Abstract

For three-dimensional shape measurement, phase-shifting techniques are widely used to recover the objective phase containing height information from images of projected fringes. Although such techniques can provide an accurate result in theory, there might be considerable error in practice. One main cause of such an error is distortion of fringes due to nonlinear responses of a measurement system. In this paper, a postprocessing method for compensating distortion is proposed. Compared to other compensation methods, the proposed method is flexible in two senses: (1) no specific model of nonlinearity (such as the gamma model) is needed, and (2) no special calibration data are needed (only the observed image of the fringe is required). Experiments using simulated and real data confirmed that the proposed method can compensate multiple types of nonlinearity without being concerned about the model.

© 2016 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (3)

2015 (2)

S. Zhang, “Comparative study on passive and active projector nonlinear gamma calibration,” Appl. Opt. 54, 3834–3841 (2015).
[Crossref]

C. Zhang, H. Zhao, L. Zhang, and X. Wang, “Full-field phase error detection and compensation method for digital phase-shifting fringe projection profilometry,” Meas. Sci. Technol. 26, 035201 (2015).
[Crossref]

2014 (3)

P. Zhou, X. Liu, Y. He, and T. Zhu, “Phase error analysis and compensation considering ambient light for phase measuring profilometry,” Opt. Lasers Eng. 55, 99–104 (2014).
[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, 094102 (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, 050501 (2014).
[Crossref]

2012 (2)

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, 533–538 (2012).
[Crossref]

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, 1047–1058 (2012).
[Crossref]

2011 (2)

2010 (5)

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35, 1992–1994 (2010).
[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, 553–562 (2010).
[Crossref]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[Crossref]

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

L. N. Trefethen, “Householder triangularization of a quasimatrix,” IMA J. Numer. Anal. 30, 887–897 (2010).
[Crossref]

2009 (2)

M. Otani, T. Hirahara, and S. Ise, “Numerical study on source-distance dependency of head-related transfer functions,” J. Acoust. Soc. Am. 125, 3253–3261 (2009).
[Crossref]

B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34, 416–418 (2009).
[Crossref]

2007 (2)

2006 (2)

M. Otani and S. Ise, “Fast calculation system specialized for head-related transfer function based on boundary element method,” J. Acoust. Soc. Am. 119, 2589–2598 (2006).
[Crossref]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678–687 (2006).
[Crossref]

2004 (1)

2003 (1)

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

1999 (1)

P. S. Huang, F. Jin, and F. P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
[Crossref]

1996 (1)

1995 (1)

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, 561–573 (1992).
[Crossref]

1983 (1)

Asundi, A.

Baker, M. J.

M. J. Baker, J. Xi, and J. F. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233–1243 (2007).
[Crossref]

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

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, 050501 (2014).
[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, 045201 (2016).
[Crossref]

Chen, J.

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, 094102 (2014).
[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, 533–538 (2012).
[Crossref]

Chen, M.

Chiang, F. P.

P. S. Huang, F. Jin, and F. P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
[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, 163–168 (2003).
[Crossref]

Chicharo, J.

Chicharo, J. F.

M. J. Baker, J. Xi, and J. F. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233–1243 (2007).
[Crossref]

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

Chitanont, N.

Dai, J.

Ekstrand, L.

Farrant, D. I.

Gorthi, S. S.

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

Guo, H.

Hao, Q.

Hassebrook, L. G.

He, H.

He, Y.

P. Zhou, X. Liu, Y. He, and T. Zhu, “Phase error analysis and compensation considering ambient light for phase measuring profilometry,” Opt. Lasers Eng. 55, 99–104 (2014).
[Crossref]

Hibino, K.

Hirahara, T.

M. Otani, T. Hirahara, and S. Ise, “Numerical study on source-distance dependency of head-related transfer functions,” J. Acoust. Soc. Am. 125, 3253–3261 (2009).
[Crossref]

Hoang, T.

Hu, Y.

Huang, L.

Huang, P. S.

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

P. S. Huang, F. Jin, and F. P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
[Crossref]

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, 045201 (2016).
[Crossref]

Ikeda, Y.

Ise, S.

M. Otani, T. Hirahara, and S. Ise, “Numerical study on source-distance dependency of head-related transfer functions,” J. Acoust. Soc. Am. 125, 3253–3261 (2009).
[Crossref]

M. Otani and S. Ise, “Fast calculation system specialized for head-related transfer function based on boundary element method,” J. Acoust. Soc. Am. 119, 2589–2598 (2006).
[Crossref]

Ishikawa, K.

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, 045201 (2016).
[Crossref]

Jin, F.

P. S. Huang, F. Jin, and F. P. Chiang, “Quantitative evaluation of corrosion by a digital fringe projection technique,” Opt. Lasers Eng. 31, 371–380 (1999).
[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, 050501 (2014).
[Crossref]

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, 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, 553–562 (2010).
[Crossref]

Li, B.

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, 533–538 (2012).
[Crossref]

Li, E.

Li, Y.

Li, Z.

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, 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, 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, 045201 (2016).
[Crossref]

P. Zhou, X. Liu, Y. He, and T. Zhu, “Phase error analysis and compensation considering ambient light for phase measuring profilometry,” Opt. Lasers Eng. 55, 99–104 (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, 533–538 (2012).
[Crossref]

Miao, H.

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, 094102 (2014).
[Crossref]

Murata, N.

S. Sonoda and N. Murata, “Sampling hidden parameters from oracle distribution,” in Proceedings of International Conference on Artificial Neural Networks (ICANN) (Springer, 2014), pp. 539–546.

Mutoh, K.

Nguyen, D.

Niwa, H.

Oikawa, Y.

Onuma, T.

Oreb, B. F.

Otani, M.

M. Otani, T. Hirahara, and S. Ise, “Numerical study on source-distance dependency of head-related transfer functions,” J. Acoust. Soc. Am. 125, 3253–3261 (2009).
[Crossref]

M. Otani and S. Ise, “Fast calculation system specialized for head-related transfer function based on boundary element method,” J. Acoust. Soc. Am. 119, 2589–2598 (2006).
[Crossref]

Pan, B.

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, 045201 (2016).
[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, 533–538 (2012).
[Crossref]

Rastogi, P.

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

Sonoda, S.

S. Sonoda and N. Murata, “Sampling hidden parameters from oracle distribution,” in Proceedings of International Conference on Artificial Neural Networks (ICANN) (Springer, 2014), pp. 539–546.

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, 561–573 (1992).
[Crossref]

Surrel, Y.

Takeda, M.

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, 533–538 (2012).
[Crossref]

Trefethen, L. N.

L. N. Trefethen, “Householder triangularization of a quasimatrix,” IMA J. Numer. Anal. 30, 887–897 (2010).
[Crossref]

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, 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, 561–573 (1992).
[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, 050501 (2014).
[Crossref]

Wang, X.

C. Zhang, H. Zhao, L. Zhang, and X. Wang, “Full-field phase error detection and compensation method for digital phase-shifting fringe projection profilometry,” Meas. Sci. Technol. 26, 035201 (2015).
[Crossref]

Wang, Y.

Wang, Z.

Xi, J.

M. J. Baker, J. Xi, and J. F. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233–1243 (2007).
[Crossref]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678–687 (2006).
[Crossref]

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

Xiong, C.

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, 094102 (2014).
[Crossref]

Xu, Y.

Yang, Z.

Yao, J.

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, 094102 (2014).
[Crossref]

Yatabe, K.

Yau, S.

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, 045201 (2016).
[Crossref]

Yoshii, M.

Zhang, C.

C. Zhang, H. Zhao, L. Zhang, and X. Wang, “Full-field phase error detection and compensation method for digital phase-shifting fringe projection profilometry,” Meas. Sci. Technol. 26, 035201 (2015).
[Crossref]

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

Zhang, L.

C. Zhang, H. Zhao, L. Zhang, and X. Wang, “Full-field phase error detection and compensation method for digital phase-shifting fringe projection profilometry,” Meas. Sci. Technol. 26, 035201 (2015).
[Crossref]

Zhang, S.

Zhang, X.

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, 045201 (2016).
[Crossref]

Zhao, H.

C. Zhang, H. Zhao, L. Zhang, and X. Wang, “Full-field phase error detection and compensation method for digital phase-shifting fringe projection profilometry,” Meas. Sci. Technol. 26, 035201 (2015).
[Crossref]

Zhou, P.

P. Zhou, X. Liu, Y. He, and T. Zhu, “Phase error analysis and compensation considering ambient light for phase measuring profilometry,” Opt. Lasers Eng. 55, 99–104 (2014).
[Crossref]

Zhou, W. S.

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, 561–573 (1992).
[Crossref]

Zhou, Y.

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, 094102 (2014).
[Crossref]

Zhu, L.

Zhu, R.

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, 533–538 (2012).
[Crossref]

Zhu, T.

P. Zhou, X. Liu, Y. He, and T. Zhu, “Phase error analysis and compensation considering ambient light for phase measuring profilometry,” Opt. Lasers Eng. 55, 99–104 (2014).
[Crossref]

Appl. Opt. (9)

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
[Crossref]

K. Yatabe and Y. Oikawa, “Convex optimization based windowed Fourier filtering with multiple windows for wrapped phase denoising,” Appl. Opt. 55, 4632–4641 (2016).
[Crossref]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[Crossref]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678–687 (2006).
[Crossref]

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

S. Zhang, “Comparative study on passive and active projector nonlinear gamma calibration,” Appl. Opt. 54, 3834–3841 (2015).
[Crossref]

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

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Figures (5)

Fig. 1.
Fig. 1. Phase reconstruction error of the simulated three-step phase-shifting method caused by the four nonlinear distortion effects. (a)–(d) Correspond to the cases in Eqs. (26)–(29). The blue dashed lines represent error without compensation. The red dotted lines are error after compensation by the gamma model, while the green solid lines are error after compensation by the polynomial model.
Fig. 2.
Fig. 2. Phase reconstruction error of the three-step phase-shifting method of a flat surface measured by a commercial projector and camera. (a) was obtained by projecting sinusoidal fringes, while (b) was defocused by binary fringes. The blue dashed lines represent error without compensation. The red dotted lines are error after compensation by the gamma model, while the green solid lines are error after compensation by the polynomial model.
Fig. 3.
Fig. 3. One of four fringe patterns of the ear pinna of a dummy head used as test data.
Fig. 4.
Fig. 4. Measured ear pinna of a dummy head obtained by the four-step phase-shifting method. The viewing angle was adjusted so that the periodic error caused by nonlinearity became more apparent. White lines (a) and (b) correspond to Figs. 5(a) and 5(b), respectively.
Fig. 5.
Fig. 5. Cross sections of the measured ear pinna. (a) and (b) correspond to the white lines in Fig. 4. The blue dashed lines represent the results without compensation, while the red dotted lines and the green solid lines are, respectively, the results after compensation with the gamma model and the polynomial model.

Tables (2)

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Table 1. RMS and Percentage of Error for Each Model

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Table 2. RMS and Percentage of Error the Measured Data

Equations (30)

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I ˜ ( x ) = N ( I ( x ) ) = N ( I 0 + cos [ φ ( x ) + δ ] ) ,
I ˜ ( x ) = c 0 + n c n cos [ n ( φ ( x ) + δ ) ] ,
( F I ) ( ω ) = R 2 I ( x ) e 2 π i ω , x d x ,
Ω High | ( F I ˜ ) ( ω ) | 2 d ω ,
R ( I ˜ ) = Ω High | ( F I ˜ ) ( ω ) | 2 d ω / Ω Low | ( F I ˜ ) ( ω ) | 2 d ω ,
Φ arg min Φ [ R ( Φ ( I ˜ ) ) ] ,
I ˜ ( x ) = N ( I ( x ) ) = ( I ( x ) ) Γ , ( Γ > 0 ) ,
γ arg min γ [ R ( I ˜ γ ) ] .
Φ α ( I ˜ ) = n = 1 N α n ψ n ( I ˜ ) ,
α arg min α [ R ( Φ α ( I ˜ ) ) ] .
Ψ = [ vec ( ψ 1 ( I ˜ ) ) , vec ( ψ 2 ( I ˜ ) ) , , vec ( ψ N ( I ˜ ) ) ] ,
Ψ α = Φ α ( I ˜ ) = n = 1 N α n ψ n ( I ˜ ) ,
α arg min α [ R ( Ψ α ) ] .
α arg min α [ S H F Ψ α 2 2 S L F Ψ α 2 2 ] ,
A = S H F Ψ , B = S L F Ψ ,
α arg min α [ A α 2 2 B α 2 2 ] ,
α arg min α [ α T A T A α α T B T B α ] ,
A T A α = λ B T B α ,
z 2 2 = | z | 2 = Re ( z ) 2 + Im ( z ) 2 ,
A α 2 2 = A ˜ α 2 2 , B α 2 2 = B ˜ α 2 2 ,
A ˜ = [ Re ( A ) T , Im ( A ) T ] T , B ˜ = [ Re ( B ) T , Im ( B ) T ] T .
A ˜ T A ˜ α = λ B ˜ T B ˜ α .
ψ n ( x ) = 2 n + 1 2 1 2 n n ! d n d x n ( x 2 1 ) n ,
I ˜ [ n ] = Ξ [ N ( 0.5 + 0.5 cos [ 2 π x + δ [ n ] ] ) ] ,
Ξ [ I ] = ( I min { I } ) / max { ( I min { I } ) } ,
N ( I ) = I 2 ,
N ( I ) = ( I + 1 ) 4 ,
N ( I ) = arctan [ 20 ( I 0.5 ) ] ,
N ( I ) = arctan [ 2 { ( I + 0.5 ) 5 0.5 } ] .
φ = Arg [ ( n I ˜ [ n ] cos [ δ [ n ] ] ) i ( n I ˜ [ n ] sin [ δ [ n ] ] ) ] ,

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