Abstract

This paper presents a method that expresses the fringe pattern as an exponential function and a mathematical model for gamma-independent phase computation. The method was compared to: (i) conventional phase measurement without nonlinearity correction, and (ii) conventional gamma correction by pattern pre-distortion based on an input-to-projector camera-output look-up table. The pre-distorted and exponential methods achieved large reduction in error compared to conventional computation with no gamma correction. The advantage of the exponential method is that no system gamma nonlinearity calibration procedure or information is required. This reduces optical system setup before measurement and permits easier use of off-the-shelf projectors.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2017 (1)

W. Zhang, L. Yu, W. Li, H. Xia, H. Deng, and J. Zhang, “Black-box phase error compensation for digital phase-shifting profilometry,” IEEE Trans. Instrum. Meas. 99, 1–7 (2017).

2016 (3)

C. Xiong, J. Yao, J. Chen, and H. Miao, “A convenient look-up-table based method for the compensation of non-linear error in digital fringe projection,” Theor. Appl. Mech. Lett. 6(1), 49–53 (2016).
[Crossref]

Z. Cai, X. Liu, X. Peng, Z. Zhang, H. Jiang, Y. Yin, and S. Huang, “Phase error compensation methods for high-accuracy profile measurement,” Meas. Sci. Technol. 27(4), 045201 (2016).
[Crossref]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Compensation of fringe distortion for phase-shifting three-dimensional shape measurement by inverse map estimation,” Appl. Opt. 55(22), 6017–6024 (2016).
[Crossref] [PubMed]

2014 (5)

J. Yao, C. Xiong, Y. Zhou, H. Miao, and J. Chen, “Phase error elimination considering gamma nonlinearity, system vibration, and noise for fringe projection profilometry,” Opt. Eng. 53(9), 094102 (2014).
[Crossref]

J. Z. Peng, H. K. Ouyang, Q. Yu, Y. J. Yu, and K. S. Wang, “Phase error correction for fringe projection profilometry by using constrained cubic spline,” Adv. Manuf. 2(1), 39–47 (2014).
[Crossref]

K. Liu, S. Wang, D. L. Lau, K. E. Barner, and F. Kiamilev, “Nonlinearity calibrating algorithm for structured light illumination,” Opt. Eng. 53(5), 050501 (2014).
[Crossref]

S. Zhang, “Active versus passive projector nonlinear gamma compensation method for high-quality fringe pattern generation,” Proc. SPIE 9110, 911002 (2014).
[Crossref]

C. Waddington and J. Kofman, “Modified sinusoidal fringe-pattern projection for variable illuminance in phase-shifting three-dimensional surface-shape metrology,” Opt. Eng. 53(8), 084109 (2014).
[Crossref]

2012 (1)

2010 (3)

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]

Y. Liu, J. Xi, Y. Yu, and J. Chicharo, “Phase error correction based on Inverse Function Shift Estimation in Phase Shifting Profilometry using a digital video projector,” Proc. SPIE 7855, 78550W (2010).
[Crossref]

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]

2009 (3)

2008 (2)

P. Jia, J. Kofman, and C. English, “Error compensation in two-step triangular-pattern phase-shifting profilometry,” Opt. Lasers Eng. 46(4), 311–320 (2008).
[Crossref]

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).
[Crossref]

2007 (4)

S. Zhang and S. T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46(1), 36–43 (2007).
[Crossref] [PubMed]

P. Jia, J. Kofman, and C. English, “Multiple-Step Triangular-Pattern Phase Shifting and the Influence of Number of Steps and Pitch on Measurement Accuracy,” Appl. Opt. 46(16), 3253–3262 (2007).
[Crossref] [PubMed]

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46(6), 063601 (2007).
[Crossref]

H. Guo and Z. Zhao, “Nonlinearity correction in digital fringe projection profilometry by using histogram matching technique,” Proc. SPIE 6616, 66162I (2007).
[Crossref]

2004 (1)

2003 (2)

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

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

2002 (2)

Y. Cao, X. Su, L. Xiang, and W. Chen, “Intensity transfer function of DMD and its application in PMP,” Proc. SPIE 4778, 83–88 (2002).
[Crossref]

P. S. Huang, Q. J. Hu, and F.-P. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41(22), 4503–4509 (2002).
[Crossref] [PubMed]

2001 (1)

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10(10), 1428–1433 (2001).
[Crossref] [PubMed]

1999 (1)

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]

1996 (1)

1995 (1)

1994 (1)

T. R. Judge and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21(4), 199–239 (1994).
[Crossref]

Asundi, A.

Barner, K. E.

K. Liu, S. Wang, D. L. Lau, K. E. Barner, and F. Kiamilev, “Nonlinearity calibrating algorithm for structured light illumination,” Opt. Eng. 53(5), 050501 (2014).
[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]

Bryanston-Cross, P. J.

T. R. Judge and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21(4), 199–239 (1994).
[Crossref]

Busca, G.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).
[Crossref]

Cai, Z.

Z. Cai, X. Liu, X. Peng, Z. Zhang, H. Jiang, Y. Yin, and S. Huang, “Phase error compensation methods for high-accuracy profile measurement,” Meas. Sci. Technol. 27(4), 045201 (2016).
[Crossref]

Cao, Y.

Y. Cao, X. Su, L. Xiang, and W. Chen, “Intensity transfer function of DMD and its application in PMP,” Proc. SPIE 4778, 83–88 (2002).
[Crossref]

Chen, J.

C. Xiong, J. Yao, J. Chen, and H. Miao, “A convenient look-up-table based method for the compensation of non-linear error in digital fringe projection,” Theor. Appl. Mech. Lett. 6(1), 49–53 (2016).
[Crossref]

J. Yao, C. Xiong, Y. Zhou, H. Miao, and J. Chen, “Phase error elimination considering gamma nonlinearity, system vibration, and noise for fringe projection profilometry,” Opt. Eng. 53(9), 094102 (2014).
[Crossref]

Chen, M.

Chen, W.

Y. Cao, X. Su, L. Xiang, and W. Chen, “Intensity transfer function of DMD and its application in PMP,” Proc. SPIE 4778, 83–88 (2002).
[Crossref]

Chiang, F.-P.

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

P. S. Huang, Q. J. Hu, and F.-P. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41(22), 4503–4509 (2002).
[Crossref] [PubMed]

Chicharo, J.

Y. Liu, J. Xi, Y. Yu, and J. Chicharo, “Phase error correction based on Inverse Function Shift Estimation in Phase Shifting Profilometry using a digital video projector,” Proc. SPIE 7855, 78550W (2010).
[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]

Deng, H.

W. Zhang, L. Yu, W. Li, H. Xia, H. Deng, and J. Zhang, “Black-box phase error compensation for digital phase-shifting profilometry,” IEEE Trans. Instrum. Meas. 99, 1–7 (2017).

English, C.

P. Jia, J. Kofman, and C. English, “Error compensation in two-step triangular-pattern phase-shifting profilometry,” Opt. Lasers Eng. 46(4), 311–320 (2008).
[Crossref]

P. Jia, J. Kofman, and C. English, “Multiple-Step Triangular-Pattern Phase Shifting and the Influence of Number of Steps and Pitch on Measurement Accuracy,” Appl. Opt. 46(16), 3253–3262 (2007).
[Crossref] [PubMed]

Farid, H.

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10(10), 1428–1433 (2001).
[Crossref] [PubMed]

Farrant, D. I.

Guo, H.

H. Guo and Z. Zhao, “Nonlinearity correction in digital fringe projection profilometry by using histogram matching technique,” Proc. SPIE 6616, 66162I (2007).
[Crossref]

H. Guo, H. He, and M. Chen, “Gamma correction for digital fringe projection profilometry,” Appl. Opt. 43(14), 2906–2914 (2004).
[Crossref] [PubMed]

Hao, Q.

Hassebrook, L. G.

He, H.

Hibino, K.

Hu, Q. J.

Huang, L.

Huang, P. S.

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46(6), 063601 (2007).
[Crossref]

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

P. S. Huang, Q. J. Hu, and F.-P. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41(22), 4503–4509 (2002).
[Crossref] [PubMed]

Huang, S.

Z. Cai, X. Liu, X. Peng, Z. Zhang, H. Jiang, Y. Yin, and S. Huang, “Phase error compensation methods for high-accuracy profile measurement,” Meas. Sci. Technol. 27(4), 045201 (2016).
[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]

Ishikawa, K.

Jia, P.

P. Jia, J. Kofman, and C. English, “Error compensation in two-step triangular-pattern phase-shifting profilometry,” Opt. Lasers Eng. 46(4), 311–320 (2008).
[Crossref]

P. Jia, J. Kofman, and C. English, “Multiple-Step Triangular-Pattern Phase Shifting and the Influence of Number of Steps and Pitch on Measurement Accuracy,” Appl. Opt. 46(16), 3253–3262 (2007).
[Crossref] [PubMed]

Jia, S.

Jiang, H.

Z. Cai, X. Liu, X. Peng, Z. Zhang, H. Jiang, Y. Yin, and S. Huang, “Phase error compensation methods for high-accuracy profile measurement,” Meas. Sci. Technol. 27(4), 045201 (2016).
[Crossref]

Judge, T. R.

T. R. Judge and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21(4), 199–239 (1994).
[Crossref]

Kemao, Q.

Kiamilev, F.

K. Liu, S. Wang, D. L. Lau, K. E. Barner, and F. Kiamilev, “Nonlinearity calibrating algorithm for structured light illumination,” Opt. Eng. 53(5), 050501 (2014).
[Crossref]

Kofman, J.

C. Waddington and J. Kofman, “Modified sinusoidal fringe-pattern projection for variable illuminance in phase-shifting three-dimensional surface-shape metrology,” Opt. Eng. 53(8), 084109 (2014).
[Crossref]

P. Jia, J. Kofman, and C. English, “Error compensation in two-step triangular-pattern phase-shifting profilometry,” Opt. Lasers Eng. 46(4), 311–320 (2008).
[Crossref]

P. Jia, J. Kofman, and C. English, “Multiple-Step Triangular-Pattern Phase Shifting and the Influence of Number of Steps and Pitch on Measurement Accuracy,” Appl. Opt. 46(16), 3253–3262 (2007).
[Crossref] [PubMed]

Larkin, K. G.

Lau, D. L.

K. Liu, S. Wang, D. L. Lau, K. E. Barner, and F. Kiamilev, “Nonlinearity calibrating algorithm for structured light illumination,” Opt. Eng. 53(5), 050501 (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]

Li, W.

W. Zhang, L. Yu, W. Li, H. Xia, H. Deng, and J. Zhang, “Black-box phase error compensation for digital phase-shifting profilometry,” IEEE Trans. Instrum. Meas. 99, 1–7 (2017).

Liu, K.

K. Liu, S. Wang, D. L. Lau, K. E. Barner, and F. Kiamilev, “Nonlinearity calibrating algorithm for structured light illumination,” Opt. Eng. 53(5), 050501 (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]

Liu, X.

Z. Cai, X. Liu, X. Peng, Z. Zhang, H. Jiang, Y. Yin, and S. Huang, “Phase error compensation methods for high-accuracy profile measurement,” Meas. Sci. Technol. 27(4), 045201 (2016).
[Crossref]

Liu, Y.

Y. Liu, J. Xi, Y. Yu, and J. Chicharo, “Phase error correction based on Inverse Function Shift Estimation in Phase Shifting Profilometry using a digital video projector,” Proc. SPIE 7855, 78550W (2010).
[Crossref]

Miao, H.

C. Xiong, J. Yao, J. Chen, and H. Miao, “A convenient look-up-table based method for the compensation of non-linear error in digital fringe projection,” Theor. Appl. Mech. Lett. 6(1), 49–53 (2016).
[Crossref]

J. Yao, C. Xiong, Y. Zhou, H. Miao, and J. Chen, “Phase error elimination considering gamma nonlinearity, system vibration, and noise for fringe projection profilometry,” Opt. Eng. 53(9), 094102 (2014).
[Crossref]

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]

Oikawa, Y.

Oreb, B. F.

Ouyang, H. K.

J. Z. Peng, H. K. Ouyang, Q. Yu, Y. J. Yu, and K. S. Wang, “Phase error correction for fringe projection profilometry by using constrained cubic spline,” Adv. Manuf. 2(1), 39–47 (2014).
[Crossref]

Pan, B.

Pan, T.

Peng, J. Z.

J. Z. Peng, H. K. Ouyang, Q. Yu, Y. J. Yu, and K. S. Wang, “Phase error correction for fringe projection profilometry by using constrained cubic spline,” Adv. Manuf. 2(1), 39–47 (2014).
[Crossref]

Peng, X.

Z. Cai, X. Liu, X. Peng, Z. Zhang, H. Jiang, Y. Yin, and S. Huang, “Phase error compensation methods for high-accuracy profile measurement,” Meas. Sci. Technol. 27(4), 045201 (2016).
[Crossref]

Su, X.

Y. Cao, X. Su, L. Xiang, and W. Chen, “Intensity transfer function of DMD and its application in PMP,” Proc. SPIE 4778, 83–88 (2002).
[Crossref]

Surrel, Y.

Vo, M.

Waddington, C.

C. Waddington and J. Kofman, “Modified sinusoidal fringe-pattern projection for variable illuminance in phase-shifting three-dimensional surface-shape metrology,” Opt. Eng. 53(8), 084109 (2014).
[Crossref]

Wang, K. S.

J. Z. Peng, H. K. Ouyang, Q. Yu, Y. J. Yu, and K. S. Wang, “Phase error correction for fringe projection profilometry by using constrained cubic spline,” Adv. Manuf. 2(1), 39–47 (2014).
[Crossref]

Wang, S.

K. Liu, S. Wang, D. L. Lau, K. E. Barner, and F. Kiamilev, “Nonlinearity calibrating algorithm for structured light illumination,” Opt. Eng. 53(5), 050501 (2014).
[Crossref]

Wang, Y.

Wang, Z.

M. Vo, Z. Wang, B. Pan, and T. Pan, “Hyper-accurate flexible calibration technique for fringe-projection-based three-dimensional imaging,” Opt. Express 20(15), 16926–16941 (2012).
[Crossref]

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]

Xi, J.

Y. Liu, J. Xi, Y. Yu, and J. Chicharo, “Phase error correction based on Inverse Function Shift Estimation in Phase Shifting Profilometry using a digital video projector,” Proc. SPIE 7855, 78550W (2010).
[Crossref]

Xia, H.

W. Zhang, L. Yu, W. Li, H. Xia, H. Deng, and J. Zhang, “Black-box phase error compensation for digital phase-shifting profilometry,” IEEE Trans. Instrum. Meas. 99, 1–7 (2017).

Xiang, L.

Y. Cao, X. Su, L. Xiang, and W. Chen, “Intensity transfer function of DMD and its application in PMP,” Proc. SPIE 4778, 83–88 (2002).
[Crossref]

Xiong, C.

C. Xiong, J. Yao, J. Chen, and H. Miao, “A convenient look-up-table based method for the compensation of non-linear error in digital fringe projection,” Theor. Appl. Mech. Lett. 6(1), 49–53 (2016).
[Crossref]

J. Yao, C. Xiong, Y. Zhou, H. Miao, and J. Chen, “Phase error elimination considering gamma nonlinearity, system vibration, and noise for fringe projection profilometry,” Opt. Eng. 53(9), 094102 (2014).
[Crossref]

Xiong, L.

Yao, J.

C. Xiong, J. Yao, J. Chen, and H. Miao, “A convenient look-up-table based method for the compensation of non-linear error in digital fringe projection,” Theor. Appl. Mech. Lett. 6(1), 49–53 (2016).
[Crossref]

J. Yao, C. Xiong, Y. Zhou, H. Miao, and J. Chen, “Phase error elimination considering gamma nonlinearity, system vibration, and noise for fringe projection profilometry,” Opt. Eng. 53(9), 094102 (2014).
[Crossref]

Yatabe, K.

Yau, S. T.

Yin, Y.

Z. Cai, X. Liu, X. Peng, Z. Zhang, H. Jiang, Y. Yin, and S. Huang, “Phase error compensation methods for high-accuracy profile measurement,” Meas. Sci. Technol. 27(4), 045201 (2016).
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Yu, L.

W. Zhang, L. Yu, W. Li, H. Xia, H. Deng, and J. Zhang, “Black-box phase error compensation for digital phase-shifting profilometry,” IEEE Trans. Instrum. Meas. 99, 1–7 (2017).

Yu, Q.

J. Z. Peng, H. K. Ouyang, Q. Yu, Y. J. Yu, and K. S. Wang, “Phase error correction for fringe projection profilometry by using constrained cubic spline,” Adv. Manuf. 2(1), 39–47 (2014).
[Crossref]

Yu, Y.

Y. Liu, J. Xi, Y. Yu, and J. Chicharo, “Phase error correction based on Inverse Function Shift Estimation in Phase Shifting Profilometry using a digital video projector,” Proc. SPIE 7855, 78550W (2010).
[Crossref]

Yu, Y. J.

J. Z. Peng, H. K. Ouyang, Q. Yu, Y. J. Yu, and K. S. Wang, “Phase error correction for fringe projection profilometry by using constrained cubic spline,” Adv. Manuf. 2(1), 39–47 (2014).
[Crossref]

Zappa, E.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).
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Zhang, C.

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

Zhang, J.

W. Zhang, L. Yu, W. Li, H. Xia, H. Deng, and J. Zhang, “Black-box phase error compensation for digital phase-shifting profilometry,” IEEE Trans. Instrum. Meas. 99, 1–7 (2017).

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S. Zhang, “Active versus passive projector nonlinear gamma compensation method for high-quality fringe pattern generation,” Proc. SPIE 9110, 911002 (2014).
[Crossref]

S. Zhang, “Digital multiple wavelength phase shifting algorithm,” Proc. SPIE 7432, 74320N (2009).
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S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46(6), 063601 (2007).
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S. Zhang and S. T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46(1), 36–43 (2007).
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W. Zhang, L. Yu, W. Li, H. Xia, H. Deng, and J. Zhang, “Black-box phase error compensation for digital phase-shifting profilometry,” IEEE Trans. Instrum. Meas. 99, 1–7 (2017).

Zhang, Z.

Z. Cai, X. Liu, X. Peng, Z. Zhang, H. Jiang, Y. Yin, and S. Huang, “Phase error compensation methods for high-accuracy profile measurement,” Meas. Sci. Technol. 27(4), 045201 (2016).
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H. Guo and Z. Zhao, “Nonlinearity correction in digital fringe projection profilometry by using histogram matching technique,” Proc. SPIE 6616, 66162I (2007).
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J. Yao, C. Xiong, Y. Zhou, H. Miao, and J. Chen, “Phase error elimination considering gamma nonlinearity, system vibration, and noise for fringe projection profilometry,” Opt. Eng. 53(9), 094102 (2014).
[Crossref]

Adv. Manuf. (1)

J. Z. Peng, H. K. Ouyang, Q. Yu, Y. J. Yu, and K. S. Wang, “Phase error correction for fringe projection profilometry by using constrained cubic spline,” Adv. Manuf. 2(1), 39–47 (2014).
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Appl. Opt. (6)

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IEEE Trans. Instrum. Meas. (1)

W. Zhang, L. Yu, W. Li, H. Xia, H. Deng, and J. Zhang, “Black-box phase error compensation for digital phase-shifting profilometry,” IEEE Trans. Instrum. Meas. 99, 1–7 (2017).

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

Z. Cai, X. Liu, X. Peng, Z. Zhang, H. Jiang, Y. Yin, and S. Huang, “Phase error compensation methods for high-accuracy profile measurement,” Meas. Sci. Technol. 27(4), 045201 (2016).
[Crossref]

Opt. Eng. (6)

J. Yao, C. Xiong, Y. Zhou, H. Miao, and J. Chen, “Phase error elimination considering gamma nonlinearity, system vibration, and noise for fringe projection profilometry,” Opt. Eng. 53(9), 094102 (2014).
[Crossref]

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46(6), 063601 (2007).
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C. Waddington and J. Kofman, “Modified sinusoidal fringe-pattern projection for variable illuminance in phase-shifting three-dimensional surface-shape metrology,” Opt. Eng. 53(8), 084109 (2014).
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Opt. Express (1)

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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).
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P. Jia, J. Kofman, and C. English, “Error compensation in two-step triangular-pattern phase-shifting profilometry,” Opt. Lasers Eng. 46(4), 311–320 (2008).
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Opt. Lett. (2)

Proc. SPIE (6)

H. Guo and Z. Zhao, “Nonlinearity correction in digital fringe projection profilometry by using histogram matching technique,” Proc. SPIE 6616, 66162I (2007).
[Crossref]

S. Zhang, “Digital multiple wavelength phase shifting algorithm,” Proc. SPIE 7432, 74320N (2009).
[Crossref]

Y. Liu, J. Xi, Y. Yu, and J. Chicharo, “Phase error correction based on Inverse Function Shift Estimation in Phase Shifting Profilometry using a digital video projector,” Proc. SPIE 7855, 78550W (2010).
[Crossref]

S. Zhang, “Active versus passive projector nonlinear gamma compensation method for high-quality fringe pattern generation,” Proc. SPIE 9110, 911002 (2014).
[Crossref]

Y. Cao, X. Su, L. Xiang, and W. Chen, “Intensity transfer function of DMD and its application in PMP,” Proc. SPIE 4778, 83–88 (2002).
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C. Xiong, J. Yao, J. Chen, and H. Miao, “A convenient look-up-table based method for the compensation of non-linear error in digital fringe projection,” Theor. Appl. Mech. Lett. 6(1), 49–53 (2016).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Conventional and exponential fringe patterns, with and without gamma distortion.
Fig. 2
Fig. 2 Phase with corresponding fringe patterns: (top) column of phase map, (bottom) column of phase-shifted fringe patterns.
Fig. 3
Fig. 3 Projector input to camera output gamma function (blue) and its inverse function (red).
Fig. 4
Fig. 4 Cross section of camera-captured fringe patterns: conventional, pre-distorted (corrected), and exponential.
Fig. 5
Fig. 5 Phase maps generated from fringe patterns: (a) conventional fringe patterns, (b) exponential fringe patterns before outlier removal, and (c) exponential fringe patterns after outlier removal.
Fig. 6
Fig. 6 Phase error of one column of the phase map: conventional, pre-distorted (corrected) and exponential.
Fig. 7
Fig. 7 Comparison of phase RMSE for conventional, pre-distorted (corrected), and exponential methods at different plate positions.
Fig. 8
Fig. 8 Phase maps of the measured mask: a) conventional, b) pre-distorted (corrected), and c) exponential.

Tables (2)

Tables Icon

Table 1 RMSE (rad) for different amounts of noise.

Tables Icon

Table 2 Mean phase RMSE and standard deviation (SD) of phase RMSE for conventional, pre-distorted (corrected), and exponential methods over all plate positions.

Equations (19)

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I n p ( x , y ) = A p ( x , y ) + B p ( x , y ) [ 1 2 + 1 2 cos ( 2 π f x + δ n ) ] ,
I n c ( x , y ) = ( A p ( x , y ) + B p ( x , y ) [ 1 2 + 1 2 cos ( ϕ ( x , y ) + δ n ) ] ) γ ,
ϕ ( x , y ) = tan 1 I 4 ( x , y ) I 2 ( x , y ) I 1 ( x , y ) I 3 ( x , y ) .
I n p ( x , y ) = A p ( x , y ) + B p ( x , y ) exp [ 1 2 + 1 2 cos ( ϕ ( x , y ) + δ n ) ] ,
I n ( x , y ) = A ( x , y ) + B ( x , y ) exp [ γ ( 1 2 + 1 2 cos ( ϕ ( x , y ) + δ n ) ) ] .
I n ( x , y ) = A ( x , y ) + B ( x , y ) exp ( γ 2 ) exp [ γ 2 cos ( ϕ ( x , y ) + δ n ) ] .
I n ( x , y ) = A ( x , y ) + B ( x , y ) exp [ 1 2 γ cos ( ϕ ( x , y ) + δ n ) ] ,
I 1 ( x , y ) = A ( x , y ) + B ( x , y ) exp [ 1 2 γ cos ( ϕ ( x , y ) ) ] ,
I 2 ( x , y ) = A ( x , y ) + B ( x , y ) exp [ 1 2 γ cos ( ϕ ( x , y ) + π 2 ) ] = A ( x , y ) + B ( x , y ) exp [ 1 2 γ sin ( ϕ ( x , y ) ) ] ,
I 3 ( x , y ) = A ( x , y ) + B ( x , y ) exp [ 1 2 γ cos ( ϕ ( x , y ) + π ) ] = A ( x , y ) + B ( x , y ) exp [ 1 2 γ cos ( ϕ ( x , y ) ) ] ,
I 4 ( x , y ) = A ( x , y ) + B ( x , y ) exp [ 1 2 γ cos ( ϕ ( x , y ) + 3 π 2 ) ] = A ( x , y ) + B ( x , y ) exp [ 1 2 γ sin ( ϕ ( x , y ) ) ] .
I 1 ( x , y ) = A ( x , y ) + B ( x , y ) t ( x , y ) ,
I 2 ( x , y ) = A ( x , y ) + B ( x , y ) / s ( x , y ) ,
I 3 ( x , y ) = A ( x , y ) + B ( x , y ) / t ( x , y ) ,
I 4 ( x , y ) = A ( x , y ) + B ( x , y ) s ( x , y ) ,
s ( x , y ) = [ I 4 ( x , y ) I 1 ( x , y ) ] [ I 4 ( x , y ) I 3 ( x , y ) ] [ I 1 ( x , y ) I 2 ( x , y ) ] [ I 3 ( x , y ) I 2 ( x , y ) ] ,
t ( x , y ) = [ I 4 ( x , y ) I 1 ( x , y ) ] [ I 1 ( x , y ) I 2 ( x , y ) ] [ I 4 ( x , y ) I 3 ( x , y ) ] [ I 3 ( x , y ) I 2 ( x , y ) ] .
γ ( x , y ) = ± [ ln ( s ( x , y ) ) ] 2 + [ ln ( t ( x , y ) ) ] 2 ,
ϕ ( x , y ) = tan 1 [ ln ( s ( x , y ) ) / ln ( t ( x , y ) ) ] .

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