt. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5439","DOI":"10.1364\/AO.53.005439","article-title":"Phase discontinuity predictions using a machine-learning trained kernel","volume":"53","author":"Sawaf","year":"2014","journal-title":"Appl. Opt."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3188","DOI":"10.1364\/JOSAA.14.003188","article-title":"Error-reduction methods for shape measurement by temporal phase unwrapping","volume":"14","author":"Huntley","year":"1997","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2470","DOI":"10.1364\/AO.21.002470","article-title":"Analysis of the phase unwrapping algorithm","volume":"21","author":"Itoh","year":"1982","journal-title":"Appl. Opt."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3047","DOI":"10.1364\/AO.32.003047","article-title":"Temporal phase-unwrapping algorithm for automated interferogram analysis","volume":"32","author":"Huntley","year":"1993","journal-title":"Appl. Opt."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"8722","DOI":"10.1364\/AO.45.008722","article-title":"Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform","volume":"45","author":"Gdeisat","year":"2006","journal-title":"Appl. Opt."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1016\/j.optlaseng.2009.07.009","article-title":"Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: Recent progress, challenges, and suggested developments","volume":"47","author":"Gdeisat","year":"2009","journal-title":"Opt. Lasers Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2770","DOI":"10.1364\/AO.36.002770","article-title":"Temporal phase unwrapping: Application to surface profiling of discontinuous objects","volume":"36","author":"Saldner","year":"1997","journal-title":"Appl. Opt."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2810","DOI":"10.1364\/AO.26.002810","article-title":"Step height measurement using two-wavelength phase-shifting interferometry","volume":"26","author":"Creath","year":"1987","journal-title":"Appl. Opt."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zhang, S. (2009, January 3\u20134). Digital multiple wavelength phase shifting algorithm. Proceedings of the 2009 SPIE Meeting on Optical Inspection and Metrology for Non-Optics Industries, San Diego, CA, USA.","DOI":"10.1117\/12.823903"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5149","DOI":"10.1364\/OE.19.005149","article-title":"Superfast multifrequency phase-shifting technique with optimal pulse width modulation","volume":"19","author":"Wang","year":"2011","journal-title":"Opt. Express"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"035304","DOI":"10.1088\/0957-0233\/22\/3\/035304","article-title":"Phase invalidity identification framework with the temporal phase unwrapping method","volume":"22","author":"Huang","year":"2011","journal-title":"Meas. Sci. Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.optlaseng.2016.04.022","article-title":"Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review","volume":"85","author":"Zuo","year":"2016","journal-title":"Opt. Lasers Eng."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Cao, P., Xi, J., Yu, Y., and Guo, Q. (2014, January 6). A multiple wavelength unwrapping algorithm for digital fringe profilometry based on spatial shift estimation. Proceedings of the Three-Dimensional Image Processing, Measurement (3DIPM), and Application 2014, San Francisco, CA, USA.","DOI":"10.1117\/12.2038954"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1016\/j.optlaseng.2013.02.012","article-title":"High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection","volume":"51","author":"Zuo","year":"2013","journal-title":"Opt. Lasers Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/0143-8166(91)90055-X","article-title":"Automatic processing of fringe patterns in integer interferometers","volume":"14","author":"Gushov","year":"1991","journal-title":"Opt. Lasers Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"16819","DOI":"10.1364\/OE.22.016819","article-title":"Recovery of absolute height from wrapped phase maps for fringe projection profilometry","volume":"22","author":"Xu","year":"2014","journal-title":"Opt. Express"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"8710","DOI":"10.1364\/AO.56.008710","article-title":"Absolute three-dimensional shape measurement with two-frequency square binary patterns","volume":"56","author":"Jiang","year":"2017","journal-title":"Appl. Opt."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.optlaseng.2018.03.003","article-title":"Absolute phase retrieval methods for digital fringe projection profilometry: A review","volume":"107","author":"Zhang","year":"2018","journal-title":"Opt. Lasers Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.optlaseng.2018.02.012","article-title":"Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping","volume":"106","author":"Servin","year":"2018","journal-title":"Opt. Lasers Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1364\/AO.46.000036","article-title":"Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector","volume":"46","author":"Zhang","year":"2007","journal-title":"Appl. Opt."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1364\/OL.34.000416","article-title":"Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry","volume":"34","author":"Pan","year":"2009","journal-title":"Opt. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"5721","DOI":"10.1364\/AO.55.005721","article-title":"Phase error analysis and compensation for phase shifting profilometry with projector defocusing","volume":"55","author":"Zheng","year":"2016","journal-title":"Appl. Opt."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.optlaseng.2016.06.009","article-title":"Microscopic structured light 3D profilometry: Binary defocusing technique vs. sinusoidal fringe projection","volume":"96","author":"Li","year":"2017","journal-title":"Opt. Lasers Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4258","DOI":"10.1364\/OE.26.004258","article-title":"Instability of projection light source and real-time phase error correction method for phase-shifting profilometry","volume":"26","author":"Chen","year":"2018","journal-title":"Opt. Express"},{"key":"ref_36","unstructured":"Wikipedia (2019, August 26). \u201cCauchy\u2013Schwarz Inequality\u201d. Available online: https:\/\/en.wikipedia.org\/wiki\/Cauchy\u2013Schwarz_inequality."},{"key":"ref_37","unstructured":"Oscar, L. (2019). Discrete Mathematics: An Open Introduction, University of Northern Colorado. [3rd ed.]."},{"key":"ref_38","unstructured":"Rosen, K.H. (2011). Elementary Number Theory, Pearson Education, Inc.\/Addison Wesley."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/21\/4683\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:29:59Z","timestamp":1760189399000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/21\/4683"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,28]]},"references-count":38,"journal-issue":{"issue":"21","published-online":{"date-parts":[[2019,11]]}},"alternative-id":["s19214683"],"URL":"https:\/\/doi.org\/10.3390\/s19214683","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,28]]}}}