Abstract

In fringe projection profilometry, the projector parameters are difficult to calibrate accurately thus inducing errors in measurement results. To solve this problem, this paper analyzes the epipolar geometry of the fringe projection system, revealing that, on an epipolar plane, the depth variation along an incident ray induces the pixel movement along the epipolar line on the image plane of the camera. The depth variation and the pixel movement can be related to each other by using projective transformations. Under this condition, their cross-ratio keeps invariant. By use of this cross-ratio invariance, we suggest a depth recovering method immune to projector errors. To calibrate the measurement system, we shift a reference board perpendicularly to three positions with known depths and measure its phase maps as the reference phase maps. When measuring an object, we calculate the object depth at each pixel by equating the cross-ratio of the depths to that of the corresponding pixels having the same phase. Experimental results demonstrate that, with this technique, the errors associated with the projector, including its errors in geometry parameters, its lens distortions, and its luminance nonlinearity, do not affect the measurement results.

© 2017 Optical Society of America

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References

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2017 (2)

2016 (3)

2012 (1)

I. leandry, C. Brequei, and V. Valle, “Calibration of a structured-light projection system: development to large dimension objects,” Opt. Laser Technol. 50, 373–378 (2012).

2010 (4)

2009 (1)

2008 (3)

R. Yang, S. Cheng, and Y. Chen, “Flexible and accurate implementation of a binocular structured light system,” Opt. Lasers Eng. 46, 373–379 (2008).

W. Gao, L. Wang, and Z. Hu, “Flexible method for structured light system calibration,” Opt. Eng. 47, 083602 (2008).

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

2007 (2)

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).
[PubMed]

P. J. Tavares and M. A. Vaz, “Linear calibration procedure for the phase-to-height relationship in phase measurement profilometry,” Opt. Commun. 274, 307–314 (2007).

2006 (2)

2005 (1)

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44, 033603 (2005).

2004 (2)

R. Legarda-sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2004).

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

2003 (3)

H. Liu, W. Su, and K. Reichard, “Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement,” Opt. Commun. 216, 65–80 (2003).

Q. Hu, P. S. Huang, Q. Fu, and F.-P. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487–493 (2003).

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).

2002 (1)

R. Sitnik, M. Kujawińska, and J. Woźnicki, “Digital fringe projection system for large-volume 360-deg shape measurement,” Opt. Eng. 41, 443–449 (2002).

2000 (3)

Y. Y. Hung, L. Lin, H. M. Shang, and B. G. Park, “Practical three dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).

J. Guehring, “Dense 3D surface acquisition by structured light using off-the-shelf components,” Proc. SPIE 4309, 220–231 (2000).

1999 (2)

C. R. Coggrave and J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38, 1573–1581 (1999).

S. Kakunai, T. Sakamoto, and K. Iwata, “Profile measurement taken with liquid-crystal gratings,” Appl. Opt. 38(13), 2824–2828 (1999).
[PubMed]

1994 (2)

W. Zhou and X. Y. Su, “A direct mapping algorithm for phase measuring profilometry,” J. Mod. Opt. 41, 89–94 (1994).

H. Zhao, W. Chen, and Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33(20), 4497–4500 (1994).
[PubMed]

1985 (1)

1984 (1)

1983 (1)

Asundi, A.

Asundi, A. K.

Barnes, J.

Z. Wang, D. Nguyen, and J. Barnes, “Recent advances in 3D shape measurement and imaging using fringe projection technique,” in Proceedings of the SEM Annual Congress and Exposition on Experimental and Applied Mechanics 2009 (Society for Experimental Mechanics, 2009), pp. 2644–2653.

Bothe, T.

R. Legarda-sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2004).

Brequei, C.

I. leandry, C. Brequei, and V. Valle, “Calibration of a structured-light projection system: development to large dimension objects,” Opt. Laser Technol. 50, 373–378 (2012).

Brohinsky, W. R.

Chen, M.

H. Guo, M. Chen, and P. Zheng, “Least-squares fitting of carrier phase distribution by using a rational function in fringe projection profilometry [corrected],” Opt. Lett. 31(24), 3588–3590 (2006).
[PubMed]

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44, 033603 (2005).

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

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).

Chen, Q.

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).

Chen, W.

Chen, Y.

R. Yang, S. Cheng, and Y. Chen, “Flexible and accurate implementation of a binocular structured light system,” Opt. Lasers Eng. 46, 373–379 (2008).

Cheng, S.

R. Yang, S. Cheng, and Y. Chen, “Flexible and accurate implementation of a binocular structured light system,” Opt. Lasers Eng. 46, 373–379 (2008).

Chiang, F.-P.

Q. Hu, P. S. Huang, Q. Fu, and F.-P. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487–493 (2003).

Chua, P. S.

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, 1573–1581 (1999).

Fu, Q.

Q. Hu, P. S. Huang, Q. Fu, and F.-P. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487–493 (2003).

Gao, W.

W. Gao, L. Wang, and Z. Hu, “Flexible method for structured light system calibration,” Opt. Eng. 47, 083602 (2008).

Gorthi, S. S.

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

Guehring, J.

J. Guehring, “Dense 3D surface acquisition by structured light using off-the-shelf components,” Proc. SPIE 4309, 220–231 (2000).

Guo, H.

Halioua, M.

Hao, Q.

Hassebrook, L. G.

He, H.

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44, 033603 (2005).

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

Hoang, T.

Hu, Q.

Q. Hu, P. S. Huang, Q. Fu, and F.-P. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487–493 (2003).

Hu, Z.

W. Gao, L. Wang, and Z. Hu, “Flexible method for structured light system calibration,” Opt. Eng. 47, 083602 (2008).

Huang, L.

Huang, P. S.

S. Zhang and P. S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45, 083601 (2006).

Q. Hu, P. S. Huang, Q. Fu, and F.-P. Chiang, “Calibration of a three-dimensional shape measurement system,” Opt. Eng. 42, 487–493 (2003).

Hung, Y. Y.

Y. Y. Hung, L. Lin, H. M. Shang, and B. G. Park, “Practical three dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).

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, 1573–1581 (1999).

Iwata, K.

Jüptner, W. P.

R. Legarda-sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2004).

Kakunai, S.

Kemao, Q.

Kujawinska, M.

R. Sitnik, M. Kujawińska, and J. Woźnicki, “Digital fringe projection system for large-volume 360-deg shape measurement,” Opt. Eng. 41, 443–449 (2002).

Lau, D. L.

leandry, I.

I. leandry, C. Brequei, and V. Valle, “Calibration of a structured-light projection system: development to large dimension objects,” Opt. Laser Technol. 50, 373–378 (2012).

Legarda-sáenz, R.

R. Legarda-sáenz, T. Bothe, and W. P. Jüptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2004).

Li, Z.

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

Lin, L.

Y. Y. Hung, L. Lin, H. M. Shang, and B. G. Park, “Practical three dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).

Liu, H.

H. Liu, W. Su, and K. Reichard, “Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement,” Opt. Commun. 216, 65–80 (2003).

Liu, H. C.

Liu, K.

Lu, Y.

Lü, F.

Mutoh, K.

Nguyen, D.

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[PubMed]

Z. Wang, D. Nguyen, and J. Barnes, “Recent advances in 3D shape measurement and imaging using fringe projection technique,” in Proceedings of the SEM Annual Congress and Exposition on Experimental and Applied Mechanics 2009 (Society for Experimental Mechanics, 2009), pp. 2644–2653.

Pan, B.

Park, B. G.

Y. Y. Hung, L. Lin, H. M. Shang, and B. G. Park, “Practical three dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).

Rastogi, P.

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

Reichard, K.

H. Liu, W. Su, and K. Reichard, “Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement,” Opt. Commun. 216, 65–80 (2003).

Sakamoto, T.

Shang, H. M.

Y. Y. Hung, L. Lin, H. M. Shang, and B. G. Park, “Practical three dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).

Shi, Y.

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

Sitnik, R.

R. Sitnik, M. Kujawińska, and J. Woźnicki, “Digital fringe projection system for large-volume 360-deg shape measurement,” Opt. Eng. 41, 443–449 (2002).

Srinivasan, V.

Stetson, K. A.

Su, W.

H. Liu, W. Su, and K. Reichard, “Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement,” Opt. Commun. 216, 65–80 (2003).

Su, X. Y.

W. Zhou and X. Y. Su, “A direct mapping algorithm for phase measuring profilometry,” J. Mod. Opt. 41, 89–94 (1994).

Takeda, M.

Tan, Y.

Tavares, P. J.

P. J. Tavares and M. A. Vaz, “Linear calibration procedure for the phase-to-height relationship in phase measurement profilometry,” Opt. Commun. 274, 307–314 (2007).

Tsai, R. Y.

R. Y. Tsai, “An efficient and accurate camera calibration technique for 3D machine vision,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1986), pp. 364–374.

Valle, V.

I. leandry, C. Brequei, and V. Valle, “Calibration of a structured-light projection system: development to large dimension objects,” Opt. Laser Technol. 50, 373–378 (2012).

Vaz, M. A.

P. J. Tavares and M. A. Vaz, “Linear calibration procedure for the phase-to-height relationship in phase measurement profilometry,” Opt. Commun. 274, 307–314 (2007).

Wang, C.

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

Wang, L.

W. Gao, L. Wang, and Z. Hu, “Flexible method for structured light system calibration,” Opt. Eng. 47, 083602 (2008).

Wang, Y.

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).
[PubMed]

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

Wang, Z.

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[PubMed]

Z. Wang, D. Nguyen, and J. Barnes, “Recent advances in 3D shape measurement and imaging using fringe projection technique,” in Proceedings of the SEM Annual Congress and Exposition on Experimental and Applied Mechanics 2009 (Society for Experimental Mechanics, 2009), pp. 2644–2653.

Woznicki, J.

R. Sitnik, M. Kujawińska, and J. Woźnicki, “Digital fringe projection system for large-volume 360-deg shape measurement,” Opt. Eng. 41, 443–449 (2002).

Xing, S.

Yang, R.

R. Yang, S. Cheng, and Y. Chen, “Flexible and accurate implementation of a binocular structured light system,” Opt. Lasers Eng. 46, 373–379 (2008).

Yau, S. T.

Yu, Y.

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44, 033603 (2005).

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).

Zhang, R.

Zhang, S.

Zhang, Z. Y.

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).

Zhao, H.

Zheng, P.

Zhou, W.

W. Zhou and X. Y. Su, “A direct mapping algorithm for phase measuring profilometry,” J. Mod. Opt. 41, 89–94 (1994).

Zuo, C.

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).

Appl. Opt. (12)

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983).
[PubMed]

V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23(18), 3105–3108 (1984).
[PubMed]

L. Huang, P. S. Chua, and A. Asundi, “Least-squares calibration method for fringe projection profilometry considering camera lens distortion,” Appl. Opt. 49(9), 1539–1548 (2010).
[PubMed]

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

S. Kakunai, T. Sakamoto, and K. Iwata, “Profile measurement taken with liquid-crystal gratings,” Appl. Opt. 38(13), 2824–2828 (1999).
[PubMed]

K. A. Stetson and W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24(21), 3631–3637 (1985).
[PubMed]

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).
[PubMed]

R. Zhang, H. Guo, and A. K. Asundi, “Geometric analysis of influence of fringe directions on phase sensitivities in fringe projection profilometry,” Appl. Opt. 55(27), 7675–7687 (2016).
[PubMed]

H. Zhao, W. Chen, and Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33(20), 4497–4500 (1994).
[PubMed]

F. Lü, S. Xing, and H. Guo, “Self-correction of projector nonlinearity in phase-shifting fringe projection profilometry,” Appl. Opt. 56(25), 7204–7216 (2017).
[PubMed]

S. Xing and H. Guo, “Temporal phase unwrapping for fringe projection profilometry aided by recursion of Chebyshev polynomials,” Appl. Opt. 56(6), 1591–1602 (2017).
[PubMed]

Y. Lu, R. Zhang, and H. Guo, “Correction of illumination fluctuations in phase-shifting technique by use of fringe histograms,” Appl. Opt. 55(1), 184–197 (2016).
[PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).

J. Mod. Opt. (1)

W. Zhou and X. Y. Su, “A direct mapping algorithm for phase measuring profilometry,” J. Mod. Opt. 41, 89–94 (1994).

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

Opt. Commun. (2)

P. J. Tavares and M. A. Vaz, “Linear calibration procedure for the phase-to-height relationship in phase measurement profilometry,” Opt. Commun. 274, 307–314 (2007).

H. Liu, W. Su, and K. Reichard, “Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement,” Opt. Commun. 216, 65–80 (2003).

Opt. Eng. (9)

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44, 033603 (2005).

R. Sitnik, M. Kujawińska, and J. Woźnicki, “Digital fringe projection system for large-volume 360-deg shape measurement,” Opt. Eng. 41, 443–449 (2002).

Y. Y. Hung, L. Lin, H. M. Shang, and B. G. Park, “Practical three dimensional computer vision techniques for full-field surface measurement,” Opt. Eng. 39, 143–149 (2000).

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Figures (6)

Fig. 1
Fig. 1 Measurement system.
Fig. 2
Fig. 2 Geometry of cross-ratio invariance for fringe projection profilometry. (a) shows the epipolar geometry of the measurement system. (b) shows the 2D plot of the epipolar plane in (a), on which the cross-ratio of the object depths, the cross-ratio of the phases, and the cross-ratio of the pixel shifts on πc are equal to one another.
Fig. 3
Fig. 3 Calibration results of the reference phase maps. Row (a) shows the captured fringe patterns on the reference plane. The three fringe patterns in each set have the same phase shift of 0, and correspond, from front to back, to the depth positions H1 = 0 mm, H2 = 30 mm, and H3 = 60 mm, respectively. Row (b) shows their corresponding phase maps in radians, which will be used as the reference phase maps in measurement. The columns correspond to different error conditions. (1) The fringes are straight, and the projector nonlinearity has been corrected. (2) The fringes are straight, but the projector nonlinearity is not corrected. (3) The fringes are slightly curved, but the projector nonlinearity has been corrected. (4) The fringes are slightly curved, and the projector nonlinearity is not corrected.
Fig. 4
Fig. 4 Measurement results of an object having a height smaller than 30mm. The columns, (1) through (4), aligned with those in Fig. 3, correspond to different situations of errors of the projector. The rows, from (a) to (d), show the fringe patterns having a phase shift of 0, the unwrapped phase maps in radians, the depth maps in millimeters calculated by use of the cross-ratio of fringe phases, and the depth maps in millimeters calculated by use of the cross-ratio of pixel shifts, respectively.
Fig. 5
Fig. 5 Measurement results of an object having a height over 80mm. The columns, (1) to (4), aligned with those in Fig. 3, correspond to different situations of errors of the projector. The rows, from (a) to (d), show the fringe patterns having a phase shift of 0, the unwrapped phase maps in radians, the depth maps in millimeters calculated by use of the cross-ratio of fringe phases, and the depth maps in millimeters calculated by use of the cross-ratio of pixel shifts, respectively.
Fig. 6
Fig. 6 Cross-sections of the measured profiles of a flat plane positioned at the depth of 45 mm. The columns, (1) through (4), are aligned with those in Fig. 3, corresponding to different situations of errors of the projector. The top row (a) is obtained by using the method based on the cross-ratio of fringe phases, and the bottom row (b) are obtained by using the method based on the cross-ratio of pixel shifts.

Tables (1)

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Table 1 The RMS values (mm) of the deviations of the measurement results.

Equations (10)

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I k (u,v)=a(u,v)+b(u,v)cos[ϕ(u,v)+2kπ/K],
ϕ wrapped (u,v)=arctan[ k=0 K1 I k (u,v)sin( 2kπ/K ) k=0 K1 I k (u,v)cos( 2kπ/K ) ].
(Q, N 1 ; N 2 , N 3 )= Q N 2 N 1 N 3 N 1 N 2 Q N 3 = [ H 2 h(u,v)]( H 3 H 1 ) ( H 2 H 1 )[ H 3 h(u,v)] =( q , n 1 ; n 2 , n 3 )= q n 2 n 1 n 3 n 1 n 2 q n 3 = [ Φ 2 (u,v)ϕ(u,v)][ Φ 3 (u,v) Φ 1 (u,v)] [ Φ 2 (u,v) Φ 1 (u,v)][ Φ 3 (u,v)ϕ(u,v)]
h(u,v)= A(u,v)+B(u,v)ϕ(u,v) C(u,v)+D(u,v)ϕ(u,v)
{ A= Φ 1 Φ 2 H 3 ( H 1 H 2 )+ Φ 2 Φ 3 H 1 ( H 2 H 3 )+ Φ 3 Φ 1 H 2 ( H 3 H 1 ) B= Φ 3 H 3 ( H 1 H 2 )+ Φ 1 H 1 ( H 2 H 3 )+ Φ 2 H 2 ( H 3 H 1 ) C= Φ 1 Φ 2 ( H 1 H 2 )+ Φ 2 Φ 3 ( H 2 H 3 )+ Φ 3 Φ 1 ( H 3 H 1 ) D= Φ 3 ( H 1 H 2 )+ Φ 1 ( H 2 H 3 )+ Φ 2 ( H 3 H 1 )
Φ 1 ( u m1 , v m1 )= Φ 2 ( u m2 , v m2 )= Φ 3 ( u m3 , v m3 )=ϕ(u,v).
(Q, M 1 ; M 2 , M 3 )= Q M 2 M 1 M 3 M 1 M 2 Q M 3 = [ H 2 h(u,v)]( H 3 H 1 ) ( H 2 H 1 )[ H 3 h(u,v)] =(q, m 1 ; m 2 , m 3 )= q m 2 m 1 m 3 m 1 m 2 q m 3 = [( u m2 +j v m2 )(u+jv)][( u m3 +j v m3 )( u m1 +j v m1 )] [( u m2 +j v m2 )( u m1 +j v m1 )][( u m3 +j v m3 )(u+jv)] = ( u m2 u)( u m3 u m1 ) ( u m2 u m1 )( u m3 u) = ( v m2 v)( v m3 v m1 ) ( v m2 v m1 )( v m3 v)
{ Φ Hκ (u,v) d κ3 + d κ4 u+ d κ5 v 1+ d κ1 u+ d κ2 v Φ Vκ (u,v) d κ6 + d κ7 u+ d κ8 v 1+ d κ1 u+ d κ2 v
{ d κ3 + d κ4 u e + d κ5 v e 1+ d κ1 u e + d κ2 v e d λ3 + d λ4 u e + d λ5 v e 1+ d λ1 u e + d λ2 v e =0 d κ6 + d κ7 u e + d κ8 v e 1+ d κ1 u e + d κ2 v e d λ6 + d λ7 u e + d λ8 v e 1+ d λ1 u e + d λ2 v e =0
h(u,v)= ( u m1 u m2 + u m3 u) H 3 ( H 1 H 2 )+( u m2 u m3 + u m1 u) H 1 ( H 2 H 3 )+( u m3 u m1 + u m2 u) H 2 ( H 3 H 1 ) ( u m1 u m2 + u m3 u)( H 1 H 2 )+( u m2 u m3 + u m1 u)( H 2 H 3 )+( u m3 u m1 + u m2 u)( H 3 H 1 ) .

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