Abstract

Three-dimensional (3D) shape reconstruction by projection of defocused binary patterns overcomes the nonlinearity introduced by the projector. Current patch-based procedures that generate dithered patterns are time consuming and are affected by the harmonics introduced through the tiling process. To overcome this problem, we propose a novel idea, to the best of our knowledge, to generate dithering patterns using the composition of two-dimensional patches as a stack of one-dimensional arrays obtained through an efficient deterministic approach. This procedure is a one-dimension optimization problem in the intensity domain, employing only a quarter of the fringe pitch. Furthermore, the unwanted distorting harmonics are eliminated using a Hilbert transform method. Both numerical simulations and experimental results verify the effectiveness of the proposal.

© 2020 Optical Society of America

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References

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  1. S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
    [Crossref]
  2. C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
    [Crossref]
  3. C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: a comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
    [Crossref]
  4. C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Lasers Eng. 109, 23–59 (2018).
    [Crossref]
  5. 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, 063601 (2007).
    [Crossref]
  6. Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
    [Crossref]
  7. S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34, 3080–3082 (2009).
    [Crossref]
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    [Crossref]
  9. Y. Wang and S. Zhang, “Optimum pulse width modulation for sinusoidal fringe generation with projector defocusing,” Opt. Lett. 35, 4121–4123 (2010).
    [Crossref]
  10. C. Zuo, Q. Chen, S. Feng, F. Feng, G. Gu, and X. Sui, “Optimized pulse width modulation pattern strategy for three-dimensional profilometry with projector defocusing,” Appl. Opt. 51, 4477–4490 (2012).
    [Crossref]
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    [Crossref]
  12. Y. Wang and S. Zhang, “Three-dimensional shape measurement with binary dithered patterns,” Appl. Opt. 51, 6631–6636 (2012).
    [Crossref]
  13. Y. Wang, J. I. Laughner, I. R. Efimov, and S. Zhang, “3D absolute shape measurement of live rabbit hearts with a superfast two-frequency phase-shifting technique,” Opt. Express 21, 5822–5832 (2013).
    [Crossref]
  14. W. Lohry and S. Zhang, “Genetic method to optimize binary dithering technique for high-quality fringe generation,” Opt. Lett. 38, 540–542 (2013).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  22. B. Li, N. Karpinsky, and S. Zhang, “Novel calibration method for structured-light system with an out-of-focus projector,” Appl. Opt. 53, 3415–3426 (2014).
    [Crossref]
  23. V. D. Madjarova, H. Kadono, and S. Toyooka, “Dynamic electronic speckle pattern interferometry (DESPI) phase analyses with temporal Hilbert transform,” Opt. Express 11, 617–623 (2003).
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  24. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Design of phase shifting algorithms: fringe contrast maximum,” Opt. Express 22, 18203–18213 (2014).
    [Crossref]
  25. M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley, 2014).

2018 (2)

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

F. Lu, C. Wu, and J. Yang, “High-quality binary fringe patterns generation by combining optimization on global and local similarity,” J. Eur. Opt. Soc. Rapid Publ. 14, 12 (2018).
[Crossref]

2017 (2)

Z. X. Xu and Y. H. Chan, “Removing harmonic distortion of measurements of a defocusing three-step phase-shifting digital fringe projection system,” Opt. Lasers Eng. 90, 139–145 (2017).
[Crossref]

A. Silva, J. L. Flores, A. Muñoz, G. A. Ayubi, and J. A. Ferrari, “Three-dimensional shape profiling by out-of-focus projection of colored pulse width modulation fringe patterns,” Appl. Opt. 56, 5198–5203 (2017).
[Crossref]

2016 (1)

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: a comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

2015 (1)

2014 (4)

B. Li, N. Karpinsky, and S. Zhang, “Novel calibration method for structured-light system with an out-of-focus projector,” Appl. Opt. 53, 3415–3426 (2014).
[Crossref]

Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Design of phase shifting algorithms: fringe contrast maximum,” Opt. Express 22, 18203–18213 (2014).
[Crossref]

J. Dai, B. Li, and S. Zhang, “Intensity-optimized dithering technique for three-dimensional shape measurement with projector defocusing,” Opt. Lasers Eng. 53, 79–85 (2014).
[Crossref]

J. Dai, B. Li, and S. Zhang, “High-quality fringe pattern generation using binary pattern optimization through symmetry and periodicity,” Opt. Lasers Eng. 52, 195–200 (2014).
[Crossref]

2013 (3)

2012 (2)

2010 (4)

G. A. Ayubi, J. A. Ayubi, J. M. Di Martino, and J. A. Ferrari, “Pulse-width modulation in defocused three-dimensional fringe projection,” Opt. Lett. 35, 3682–3684 (2010).
[Crossref]

Y. Wang and S. Zhang, “Optimum pulse width modulation for sinusoidal fringe generation with projector defocusing,” Opt. Lett. 35, 4121–4123 (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]

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[Crossref]

2009 (1)

2008 (1)

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[Crossref]

2007 (1)

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, 063601 (2007).
[Crossref]

2003 (1)

1994 (1)

Asundi, A.

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: a comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

Ayubi, G. A.

Ayubi, J. A.

Cai, Z.

Chan, Y. H.

Z. X. Xu and Y. H. Chan, “Removing harmonic distortion of measurements of a defocusing three-step phase-shifting digital fringe projection system,” Opt. Lasers Eng. 90, 139–145 (2017).
[Crossref]

Chen, Q.

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: a comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

C. Zuo, Q. Chen, S. Feng, F. Feng, G. Gu, and X. Sui, “Optimized pulse width modulation pattern strategy for three-dimensional profilometry with projector defocusing,” Appl. Opt. 51, 4477–4490 (2012).
[Crossref]

Chen, W.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[Crossref]

Dai, J.

J. Dai, B. Li, and S. Zhang, “High-quality fringe pattern generation using binary pattern optimization through symmetry and periodicity,” Opt. Lasers Eng. 52, 195–200 (2014).
[Crossref]

J. Dai, B. Li, and S. Zhang, “Intensity-optimized dithering technique for three-dimensional shape measurement with projector defocusing,” Opt. Lasers Eng. 53, 79–85 (2014).
[Crossref]

J. Dai and S. Zhang, “Phase-optimized dithering technique for high-quality 3D shape measurement,” Opt. Lasers Eng. 51, 790–795 (2013).
[Crossref]

Di Martino, J. M.

Efimov, I. R.

Feng, F.

Feng, S.

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

C. Zuo, Q. Chen, S. Feng, F. Feng, G. Gu, and X. Sui, “Optimized pulse width modulation pattern strategy for three-dimensional profilometry with projector defocusing,” Appl. Opt. 51, 4477–4490 (2012).
[Crossref]

Ferrari, J. A.

Flores, J. L.

Ghiglia, D. C.

Gu, G.

He, D.

Hibino, K.

Huang, L.

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: a comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

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, 063601 (2007).
[Crossref]

Huang, S.

Jiang, H.

Kadono, H.

Karpinsky, N.

Kim, Y.

Laughner, J. I.

Lei, S.

Li, B.

B. Li, N. Karpinsky, and S. Zhang, “Novel calibration method for structured-light system with an out-of-focus projector,” Appl. Opt. 53, 3415–3426 (2014).
[Crossref]

J. Dai, B. Li, and S. Zhang, “Intensity-optimized dithering technique for three-dimensional shape measurement with projector defocusing,” Opt. Lasers Eng. 53, 79–85 (2014).
[Crossref]

J. Dai, B. Li, and S. Zhang, “High-quality fringe pattern generation using binary pattern optimization through symmetry and periodicity,” Opt. Lasers Eng. 52, 195–200 (2014).
[Crossref]

Li, Z.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[Crossref]

Liu, X.

Lohry, W.

Lu, F.

F. Lu, C. Wu, and J. Yang, “High-quality binary fringe patterns generation by combining optimization on global and local similarity,” J. Eur. Opt. Soc. Rapid Publ. 14, 12 (2018).
[Crossref]

Madjarova, V. D.

Mitsuishi, M.

Muñoz, A.

Padilla, M.

M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley, 2014).

Peng, X.

Quan, C.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[Crossref]

Quiroga, J. A.

M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley, 2014).

Romero, L. A.

Servin, M.

M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley, 2014).

Shi, Y.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[Crossref]

Silva, A.

Sugita, N.

Sui, X.

Tao, T.

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

Tay, C. J.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[Crossref]

Toyooka, S.

Wang, C.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[Crossref]

Wang, Y.

Wu, C.

F. Lu, C. Wu, and J. Yang, “High-quality binary fringe patterns generation by combining optimization on global and local similarity,” J. Eur. Opt. Soc. Rapid Publ. 14, 12 (2018).
[Crossref]

Xu, Z. X.

Z. X. Xu and Y. H. Chan, “Removing harmonic distortion of measurements of a defocusing three-step phase-shifting digital fringe projection system,” Opt. Lasers Eng. 90, 139–145 (2017).
[Crossref]

Yang, J.

F. Lu, C. Wu, and J. Yang, “High-quality binary fringe patterns generation by combining optimization on global and local similarity,” J. Eur. Opt. Soc. Rapid Publ. 14, 12 (2018).
[Crossref]

Yin, W.

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

Zhang, M.

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: a comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

Zhang, S.

B. Li, N. Karpinsky, and S. Zhang, “Novel calibration method for structured-light system with an out-of-focus projector,” Appl. Opt. 53, 3415–3426 (2014).
[Crossref]

J. Dai, B. Li, and S. Zhang, “High-quality fringe pattern generation using binary pattern optimization through symmetry and periodicity,” Opt. Lasers Eng. 52, 195–200 (2014).
[Crossref]

J. Dai, B. Li, and S. Zhang, “Intensity-optimized dithering technique for three-dimensional shape measurement with projector defocusing,” Opt. Lasers Eng. 53, 79–85 (2014).
[Crossref]

Y. Wang, J. I. Laughner, I. R. Efimov, and S. Zhang, “3D absolute shape measurement of live rabbit hearts with a superfast two-frequency phase-shifting technique,” Opt. Express 21, 5822–5832 (2013).
[Crossref]

J. Dai and S. Zhang, “Phase-optimized dithering technique for high-quality 3D shape measurement,” Opt. Lasers Eng. 51, 790–795 (2013).
[Crossref]

W. Lohry and S. Zhang, “Genetic method to optimize binary dithering technique for high-quality fringe generation,” Opt. Lett. 38, 540–542 (2013).
[Crossref]

Y. Wang and S. Zhang, “Three-dimensional shape measurement with binary dithered patterns,” Appl. Opt. 51, 6631–6636 (2012).
[Crossref]

Y. Wang and S. Zhang, “Optimum pulse width modulation for sinusoidal fringe generation with projector defocusing,” Opt. Lett. 35, 4121–4123 (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. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34, 3080–3082 (2009).
[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, 063601 (2007).
[Crossref]

Zhang, Z.

Zuo, C.

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: a comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

C. Zuo, Q. Chen, S. Feng, F. Feng, G. Gu, and X. Sui, “Optimized pulse width modulation pattern strategy for three-dimensional profilometry with projector defocusing,” Appl. Opt. 51, 4477–4490 (2012).
[Crossref]

Appl. Opt. (4)

J. Eur. Opt. Soc. Rapid Publ. (1)

F. Lu, C. Wu, and J. Yang, “High-quality binary fringe patterns generation by combining optimization on global and local similarity,” J. Eur. Opt. Soc. Rapid Publ. 14, 12 (2018).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

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, 063601 (2007).
[Crossref]

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47, 053604 (2008).
[Crossref]

Opt. Express (4)

Opt. Lasers Eng. (8)

J. Dai and S. Zhang, “Phase-optimized dithering technique for high-quality 3D shape measurement,” Opt. Lasers Eng. 51, 790–795 (2013).
[Crossref]

J. Dai, B. Li, and S. Zhang, “Intensity-optimized dithering technique for three-dimensional shape measurement with projector defocusing,” Opt. Lasers Eng. 53, 79–85 (2014).
[Crossref]

J. Dai, B. Li, and S. Zhang, “High-quality fringe pattern generation using binary pattern optimization through symmetry and periodicity,” Opt. Lasers Eng. 52, 195–200 (2014).
[Crossref]

Z. X. Xu and Y. H. Chan, “Removing harmonic distortion of measurements of a defocusing three-step phase-shifting digital fringe projection system,” Opt. Lasers Eng. 90, 139–145 (2017).
[Crossref]

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

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: a comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Lasers Eng. 109, 23–59 (2018).
[Crossref]

Opt. Lett. (4)

Other (1)

M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley, 2014).

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Figures (10)

Fig. 1.
Fig. 1. Schematic setup for digital fringe projection profilometry.
Fig. 2.
Fig. 2. Binary patch generated for $T = 36\;{\rm pixels}$ and $m = {5}$, 7, 9, 11, and 13.
Fig. 3.
Fig. 3. Binary fringe pattern (a) synthesized by the present approach by tiling and repeating the binary patch shown in Fig. 2; (b) quasi-sinusoidal patterns by filtering it using a 2D Gaussian kernel, and its intensity cut along the horizontal axis (red line).
Fig. 4.
Fig. 4. Phase error as the difference between the phase map retrieved from ideal sinusoidal patterns and smoothed binary patterns (black line), and smoothed binary dithered patterns plus the HT compensation (red line); (a) a pitch of 72 pixels and (b) a pitch of 120 pixels.
Fig. 5.
Fig. 5. Numerical results. The rms error as a function of the fringe pitch for three defocusing levels, which were simulated by a Gaussian low-pass filter with different sizes: (a) ${5} \times {5}\;{\rm pixels}$ and $\sigma = {5}/{3}\;{\rm pixels}$, (b) ${9} \times {9}\;{\rm pixels}$ and $\sigma = 3\;{\rm pixels}$, and (c) ${13} \times {13}\;{\rm pixels}$ and $\sigma = {13/3}\;{\rm pixels}$.
Fig. 6.
Fig. 6. Experimental results. The rms error as a function of the defocusing level for three different fringe pitches.
Fig. 7.
Fig. 7. Retrieved phase map by (a) nine-step and (b) three-step PSA from sinusoidal fringe patterns, respectively; (c) three-step PSA from the defocused EDA patterns. (d) Horizontal phase cuts, where the black, blue, and red lines correspond to the phase shown in (a)–(c), respectively.
Fig. 8.
Fig. 8. Phase retrieved in (a) the nine-step PSA from sinusoidal fringe patterns; (b) and (c) three-step PSA and HT from the sinusoidal and defocused EDA fringe patterns, respectively. (d) Horizontal phase cuts, where the black, blue, and red lines correspond to the phase shown in (a)–(c), respectively.
Fig. 9.
Fig. 9. Experimental result: differences between the ground-truth nine-step PSA and our proposed ${\rm EDA} + {\rm HT}$. We also include the three-step with sinusoidal fringe comparison.
Fig. 10.
Fig. 10. 3D shape retrieved by using fringe patterns employing nine- and three-step PSAs together with sinusoidal fringe patterns on the left and middle columns, respectively. The right column using the three-step PSA with EDA defocused fringe patterns. The fringe pitch used was 24, 72, and 120 pixels for the first, second, and third row, respectively.

Tables (1)

Tables Icon

Algorithm 1. Exhaustive Dithering Algorithm

Equations (11)

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I k ( x , y ) = a ( x , y ) + b ( x , y ) cos [ 2 π f x + ϕ ( x , y ) k δ ] , ( k = 1 , 2 , , N ) ,
ϕ ( x , y ) = tan 1 [ k = 1 N I k ( x , y ) sin ( 2 π k / N ) k = 1 N I k ( x , y ) cos ( 2 π k / N ) ] ,
I 1 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) 2 π / 3 ] ,
I 2 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) ] ,
I 3 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + 2 π / 3 ] ,
ϕ ( x , y ) = tan 1 { 3 [ I 1 ( x , y ) I 3 ( x , y ) ] 2 I 2 ( x , y ) I 1 ( x , y ) I 3 ( x , y ) } .
min G , B I ( x , y ) G ( x , y ) B ( x , y ) 2 ,
min b s ( x ) g ( x ) b ( x ) 2 ,
I k H ( x , y ) = a ( x , y ) + b ( x , y ) sin [ 2 π f x + ϕ ( x , y ) k δ ] ,
ϕ F ( x , y ) = ϕ ( x , y ) + ϕ H ( x , y ) 2 .
r m s = ( 1 N × M y = 1 N x = 1 M [ ϕ s ( x , y ) ϕ d ( x , y ) ] 2 ) 1 / 2 ,

Metrics