Abstract

A new 3D measuring method based on computer-generated moiré fringes is proposed. The two AC components of the 0-degree and 90-degree phase-shifted fringe patterns on reference plane are prepared in advance. While the AC component of the single-shot deformed pattern is multiplied by the two prepared AC components, respectively, two computer-generated moiré fringes can be retrieved. The ratio of the two computer-generated moiré fringes is just the tangent of the phase modulated by the object. It is of great potential in real-time or even dynamical 3D measurement due to its single-shot deformed pattern feature, and it avoids the influences of the object’s reflectivity simultaneously. Compared to the Fourier transform profilometry, its error is smaller due to its higher first-order spectrum. Experimental results show the feasibility and validity of the proposed method.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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  1. J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).
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    [PubMed]
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  25. X. Su, W. Song, Y. P. Cao, and L. Xiang, “Both phase height mapping and coordinates calibration in PMP,” Proc. SPIE 4839, 874–875 (2002).

2017 (1)

L. Zhu, Y. Cao, D. He, and C. Chen, “Real-time tricolor phase measuring profilometry based on CCD sensitivity calibration,” J. Mod. Opt. 64(4), 379–387 (2017).

2016 (7)

S. Wang, K. Yan, and L. Xue, “High speed moiré based phase retrieval method for quantitative phase imaging of thin objects without phase unwrapping or aberration compensation,” Opt. Commun. 359, 272–278 (2016).

L. Zhu, Y. Cao, D. He, and C. Chen, “Gray scale imbalance correction in real-time phase measuring profilometry,” Opt. Commun. 376, 72–80 (2016).

K. Peng, Y. Cao, Y. Wu, and M. Lu, “A new method using orthogonal two-frequency grating in online 3D measurement,” Opt. Laser Technol. 83, 81–88 (2016).

Y. Wan, Y. Cao, C. Chen, and K. Peng, “An online triple-frequency color-encoded fringe projection profilometry for discontinuous object,” J. Mod. Opt. 63(14), 1–8 (2016).

S. V. D. Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).

C. Jiang, T. Bell, and S. Zhang, “High dynamic range real-time 3D shape measurement,” Opt. Express 24(7), 7337–7346 (2016).
[PubMed]

K. Zhong, Z. Li, R. Li, Y. Shi, and C. Wang, “Pre-calibration-free 3D shape measurement method based on fringe projection,” Opt. Express 24(13), 14196–14207 (2016).
[PubMed]

2015 (1)

2014 (1)

2013 (1)

J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

2011 (1)

J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).

2010 (2)

Y. He and Y. Cao, “Three-dimensional measurement method with orthogonal composite grating aided by fringe contrast and background calibration,” Opt. Eng. 49(7), 717–720 (2010).

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).

2008 (1)

C. Guan, L. G. Hassebrook, D. L. Lau, D. L. Lau, V. G. Yalla, and C. J. Casey, “Improved composite-pattern structured-light profilometry by means of post-processing,” Opt. Eng. 47(47), 7203 (2008).

2007 (1)

2006 (2)

J. Pan, P. S. Huang, and F. P. Chiang, “Color phase-shifting technique for three-dimensional shape measurement,” Opt. Eng. 45(1), 013602 (2006).

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 1269–1278 (2006).

2004 (1)

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42(3), 245–261 (2004).

2002 (1)

X. Su, W. Song, Y. P. Cao, and L. Xiang, “Both phase height mapping and coordinates calibration in PMP,” Proc. SPIE 4839, 874–875 (2002).

2001 (2)

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40(4), 637–643 (2001).

X. Su, W. Chen, Q. Zhang, and Y. Cao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).

1988 (1)

1984 (1)

1983 (1)

Aguilar, L. M. A.

J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

Bao, Q.

Bell, T.

Cao, Y.

L. Zhu, Y. Cao, D. He, and C. Chen, “Real-time tricolor phase measuring profilometry based on CCD sensitivity calibration,” J. Mod. Opt. 64(4), 379–387 (2017).

L. Zhu, Y. Cao, D. He, and C. Chen, “Gray scale imbalance correction in real-time phase measuring profilometry,” Opt. Commun. 376, 72–80 (2016).

K. Peng, Y. Cao, Y. Wu, and M. Lu, “A new method using orthogonal two-frequency grating in online 3D measurement,” Opt. Laser Technol. 83, 81–88 (2016).

Y. Wan, Y. Cao, C. Chen, and K. Peng, “An online triple-frequency color-encoded fringe projection profilometry for discontinuous object,” J. Mod. Opt. 63(14), 1–8 (2016).

Y. He and Y. Cao, “Three-dimensional measurement method with orthogonal composite grating aided by fringe contrast and background calibration,” Opt. Eng. 49(7), 717–720 (2010).

X. Su, W. Chen, Q. Zhang, and Y. Cao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).

Cao, Y. P.

X. Su, W. Song, Y. P. Cao, and L. Xiang, “Both phase height mapping and coordinates calibration in PMP,” Proc. SPIE 4839, 874–875 (2002).

Casco-Vasquez, J. F.

J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

Casey, C. J.

C. Guan, L. G. Hassebrook, D. L. Lau, D. L. Lau, V. G. Yalla, and C. J. Casey, “Improved composite-pattern structured-light profilometry by means of post-processing,” Opt. Eng. 47(47), 7203 (2008).

Chang, W.-Y.

Chen, C.

L. Zhu, Y. Cao, D. He, and C. Chen, “Real-time tricolor phase measuring profilometry based on CCD sensitivity calibration,” J. Mod. Opt. 64(4), 379–387 (2017).

Y. Wan, Y. Cao, C. Chen, and K. Peng, “An online triple-frequency color-encoded fringe projection profilometry for discontinuous object,” J. Mod. Opt. 63(14), 1–8 (2016).

L. Zhu, Y. Cao, D. He, and C. Chen, “Gray scale imbalance correction in real-time phase measuring profilometry,” Opt. Commun. 376, 72–80 (2016).

Chen, J.-H.

Chen, K.-H.

Chen, W.

X. Mao, W. Chen, and X. Su, “Improved Fourier-transform profilometry,” Appl. Opt. 46(5), 664–668 (2007).
[PubMed]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42(3), 245–261 (2004).

X. Su, W. Chen, Q. Zhang, and Y. Cao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).

Chiang, F. P.

J. Pan, P. S. Huang, and F. P. Chiang, “Color phase-shifting technique for three-dimensional shape measurement,” Opt. Eng. 45(1), 013602 (2006).

Decraemer, W. F.

Dielis, G.

Dirckx, J. J. J.

Dong, J.

Geng, J.

J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).

Guan, C.

C. Guan, L. G. Hassebrook, D. L. Lau, D. L. Lau, V. G. Yalla, and C. J. Casey, “Improved composite-pattern structured-light profilometry by means of post-processing,” Opt. Eng. 47(47), 7203 (2008).

Halioua, M.

Hassebrook, L. G.

C. Guan, L. G. Hassebrook, D. L. Lau, D. L. Lau, V. G. Yalla, and C. J. Casey, “Improved composite-pattern structured-light profilometry by means of post-processing,” Opt. Eng. 47(47), 7203 (2008).

He, D.

L. Zhu, Y. Cao, D. He, and C. Chen, “Real-time tricolor phase measuring profilometry based on CCD sensitivity calibration,” J. Mod. Opt. 64(4), 379–387 (2017).

L. Zhu, Y. Cao, D. He, and C. Chen, “Gray scale imbalance correction in real-time phase measuring profilometry,” Opt. Commun. 376, 72–80 (2016).

He, Y.

Y. He and Y. Cao, “Three-dimensional measurement method with orthogonal composite grating aided by fringe contrast and background calibration,” Opt. Eng. 49(7), 717–720 (2010).

Hsu, F.-H.

Hsu, K. Y.

Huang, P. S.

J. Pan, P. S. Huang, and F. P. Chiang, “Color phase-shifting technique for three-dimensional shape measurement,” Opt. Eng. 45(1), 013602 (2006).

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 1269–1278 (2006).

Jeught, S. V. D.

S. V. D. Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).

Jia, S.

Jiang, C.

Juarez-Salazar, R.

J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

Lau, D. L.

C. Guan, L. G. Hassebrook, D. L. Lau, D. L. Lau, V. G. Yalla, and C. J. Casey, “Improved composite-pattern structured-light profilometry by means of post-processing,” Opt. Eng. 47(47), 7203 (2008).

C. Guan, L. G. Hassebrook, D. L. Lau, D. L. Lau, V. G. Yalla, and C. J. Casey, “Improved composite-pattern structured-light profilometry by means of post-processing,” Opt. Eng. 47(47), 7203 (2008).

Li, D.

Li, R.

Li, Z.

Lian, Q.

Liu, H. C.

Lu, M.

K. Peng, Y. Cao, Y. Wu, and M. Lu, “A new method using orthogonal two-frequency grating in online 3D measurement,” Opt. Laser Technol. 83, 81–88 (2016).

Mao, X.

Meneses, C.

J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

Mutoh, K.

Pan, J.

J. Pan, P. S. Huang, and F. P. Chiang, “Color phase-shifting technique for three-dimensional shape measurement,” Opt. Eng. 45(1), 013602 (2006).

Peng, K.

Y. Wan, Y. Cao, C. Chen, and K. Peng, “An online triple-frequency color-encoded fringe projection profilometry for discontinuous object,” J. Mod. Opt. 63(14), 1–8 (2016).

K. Peng, Y. Cao, Y. Wu, and M. Lu, “A new method using orthogonal two-frequency grating in online 3D measurement,” Opt. Laser Technol. 83, 81–88 (2016).

Robledo-Sanchez, C.

J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

Rodriguez-Zurita, G.

J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

Sanchez, F. G.

J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

Shi, Y.

Song, W.

X. Su, W. Song, Y. P. Cao, and L. Xiang, “Both phase height mapping and coordinates calibration in PMP,” Proc. SPIE 4839, 874–875 (2002).

Srinivasan, V.

Su, X.

X. Mao, W. Chen, and X. Su, “Improved Fourier-transform profilometry,” Appl. Opt. 46(5), 664–668 (2007).
[PubMed]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42(3), 245–261 (2004).

X. Su, W. Song, Y. P. Cao, and L. Xiang, “Both phase height mapping and coordinates calibration in PMP,” Proc. SPIE 4839, 874–875 (2002).

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40(4), 637–643 (2001).

X. Su, W. Chen, Q. Zhang, and Y. Cao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).

Takeda, M.

Wan, Y.

Y. Wan, Y. Cao, C. Chen, and K. Peng, “An online triple-frequency color-encoded fringe projection profilometry for discontinuous object,” J. Mod. Opt. 63(14), 1–8 (2016).

Wang, C.

Wang, S.

S. Wang, K. Yan, and L. Xue, “High speed moiré based phase retrieval method for quantitative phase imaging of thin objects without phase unwrapping or aberration compensation,” Opt. Commun. 359, 272–278 (2016).

Wu, Y.

K. Peng, Y. Cao, Y. Wu, and M. Lu, “A new method using orthogonal two-frequency grating in online 3D measurement,” Opt. Laser Technol. 83, 81–88 (2016).

Xiang, L.

X. Su, W. Song, Y. P. Cao, and L. Xiang, “Both phase height mapping and coordinates calibration in PMP,” Proc. SPIE 4839, 874–875 (2002).

Xue, L.

S. Wang, K. Yan, and L. Xue, “High speed moiré based phase retrieval method for quantitative phase imaging of thin objects without phase unwrapping or aberration compensation,” Opt. Commun. 359, 272–278 (2016).

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40(4), 637–643 (2001).

Yalla, V. G.

C. Guan, L. G. Hassebrook, D. L. Lau, D. L. Lau, V. G. Yalla, and C. J. Casey, “Improved composite-pattern structured-light profilometry by means of post-processing,” Opt. Eng. 47(47), 7203 (2008).

Yan, K.

S. Wang, K. Yan, and L. Xue, “High speed moiré based phase retrieval method for quantitative phase imaging of thin objects without phase unwrapping or aberration compensation,” Opt. Commun. 359, 272–278 (2016).

Yang, J.

Zhang, Q.

X. Su, W. Chen, Q. Zhang, and Y. Cao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).

Zhang, S.

C. Jiang, T. Bell, and S. Zhang, “High dynamic range real-time 3D shape measurement,” Opt. Express 24(7), 7337–7346 (2016).
[PubMed]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 1269–1278 (2006).

Zhong, K.

Zhu, L.

L. Zhu, Y. Cao, D. He, and C. Chen, “Real-time tricolor phase measuring profilometry based on CCD sensitivity calibration,” J. Mod. Opt. 64(4), 379–387 (2017).

L. Zhu, Y. Cao, D. He, and C. Chen, “Gray scale imbalance correction in real-time phase measuring profilometry,” Opt. Commun. 376, 72–80 (2016).

Adv. Opt. Photonics (1)

J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).

Appl. Opt. (4)

J. Mod. Opt. (2)

L. Zhu, Y. Cao, D. He, and C. Chen, “Real-time tricolor phase measuring profilometry based on CCD sensitivity calibration,” J. Mod. Opt. 64(4), 379–387 (2017).

Y. Wan, Y. Cao, C. Chen, and K. Peng, “An online triple-frequency color-encoded fringe projection profilometry for discontinuous object,” J. Mod. Opt. 63(14), 1–8 (2016).

Opt. Commun. (2)

L. Zhu, Y. Cao, D. He, and C. Chen, “Gray scale imbalance correction in real-time phase measuring profilometry,” Opt. Commun. 376, 72–80 (2016).

S. Wang, K. Yan, and L. Xue, “High speed moiré based phase retrieval method for quantitative phase imaging of thin objects without phase unwrapping or aberration compensation,” Opt. Commun. 359, 272–278 (2016).

Opt. Eng. (6)

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40(4), 637–643 (2001).

J. Pan, P. S. Huang, and F. P. Chiang, “Color phase-shifting technique for three-dimensional shape measurement,” Opt. Eng. 45(1), 013602 (2006).

C. Guan, L. G. Hassebrook, D. L. Lau, D. L. Lau, V. G. Yalla, and C. J. Casey, “Improved composite-pattern structured-light profilometry by means of post-processing,” Opt. Eng. 47(47), 7203 (2008).

Y. He and Y. Cao, “Three-dimensional measurement method with orthogonal composite grating aided by fringe contrast and background calibration,” Opt. Eng. 49(7), 717–720 (2010).

J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 1269–1278 (2006).

Opt. Express (4)

Opt. Laser Technol. (1)

K. Peng, Y. Cao, Y. Wu, and M. Lu, “A new method using orthogonal two-frequency grating in online 3D measurement,” Opt. Laser Technol. 83, 81–88 (2016).

Opt. Lasers Eng. (4)

S. V. D. Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).

X. Su, W. Chen, Q. Zhang, and Y. Cao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42(3), 245–261 (2004).

Proc. SPIE (1)

X. Su, W. Song, Y. P. Cao, and L. Xiang, “Both phase height mapping and coordinates calibration in PMP,” Proc. SPIE 4839, 874–875 (2002).

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Figures (8)

Fig. 1
Fig. 1

Optical measurement system.

Fig. 2
Fig. 2

The process flow chart of computer generated moiré profilometry.

Fig. 3
Fig. 3

Stimulated results: (a) simulated object; (b) the deformed pattern; (c) 0-degree moiré fringe; (d) 90-degree moiré fringe; (e) wrapped phase; (f) reconstructed object.

Fig. 4
Fig. 4

Error distributions: (a) with FTP; (b) with CGPM.

Fig. 5
Fig. 5

Measuring results of CGMP: (a) measured object; (b) deformed pattern; (c) 0-degree moiré fringe; (d) 90-degree moiré fringe; (e) wrapped phase; (f) reconstructed object.

Fig. 6
Fig. 6

Measuring results of a colored object: (a) the measured object with color painted; (b) the deformed pattern; (c) wrapped phase of the object; (d) reconstructed object; (e) cutaway views at row 420; (f) cutaway views at row 535.

Fig. 7
Fig. 7

Comparison between FTP and CGMP for measuring the steep object: (a) the measured object; (b) the deformed pattern; (c) the cutaway views at column 450 with PMP, FTP and CGMP.

Fig. 8
Fig. 8

Spectrum distributions in different conditions: (a) with FTP; (b) with CGMP.

Tables (1)

Tables Icon

Table 1 Experimental results for different known heights (/ mm).

Equations (14)

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I 1 (x,y)=R(x,y){A+Bcos[2π f 0 x+ φ 0 (x,y)]},
I 2 (x,y)=R(x,y){A+Bcos[2π f 0 x+ φ 0 (x,y)+π/2]},
I 3 (x,y)=R(x,y){A+Bcos[2π f 0 x+ φ 0 (x,y)+π]},
I 4 (x,y)=R(x,y){A+Bcos[2π f 0 x+ φ 0 (x,y)]+3π/2}.
I ˜ 0 R (x,y)= 1 2 [ I 1 (x,y) I 3 (x,y)]=R(x,y)Bcos[2π f 0 x+ φ 0 (x,y)].
I ˜ 90 R (x,y)= 1 2 [ I 2 (x,y) I 4 (x,y)]=R(x,y)Bcos[2π f 0 x+ φ 0 (x,y)+π/2].
I o (x,y)= R ' (x,y){A+Bcos[2π f 0 x+φ(x,y)]} = R ' (x,y)A+ R ' (x,y)Bcos[2π f 0 x+φ(x,y)].
I ˜ 0 O (x,y)= R ' (x,y)Bcos[2π f 0 x+φ(x,y)] = I o (x,y)abs(FF T 1 {FFT{ I o (x,y)}rect(x/ f xmax O ,y/ f ymax O )}).
I 0 OR (x,y)= I ˜ 0 O (x,y)× I ˜ 0 R (x,y)= 1 2 R(x,y) R ' (x,y) B 2 cos[4π f 0 x+ φ 0 (x,y)+φ(x,y)], + 1 2 R(x,y) R ' (x,y) B 2 cos[φ(x,y) φ 0 (x,y)]
I 90 OR (x,y)= I ˜ 0 O (x,y)× I ˜ 0 R (x,y)= 1 2 R(x,y) R ' (x,y) B 2 sin[4π f 0 x+ φ 0 (x,y)+φ(x,y)]. + 1 2 R(x,y) R ' (x,y) B 2 sin[φ(x,y) φ 0 (x,y)]
I moire0 (x,y)= 1 2 R(x,y) R ' (x,y) B 2 cos[φ(x,y) φ 0 (x,y)], =FF T 1 {FFT{ I 0 OR (x,y)}rect(x/ f xmax m ,y/ f ymax m )}
I moire90 (x,y)= 1 2 R(x,y) R ' (x,y) B 2 sin[φ(x,y) φ 0 (x,y)] =FF T 1 {FFT{ I 90 OR (x,y)}rect(x/ f xmax m ,y/ f ymax m )}.
tan[φ(x,y) φ 0 (x,y)]= I moire90 (x,y) I moire0 (x,y) .
1 h(x,y) =a(x,y)+b(x,y) 1 ϕ(x,y) +c(x,y) 1 ϕ 2 (x,y) .

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