Abstract

Fringe-pattern profilometry (FPP) has been widely used for phase reconstruction. It involves the use of phase shifting for phase retrieval. Phase-shift errors can affect the accuracy of phase reconstruction, and limited studies have been dedicated to studying phase-shift errors due to experimental, human, or environmental factors. We propose a simple and yet accurate phase-shift estimation method. Our study shows that the method is able to accurately estimate the actual phase shifts used in the FPP technique. The proposed method can find its applications in FPP and other phase shifting-based three-dimensional imaging techniques.

© 2019 Optical Society of America

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References

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  1. S. S. Gorthi and P. Pastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
    [Crossref]
  2. J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [Crossref]
  3. I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [Crossref]
  4. S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
    [Crossref]
  5. H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 547–612.
  6. L. Song, X. Dong, J. Xi, Y. Yu, and C. Yang, “A new phase unwrapping algorithm based on three wavelength phase shift profilometry method,” Opt. Lasers Technol. 45, 319–329 (2013).
    [Crossref]
  7. P. Huang and S. Zhang, “Fast three-step phase shifting algorithm,” Appl. Opt. 45, 5086–5091 (2006).
    [Crossref]
  8. Z. Wang, “Robust measurement of the diffuse surface by phase shift profilometry,” J. Opt. 16, 105407 (2014).
    [Crossref]
  9. J. L. Flores, J. A. Ferrari, G. G. Torales, R. Legarda-Saenz, and A. Silva, “Color-fringe pattern profilometry using a generalized phase shifting algorithm,” Appl. Opt. 54, 8827–8834 (2015).
    [Crossref]
  10. F. Chen, G. M. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10–23 (2000).
    [Crossref]
  11. Z. Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” in Proceedings of the Seventh IEEE International Conference on Computer Vision (ICCV) (1999), Vol. 1, pp. 666–673.
  12. J. Vargas, J. A. Quiroga, and M. J. Terron-Lopez, “Flexible calibration procedure for fringe projection profilometry,” Opt. Eng. 46, 023601 (2007).
    [Crossref]
  13. R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2007).
    [Crossref]
  14. Y. Yin, X. Peng, A. Li, X. Liu, and B. Gao, “Calibration of fringe projection profilometry with bundle adjustment strategy,” Opt. Lett. 37, 542–544 (2012).
    [Crossref]
  15. G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–381 (2003).
    [Crossref]
  16. L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
    [Crossref]
  17. L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. 34, 2363–2365 (2009).
    [Crossref]
  18. H. Du and Z. Wang, “Three-dimensional shape measurement with an arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32, 2438–2440 (2007).
    [Crossref]
  19. B. Pan, Q. Kemao, L. Huang, and A. K. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase shifting digital fringe projection profilometry,” Opt. Lett. 34, 416–418 (2009).
    [Crossref]
  20. J. Pan, P. S. Huang, and F. P. Chiang, “Color phase shifting technique for three-dimensional shape measurement,” Opt. Eng. 45, 013602 (2006).
    [Crossref]
  21. L. Wang, H. Zhang, and Y. Xin, “A simple fringe pattern profilometry phase shift error quantification method,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2018), paper JW4A.25.
  22. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [Crossref]
  23. P. De Groot, “Derivation of algorithms for phase shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [Crossref]
  24. J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase shifting interferometry based on principal component analysis,” Opt. Lett. 36, 1326–1328 (2011).
    [Crossref]
  25. J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase shifting interferometry,” Opt. Lett. 36, 2215–2217 (2011).
    [Crossref]
  26. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase shifted interferograms,” Opt. Lett. 29, 1671–1673 (2004).
    [Crossref]
  27. Q. Hao, Q. Zhu, and Y. Hu, “Random phase shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34, 1288–1290 (2009).
    [Crossref]
  28. O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express 13, 3743–3753 (2005).
    [Crossref]
  29. R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express 22, 4738–4750 (2014).
    [Crossref]

2015 (1)

2014 (2)

2013 (1)

L. Song, X. Dong, J. Xi, Y. Yu, and C. Yang, “A new phase unwrapping algorithm based on three wavelength phase shift profilometry method,” Opt. Lasers Technol. 45, 319–329 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (3)

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[Crossref]

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

2009 (3)

2007 (3)

H. Du and Z. Wang, “Three-dimensional shape measurement with an arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32, 2438–2440 (2007).
[Crossref]

J. Vargas, J. A. Quiroga, and M. J. Terron-Lopez, “Flexible calibration procedure for fringe projection profilometry,” Opt. Eng. 46, 023601 (2007).
[Crossref]

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2007).
[Crossref]

2006 (2)

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

P. Huang and S. Zhang, “Fast three-step phase shifting algorithm,” Appl. Opt. 45, 5086–5091 (2006).
[Crossref]

2005 (1)

2004 (1)

2003 (1)

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–381 (2003).
[Crossref]

2000 (1)

F. Chen, G. M. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10–23 (2000).
[Crossref]

1997 (1)

1995 (1)

1993 (1)

1987 (1)

Asundi, A. K.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[Crossref]

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

Belenguer, T.

Bothe, T.

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2007).
[Crossref]

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10–23 (2000).
[Crossref]

Bruning, J. H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 547–612.

Chen, F.

F. Chen, G. M. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10–23 (2000).
[Crossref]

Chiang, F. P.

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

De Groot, P.

Dong, X.

L. Song, X. Dong, J. Xi, Y. Yu, and C. Yang, “A new phase unwrapping algorithm based on three wavelength phase shift profilometry method,” Opt. Lasers Technol. 45, 319–329 (2013).
[Crossref]

Du, H.

Eiju, T.

Ferrari, J. A.

Flores, J. L.

Gao, B.

Gorthi, S. S.

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

Guerrero-Sanchez, F.

Han, B.

Hao, Q.

Hariharan, P.

Hu, Y.

Huang, L.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[Crossref]

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

Huang, P.

Huang, P. S.

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

Huntley, J. M.

Jia, S.

Juarez-Salazar, R.

Juptner, W. P.

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2007).
[Crossref]

Kemao, Q.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[Crossref]

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

Legarda-Saenz, R.

J. L. Flores, J. A. Ferrari, G. G. Torales, R. Legarda-Saenz, and A. Silva, “Color-fringe pattern profilometry using a generalized phase shifting algorithm,” Appl. Opt. 54, 8827–8834 (2015).
[Crossref]

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2007).
[Crossref]

Li, A.

Liu, X.

Notni, G.

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–381 (2003).
[Crossref]

Notni, G. H.

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–381 (2003).
[Crossref]

Oreb, B. F.

Pan, B.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[Crossref]

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

Pan, J.

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

Pastogi, P.

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

Peng, X.

Quiroga, J. A.

Rangel-Huerta, A.

Robledo-Sanchez, C.

Saldner, H.

Schreiber, H.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 547–612.

Silva, A.

Soloviev, O.

Song, L.

L. Song, X. Dong, J. Xi, Y. Yu, and C. Yang, “A new phase unwrapping algorithm based on three wavelength phase shift profilometry method,” Opt. Lasers Technol. 45, 319–329 (2013).
[Crossref]

Song, M.

F. Chen, G. M. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10–23 (2000).
[Crossref]

Terron-Lopez, M. J.

J. Vargas, J. A. Quiroga, and M. J. Terron-Lopez, “Flexible calibration procedure for fringe projection profilometry,” Opt. Eng. 46, 023601 (2007).
[Crossref]

Torales, G. G.

Vargas, J.

Vdovin, G.

Wang, L.

L. Wang, H. Zhang, and Y. Xin, “A simple fringe pattern profilometry phase shift error quantification method,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2018), paper JW4A.25.

Wang, Z.

Xi, J.

L. Song, X. Dong, J. Xi, Y. Yu, and C. Yang, “A new phase unwrapping algorithm based on three wavelength phase shift profilometry method,” Opt. Lasers Technol. 45, 319–329 (2013).
[Crossref]

Xin, Y.

L. Wang, H. Zhang, and Y. Xin, “A simple fringe pattern profilometry phase shift error quantification method,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2018), paper JW4A.25.

Xiong, L.

Yamaguchi, I.

Yang, C.

L. Song, X. Dong, J. Xi, Y. Yu, and C. Yang, “A new phase unwrapping algorithm based on three wavelength phase shift profilometry method,” Opt. Lasers Technol. 45, 319–329 (2013).
[Crossref]

Yin, Y.

Yu, Y.

L. Song, X. Dong, J. Xi, Y. Yu, and C. Yang, “A new phase unwrapping algorithm based on three wavelength phase shift profilometry method,” Opt. Lasers Technol. 45, 319–329 (2013).
[Crossref]

Zhang, H.

L. Wang, H. Zhang, and Y. Xin, “A simple fringe pattern profilometry phase shift error quantification method,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2018), paper JW4A.25.

Zhang, S.

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

P. Huang and S. Zhang, “Fast three-step phase shifting algorithm,” Appl. Opt. 45, 5086–5091 (2006).
[Crossref]

Zhang, T.

Zhang, Z.

Z. Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” in Proceedings of the Seventh IEEE International Conference on Computer Vision (ICCV) (1999), Vol. 1, pp. 666–673.

Zhu, Q.

Appl. Opt. (5)

J. Opt. (1)

Z. Wang, “Robust measurement of the diffuse surface by phase shift profilometry,” J. Opt. 16, 105407 (2014).
[Crossref]

Opt. Eng. (4)

F. Chen, G. M. Brown, and M. Song, “Overview of 3-D shape measurement using optical methods,” Opt. Eng. 39, 10–23 (2000).
[Crossref]

J. Vargas, J. A. Quiroga, and M. J. Terron-Lopez, “Flexible calibration procedure for fringe projection profilometry,” Opt. Eng. 46, 023601 (2007).
[Crossref]

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2007).
[Crossref]

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

Opt. Express (2)

Opt. Lasers Eng. (3)

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (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. Pastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[Crossref]

Opt. Lasers Technol. (1)

L. Song, X. Dong, J. Xi, Y. Yu, and C. Yang, “A new phase unwrapping algorithm based on three wavelength phase shift profilometry method,” Opt. Lasers Technol. 45, 319–329 (2013).
[Crossref]

Opt. Lett. (9)

I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
[Crossref]

H. Du and Z. Wang, “Three-dimensional shape measurement with an arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32, 2438–2440 (2007).
[Crossref]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase shifted interferograms,” Opt. Lett. 29, 1671–1673 (2004).
[Crossref]

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

Q. Hao, Q. Zhu, and Y. Hu, “Random phase shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34, 1288–1290 (2009).
[Crossref]

L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. 34, 2363–2365 (2009).
[Crossref]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase shifting interferometry based on principal component analysis,” Opt. Lett. 36, 1326–1328 (2011).
[Crossref]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase shifting interferometry,” Opt. Lett. 36, 2215–2217 (2011).
[Crossref]

Y. Yin, X. Peng, A. Li, X. Liu, and B. Gao, “Calibration of fringe projection profilometry with bundle adjustment strategy,” Opt. Lett. 37, 542–544 (2012).
[Crossref]

Proc. SPIE (1)

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–381 (2003).
[Crossref]

Other (3)

Z. Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” in Proceedings of the Seventh IEEE International Conference on Computer Vision (ICCV) (1999), Vol. 1, pp. 666–673.

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 547–612.

L. Wang, H. Zhang, and Y. Xin, “A simple fringe pattern profilometry phase shift error quantification method,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2018), paper JW4A.25.

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Figures (7)

Fig. 1.
Fig. 1. Relationship between phase-shift errors $\Delta {\delta _1},\Delta {\delta _2}$ with $\Delta {\delta _1} = \Delta {\delta _2}$ , and total sum of spectral energy. Patterns of $A^\prime$ and $B^\prime$ are shown next to the data points on the graph.
Fig. 2.
Fig. 2. Simulation results. (a) Ground truth phase; (b) representative example of one of the fringe images; (c) reconstructed phase; (d) differences between ground truth and reconstructed phase when the phase shifts are exact.
Fig. 3.
Fig. 3. Simulation results. (a) Ground truth phase; (b) representative example of one of the fringe images; (c) reconstructed phase; (d) differences between ground truth and reconstructed phases.
Fig. 4.
Fig. 4. Simulation of a half-sphere. (a) Ground truth phase of the half-sphere covering a plane; (b) reconstruction phase with incorrect phase-shift values.
Fig. 5.
Fig. 5. Experimental results. (a) One of the fringe images as an example; (b) 2D image of reconstructed half ping-pong ball from top view; (c) 3D profile of the reconstructed half ping-pong ball object.
Fig. 6.
Fig. 6. Experimental results. Reconstructed ping-pong ball with incorrect phase-shift values during reconstruction. Note that there are ripples on the background as well as on the surface of the ball.
Fig. 7.
Fig. 7. Experimental results. Phase-shift estimate efficiency.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

{ I 1 ( x , y ) = A ( x , y ) + B ( x , y ) cos [ φ ( x , y ) 0 ] I 2 ( x , y ) = A ( x , y ) + B ( x , y ) cos [ φ ( x , y ) α 1 ] I 3 ( x , y ) = A ( x , y ) + B ( x , y ) cos [ φ ( x , y ) α 1 α 2 ] ,
P 1 = 1 2 sin ( α 2 / 2 ) { cos ( α 1 / 2 ) sin [ ( α 1 + α 2 ) / 2 ] ( I 1 I 3 ) cos [ ( α 1 + α 2 ) / 2 ] sin ( α 1 / 2 ) ( I 1 I 2 ) } ,
P 2 = 1 2 sin ( α 2 / 2 ) { sin ( α 1 / 2 ) sin [ ( α 1 + α 2 ) / 2 ] ( I 1 I 3 ) sin [ ( α 1 + α 2 ) / 2 ] sin ( α 1 / 2 ) ( I 1 I 2 ) } .
l φ ( x , y ) = tan 1 [ P 2 / P 1 ] = tan 1 { ( 1 cos α 1 ) ( I 1 I 3 ) [ 1 cos ( α 1 + α 2 ) ] ( I 1 I 2 ) sin α 1 ( I 1 I 3 ) sin ( α 1 + α 2 ) ( I 1 I 2 ) } .
min ( ε ) = min ( { | F { A ( x , y ) } | + | F { B ( x , y ) } | } ) φ ( x , y ) ,

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