Abstract

This paper makes use of Hilbert transform to analyze and compensate the phase error caused by the nonlinear effect in phase shifting profilometry (PSP). The characteristics of the phase error distribution in Hilbert transform domain was analyzed and compared with spatial domain. A simple and flexible phase error compensation method was proposed to directly process the phase-shifting fringe images without any auxiliary conditions or complicated computation. Experimental results demonstrated that the phase error can be reduced by about 80% in three-step PSP, and more than 95% in four or more step PSP, which verified the effectiveness, flexibility, robustness and automation of the proposed phase error compensation method.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  34. M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
    [Crossref]
  35. C. A. Poynton, “‘Gamma’ and its disguises: the nonlinear mappings of intensity in perception, CRTs, film, and video,” SMPTE Journal 102(12), 1099–1108 (1993).
    [Crossref]
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    [Crossref]

2014 (4)

H. Cui, Z. Zhao, Y. Wu, N. Dai, X. Cheng, and L. Zhang, “Digital fringe image gamma modeling and new algorithm for phase error compensation,” Optik (Stuttg.) 125(24), 7175–7181 (2014).
[Crossref]

D. Zheng and F. Da, “Absolute phase retrieval for defocused fringe projection three-dimensional measurement,” Opt. Commun. 312, 302–311 (2014).
[Crossref]

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
[Crossref]

M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
[Crossref]

2013 (1)

2012 (2)

X. Zhang, L. Zhu, Y. Li, and D. Tu, “Generic nonsinusoidal fringe model and gamma calibration in phase measuring profilometry,” J. Opt. Soc. Am. A 29(6), 1047–1058 (2012).
[Crossref] [PubMed]

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285(5), 533–538 (2012).
[Crossref]

2011 (2)

2010 (7)

2009 (2)

2007 (1)

2006 (1)

2003 (3)

V. Madjarova, H. Kadono, and S. Toyooka, “Dynamic electronic speckle pattern interferometry (DESPI) phase analyses with temporal Hilbert transform,” Opt. Express 11(6), 617–623 (2003).
[Crossref] [PubMed]

P. S. Huang, C. Zhang, and F. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42(1), 163–168 (2003).
[Crossref]

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[Crossref]

2001 (4)

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189(1–3), 21–29 (2001).
[Crossref]

M. A. Sutton, W. Zhao, S. R. McNeill, H. W. Schreier, and Y. J. Chao, “Development and assessment of a single-image fringe projection method for dynamic applications,” Exp. Mech. 41(3), 205–217 (2001).
[Crossref]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1862–1870 (2001).
[Crossref] [PubMed]

K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1871–1881 (2001).
[Crossref] [PubMed]

1999 (3)

S. L. Marple, “Computing the discrete-time ‘analytic’ signal via FFT,” IEEE Trans. Signal Process. 47(9), 2600–2603 (1999).
[Crossref]

C. R. Coggrave and J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38(9), 1573–1581 (1999).
[Crossref]

S. Kakunai, T. Sakamoto, and K. Iwata, “Profile measurement taken with liquid-crystal gratings,” Appl. Opt. 38(13), 2824–2828 (1999).
[Crossref] [PubMed]

1995 (1)

1993 (1)

C. A. Poynton, “‘Gamma’ and its disguises: the nonlinear mappings of intensity in perception, CRTs, film, and video,” SMPTE Journal 102(12), 1099–1108 (1993).
[Crossref]

1992 (1)

X.-Y. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94(6), 561–573 (1992).
[Crossref]

1990 (1)

D. A. Zweig and R. E. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” Proc. SPIE 1333, 295–302 (1990).
[Crossref]

Asundi, A.

Baker, M. J.

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of gamma non-linear luminance effects for digital video projection phase measuring profilometers”, in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE, 2008), pp. 496–501.
[Crossref]

Barnes, J. C.

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Bone, D. J.

Chao, Y. J.

M. A. Sutton, W. Zhao, S. R. McNeill, H. W. Schreier, and Y. J. Chao, “Development and assessment of a single-image fringe projection method for dynamic applications,” Exp. Mech. 41(3), 205–217 (2001).
[Crossref]

Chen, L.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285(5), 533–538 (2012).
[Crossref]

Cheng, X.

H. Cui, Z. Zhao, Y. Wu, N. Dai, X. Cheng, and L. Zhang, “Digital fringe image gamma modeling and new algorithm for phase error compensation,” Optik (Stuttg.) 125(24), 7175–7181 (2014).
[Crossref]

Chiang, F.

P. S. Huang, C. Zhang, and F. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42(1), 163–168 (2003).
[Crossref]

Chicharo, J. F.

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of gamma non-linear luminance effects for digital video projection phase measuring profilometers”, in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE, 2008), pp. 496–501.
[Crossref]

Coggrave, C. R.

C. R. Coggrave and J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38(9), 1573–1581 (1999).
[Crossref]

Cui, H.

H. Cui, Z. Zhao, Y. Wu, N. Dai, X. Cheng, and L. Zhang, “Digital fringe image gamma modeling and new algorithm for phase error compensation,” Optik (Stuttg.) 125(24), 7175–7181 (2014).
[Crossref]

Da, F.

D. Zheng and F. Da, “Absolute phase retrieval for defocused fringe projection three-dimensional measurement,” Opt. Commun. 312, 302–311 (2014).
[Crossref]

Dai, J.

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
[Crossref]

Y. Xu, L. Ekstrand, J. Dai, and S. Zhang, “Phase error compensation for three-dimensional shape measurement with projector defocusing,” Appl. Opt. 50(17), 2572–2581 (2011).
[Crossref] [PubMed]

Dai, N.

H. Cui, Z. Zhao, Y. Wu, N. Dai, X. Cheng, and L. Zhang, “Digital fringe image gamma modeling and new algorithm for phase error compensation,” Optik (Stuttg.) 125(24), 7175–7181 (2014).
[Crossref]

Efimov, I. G.

S. Zhang, Y. Gong, Y. Wang, J. Laughner, and I. G. Efimov, “Some recent advance on high-speed, high-resolution 3-D shape measurement using projector defocusing”, in 2010 International Symposium on Optomechatronic Technologies (IEEE, 2010), pp. 1–6
[Crossref]

Ekstrand, L.

Gong, Y.

S. Zhang, Y. Gong, Y. Wang, J. Laughner, and I. G. Efimov, “Some recent advance on high-speed, high-resolution 3-D shape measurement using projector defocusing”, in 2010 International Symposium on Optomechatronic Technologies (IEEE, 2010), pp. 1–6
[Crossref]

Gorthi, S. S.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010).
[Crossref]

Hao, Q.

Hassebrook, L. G.

He, X. Y.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189(1–3), 21–29 (2001).
[Crossref]

Hoang, T.

Huang, L.

Huang, P. S.

P. S. Huang, C. Zhang, and F. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42(1), 163–168 (2003).
[Crossref]

Hufnagel, R. E.

D. A. Zweig and R. E. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” Proc. SPIE 1333, 295–302 (1990).
[Crossref]

Huntley, J. M.

C. R. Coggrave and J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38(9), 1573–1581 (1999).
[Crossref]

Iwata, K.

Kadono, H.

Kakunai, S.

Kemao, Q.

Kothiyal, M. P.

Kumar, U. P.

Larkin, K. G.

Lau, D. L.

Laughner, J.

S. Zhang, Y. Gong, Y. Wang, J. Laughner, and I. G. Efimov, “Some recent advance on high-speed, high-resolution 3-D shape measurement using projector defocusing”, in 2010 International Symposium on Optomechatronic Technologies (IEEE, 2010), pp. 1–6
[Crossref]

Lei, S.

Li, B.

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
[Crossref]

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285(5), 533–538 (2012).
[Crossref]

Li, Y.

Li, Z.

Liu, K.

Lohry, W.

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
[Crossref]

Ma, S.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285(5), 533–538 (2012).
[Crossref]

Madjarova, V.

Madjarova, V. D.

Marple, S. L.

S. L. Marple, “Computing the discrete-time ‘analytic’ signal via FFT,” IEEE Trans. Signal Process. 47(9), 2600–2603 (1999).
[Crossref]

McNeill, S. R.

M. A. Sutton, W. Zhao, S. R. McNeill, H. W. Schreier, and Y. J. Chao, “Development and assessment of a single-image fringe projection method for dynamic applications,” Exp. Mech. 41(3), 205–217 (2001).
[Crossref]

Mohan, N. K.

Nguyen, D.

Nguyen, D. A.

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Notni, G.

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[Crossref]

Notni, G. H.

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[Crossref]

Oldfield, M. A.

Pan, B.

Patorski, K.

M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
[Crossref]

M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
[Crossref] [PubMed]

Pokorski, K.

Poynton, C. A.

C. A. Poynton, “‘Gamma’ and its disguises: the nonlinear mappings of intensity in perception, CRTs, film, and video,” SMPTE Journal 102(12), 1099–1108 (1993).
[Crossref]

Quan, C.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285(5), 533–538 (2012).
[Crossref]

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189(1–3), 21–29 (2001).
[Crossref]

Rastogi, P.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010).
[Crossref]

Rathjen, C.

Sakamoto, T.

Schreier, H. W.

M. A. Sutton, W. Zhao, S. R. McNeill, H. W. Schreier, and Y. J. Chao, “Development and assessment of a single-image fringe projection method for dynamic applications,” Exp. Mech. 41(3), 205–217 (2001).
[Crossref]

Shang, H. M.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189(1–3), 21–29 (2001).
[Crossref]

Su, X.-Y.

X.-Y. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94(6), 561–573 (1992).
[Crossref]

Sutton, M. A.

M. A. Sutton, W. Zhao, S. R. McNeill, H. W. Schreier, and Y. J. Chao, “Development and assessment of a single-image fringe projection method for dynamic applications,” Exp. Mech. 41(3), 205–217 (2001).
[Crossref]

Tay, C. J.

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285(5), 533–538 (2012).
[Crossref]

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189(1–3), 21–29 (2001).
[Crossref]

Toyooka, S.

Trusiak, M.

M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
[Crossref]

M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
[Crossref] [PubMed]

Tu, D.

von Bally, G.

X.-Y. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94(6), 561–573 (1992).
[Crossref]

Vukicevic, D.

X.-Y. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94(6), 561–573 (1992).
[Crossref]

Wang, C. F.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189(1–3), 21–29 (2001).
[Crossref]

Wang, Y.

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
[Crossref]

K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Gamma model and its analysis for phase measuring profilometry,” J. Opt. Soc. Am. A 27(3), 553–562 (2010).
[Crossref] [PubMed]

Y. Wang and S. Zhang, “Optimal pulse width modulation for sinusoidal fringe generation with projector defocusing,” Opt. Lett. 35(24), 4121–4123 (2010).
[Crossref] [PubMed]

S. Zhang, Y. Gong, Y. Wang, J. Laughner, and I. G. Efimov, “Some recent advance on high-speed, high-resolution 3-D shape measurement using projector defocusing”, in 2010 International Symposium on Optomechatronic Technologies (IEEE, 2010), pp. 1–6
[Crossref]

Wang, Z.

Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[Crossref] [PubMed]

Wielgus, M.

M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
[Crossref]

Wu, Y.

H. Cui, Z. Zhao, Y. Wu, N. Dai, X. Cheng, and L. Zhang, “Digital fringe image gamma modeling and new algorithm for phase error compensation,” Optik (Stuttg.) 125(24), 7175–7181 (2014).
[Crossref]

Xi, J.

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of gamma non-linear luminance effects for digital video projection phase measuring profilometers”, in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE, 2008), pp. 496–501.
[Crossref]

Xu, Y.

Yau, S. T.

Zhang, C.

P. S. Huang, C. Zhang, and F. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42(1), 163–168 (2003).
[Crossref]

Zhang, L.

H. Cui, Z. Zhao, Y. Wu, N. Dai, X. Cheng, and L. Zhang, “Digital fringe image gamma modeling and new algorithm for phase error compensation,” Optik (Stuttg.) 125(24), 7175–7181 (2014).
[Crossref]

Zhang, S.

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
[Crossref]

Y. Xu, L. Ekstrand, J. Dai, and S. Zhang, “Phase error compensation for three-dimensional shape measurement with projector defocusing,” Appl. Opt. 50(17), 2572–2581 (2011).
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S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
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S. Zhang, Y. Gong, Y. Wang, J. Laughner, and I. G. Efimov, “Some recent advance on high-speed, high-resolution 3-D shape measurement using projector defocusing”, in 2010 International Symposium on Optomechatronic Technologies (IEEE, 2010), pp. 1–6
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[Crossref]

Zhao, Z.

H. Cui, Z. Zhao, Y. Wu, N. Dai, X. Cheng, and L. Zhang, “Digital fringe image gamma modeling and new algorithm for phase error compensation,” Optik (Stuttg.) 125(24), 7175–7181 (2014).
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Zheng, D.

D. Zheng and F. Da, “Absolute phase retrieval for defocused fringe projection three-dimensional measurement,” Opt. Commun. 312, 302–311 (2014).
[Crossref]

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X.-Y. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94(6), 561–573 (1992).
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S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285(5), 533–538 (2012).
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Appl. Opt. (5)

Exp. Mech. (1)

M. A. Sutton, W. Zhao, S. R. McNeill, H. W. Schreier, and Y. J. Chao, “Development and assessment of a single-image fringe projection method for dynamic applications,” Exp. Mech. 41(3), 205–217 (2001).
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[Crossref]

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Opt. Commun. (4)

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189(1–3), 21–29 (2001).
[Crossref]

S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285(5), 533–538 (2012).
[Crossref]

X.-Y. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94(6), 561–573 (1992).
[Crossref]

D. Zheng and F. Da, “Absolute phase retrieval for defocused fringe projection three-dimensional measurement,” Opt. Commun. 312, 302–311 (2014).
[Crossref]

Opt. Eng. (2)

P. S. Huang, C. Zhang, and F. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42(1), 163–168 (2003).
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Opt. Express (2)

Opt. Lasers Eng. (4)

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
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Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010).
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S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

Opt. Lett. (5)

Optik (Stuttg.) (1)

H. Cui, Z. Zhao, Y. Wu, N. Dai, X. Cheng, and L. Zhang, “Digital fringe image gamma modeling and new algorithm for phase error compensation,” Optik (Stuttg.) 125(24), 7175–7181 (2014).
[Crossref]

Proc. SPIE (3)

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[Crossref]

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Other (2)

S. Zhang, Y. Gong, Y. Wang, J. Laughner, and I. G. Efimov, “Some recent advance on high-speed, high-resolution 3-D shape measurement using projector defocusing”, in 2010 International Symposium on Optomechatronic Technologies (IEEE, 2010), pp. 1–6
[Crossref]

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of gamma non-linear luminance effects for digital video projection phase measuring profilometers”, in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE, 2008), pp. 496–501.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Characteristics of phase error distribution: (a) the phase error with respect to phase, (b) the maximum phase error with respect to phase-shifting step.
Fig. 2
Fig. 2 One cross section of the phase error distribution with and without compensation: (a) in three-step PSP, and (b) four-step PSP.
Fig. 3
Fig. 3 The captured images: (a) in uniform illumination, and (b) fringe projection illumination.
Fig. 4
Fig. 4 3D digital reconstruction results by using three-step PSP: (a) without compensation in spatial domain, (b) without compensation in HT domain, (c) with compensation by the proposed method, (d)-(f) the corresponding 3D digital surface details in (a)-(c), and (g) a local cross section of 3D digital surface, as labeled in Fig. 3(a).
Fig. 5
Fig. 5 3D digital reconstruction results by using four-step PSP: (a) without compensation in spatial domain, (b) without compensation in HT domain, (c) with compensation by the proposed method, (d)-(f) the corresponding 3D digital surface details in (a)-(c), and (g) a local cross section of 3D digital surface, as labeled in Fig. 3(a).

Tables (2)

Tables Icon

Table 1 Maximum phase error (rad) and ratio of error reduction (%)

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Table 2 MAX and RMS of the phase error (rad)

Equations (21)

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I n ( x , y ) = A ( x , y ) + B ( x , y ) cos [ ϕ ( x , y ) + δ n ] , n = 1 , 2 , , N
I n C = ( α I n ) γ = B 0 + k = 1 { B k cos [ k ( ϕ + δ n ) ] } = B 0 + k = 1 [ B k cos ( k ϕ n ) ]
ϕ = arc tan [ n = 1 N ( I n sin δ n ) n = 1 N ( I n cos δ n ) ]
ϕ C = arc tan [ n = 1 N ( I n C sin δ n ) n = 1 N ( I n C cos δ n ) ] = arc tan { B 0 n = 1 N sin δ n n = 1 N k = 1 [ B k cos ( k ϕ n ) sin δ n ] B 0 n = 1 N cos δ n + n = 1 N k = 1 [ B k cos ( k ϕ n ) cos δ n ] } = arc tan { n = 1 N k = 1 [ B k cos ( k ϕ n ) sin δ n ] n = 1 N k = 1 [ B k cos ( k ϕ n ) cos δ n ] }
Δ ϕ = ϕ C ϕ = arc tan { cos ϕ n = 1 N k = 1 [ B k cos ( k ϕ n ) sin δ n ] sin ϕ n = 1 N k = 1 [ B k cos ( k ϕ n ) cos δ n ] cos ϕ n = 1 N k = 1 [ B k cos ( k ϕ n ) cos δ n ] sin ϕ n = 1 N k = 1 [ B k cos ( k ϕ n ) sin δ n ] } = arc tan { n = 1 N k = 1 [ B k cos ( k ϕ n ) sin ϕ n ] n = 1 N k = 1 [ B k cos ( k ϕ n ) cos ϕ n ] } = arc tan { n = 1 N k = 2 [ ( B k + 1 B k 1 ) sin ( k ϕ n ) ] N B 1 + n = 1 N k = 2 [ ( B k + 1 + B k 1 ) cos ( k ϕ n ) ] }
n = 1 N [ sin ( k ϕ n ) ] = { 0 , k m N N sin ( m N ϕ ) , k = m N , m Z + n = 1 N [ cos ( k ϕ n ) ] = { 0 , k m N N sin ( m N ϕ ) , k = m N , m Z +
Δ ϕ = arc tan { m = 1 [ ( G m N + 1 G m N 1 ) sin ( m N ϕ ) ] 1 + m = 1 [ ( G m N + 1 + G m N 1 ) cos ( m N ϕ ) ] }
G s = i = 2 s γ i + 1 γ + i
Δ ϕ = arc tan [ G N 1 sin ( N ϕ ) 1 + G N 1 cos ( N ϕ ) ]
I n H = H [ I n ] = B sin ( ϕ + δ n )
I n H C = k = 1 [ B k sin ( k ϕ n ) ]
ϕ H = arc tan [ n = 1 N ( I n H cos δ n ) n = 1 N ( I n H sin δ n ) ]
ϕ H C = arc tan [ n = 1 N ( I n H C cos δ n ) n = 1 N ( I n H C sin δ n ) ] = arc tan { n = 1 N k = 1 [ B k sin ( k ϕ n ) cos δ n ] n = 1 N k = 1 [ B k sin ( k ϕ n ) sin δ n ] }
Δ ϕ H = ϕ H C ϕ = arc tan { m = 1 [ ( G m N + 1 + G m N 1 ) sin ( m N ϕ ) ] 1 + m = 1 [ ( G m N + 1 G m N 1 ) cos ( m N ϕ ) ] }
Δ ϕ H = arc tan [ G N 1 sin ( N ϕ ) 1 G N 1 cos ( N ϕ ) ]
A Δ ϕ = | Δ ϕ | max = arc sin ( | G N 1 | )
A Δ ϕ H = | Δ ϕ H | max = arc sin ( | G N 1 | )
Δ ϕ H | ϕ + T 2 = arc tan { G N 1 sin [ N ( ϕ + π N ) ] 1 G N 1 cos [ N ( ϕ + π N ) ] } = arc tan [ G N 1 sin ( N ϕ ) 1 + G N 1 cos ( N ϕ ) ] = Δ ϕ | ϕ
Δ ϕ M = ϕ M ϕ = 1 2 ( Δ ϕ + Δ ϕ H ) = 1 2 arc tan [ G N 1 2 sin ( 2 N ϕ ) 1 G N 1 2 cos ( 2 N ϕ ) ]
| Δ ϕ M | max = 1 2 arc sin ( G N 1 2 )
r Δ ϕ = | Δ ϕ M | max | Δ ϕ | max 1 2 G N 1 2 | G N 1 | = | G N 1 | 2

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