Abstract

In fringe projection profilometry, the purpose of using two- or multi-frequency fringe patterns is to unwrap the measured phase maps temporally. Using the same patterns, this paper presents a least squares algorithm for, simultaneously with phase-unwrapping, eliminating the influences of fringe harmonics induced by various adverse factors. It is demonstrated that, for most of the points over the measured surface, projecting two sequences of phase-shifting fringe patterns having different frequencies enables providing sufficiently many equations for determining the coefficient of a high order fringe harmonic. As a result, solving these equations in the least squares sense results in a phase map having higher accuracy than that depending only on the fringe patterns of a single frequency. For the other few points which have special phases related to the two frequencies, this system of equations becomes under-determined. For coping with this case, this paper suggests an interpolation-based solution which has a low sensitivity to the variations of reflectivity and slope of the measured surface. Simulation and experimental results verify that the proposed method significantly suppresses the ripple-like artifacts in phase maps induced by fringe harmonics without capturing extra many fringe patterns or correcting the non-sinusoidal profiles of fringes. In addition, this method involves a quasi-pointwise operation, enabling correcting position-dependent phase errors and being helpful for protecting the edges and details of the measurement results from being blurred.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]

2020 (1)

2019 (3)

K. Yan, Y. Yu, C. Huang, L. Sui, K. Qian, and A. Asundi, “Fringe pattern denoising based on deep learning,” Opt. Commun. 437, 148–152 (2019).
[Crossref]

S. Ordones, M. Servin, M. Padilla, A. Muñoz, J. L. Flores, and I. Choque, “Spectral analysis for the generalized least squares phase-shifting algorithms with harmonic robustness,” Opt. Lett. 44(9), 2358–2361 (2019).
[Crossref]

S. Xing and H. Guo, “Directly recognizing and removing the projector nonliearity errors from a phase map in phase-shifting fringe projection profilometry,” Opt. Commun. 435, 212–220 (2019).
[Crossref]

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]

S. Xing and H. Guo, “Correction of projector nonlinearity in multi-frequency phase-shifting fringe projection profilometry,” Opt. Express 26(13), 16277–16291 (2018).
[Crossref]

2017 (3)

2016 (5)

2014 (2)

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
[Crossref]

B. Salahieh, Z. Chen, J. J. Rodriguez, and R. Liang, “Multi-polarization fringe projection imaging for high dynamic range objects,” Opt. Express 22(8), 10064–10071 (2014).
[Crossref]

2013 (1)

2012 (3)

Y. Wang and S. Zhang, “Three-dimensional shape measurement with binary dithered patterns,” Appl. Opt. 51(27), 6631–6636 (2012).
[Crossref]

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(5), 533–538 (2012).
[Crossref]

H. Guo and B. Lü, “Phase-shifting algorithm by use of Hough transform,” Opt. Express 20(23), 26037–26049 (2012).
[Crossref]

2011 (2)

2010 (4)

G. A. Ayubi, J. A. Ayubi, J. M. Di Martino, and J. A. Ferrari, “Pulsewidth modulation in defocused three-dimensional fringe projection,” Opt. Lett. 35(21), 3682–3684 (2010).
[Crossref]

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).
[Crossref]

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48(2), 166–171 (2010).
[Crossref]

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

2009 (2)

Z.-W. Li, Y.-S. Shi, C.-J. Wang, D.-H. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (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(4), 416–418 (2009).
[Crossref]

2007 (1)

2006 (1)

2005 (2)

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44(3), 033603 (2005).
[Crossref]

H. Guo and M. Chen, “Least-squares algorithm for phase-stepping interferometry with an unknown relative step,” Appl. Opt. 44(23), 4854–4859 (2005).
[Crossref]

2004 (2)

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

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring Deflectometry: a new approach to measuring specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[Crossref]

2003 (2)

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Optimum frequency selection in multifrequency interferometry,” Opt. Lett. 28(11), 887–889 (2003).
[Crossref]

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

2000 (1)

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(9), 1573–1581 (1999).
[Crossref]

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

1998 (1)

Y. B. Choi and S.-W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37(3), 1005–1010 (1998).
[Crossref]

1993 (1)

T. Yoshizawa and T. Tomisawa, “Shadow moiré topography by means of the phase-shift method,” Opt. Eng. 32(7), 1668–1674 (1993).
[Crossref]

1988 (1)

1985 (3)

1984 (2)

1975 (1)

1974 (1)

Asundi, A.

K. Yan, Y. Yu, C. Huang, L. Sui, K. Qian, and A. Asundi, “Fringe pattern denoising based on deep learning,” Opt. Commun. 437, 148–152 (2019).
[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]

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(4), 416–418 (2009).
[Crossref]

Asundi, A. K.

Ayubi, G. A.

Ayubi, J. A.

Brangccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

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(5), 533–538 (2012).
[Crossref]

Chen, M.

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]

Chen, Z.

Cheng, Y.-Y.

Chiang, F.-P.

Choi, Y. B.

Y. B. Choi and S.-W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37(3), 1005–1010 (1998).
[Crossref]

Choque, I.

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(9), 1573–1581 (1999).
[Crossref]

Creath, K.

K. Creath, “Temporal phase measurement method,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson and G. Reid, eds. (IOP, Bristol, UK, 1993), pp. 94–140.

Da, F.

Dai, J.

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
[Crossref]

Decraemer, W. F.

Di Martino, J. M.

Dielis, G.

Dirckx, J. J.

Feng, P.

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48(2), 166–171 (2010).
[Crossref]

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]

Ferrari, J. A.

Flores, J. L.

Gallagher, J. E.

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Guo, H.

S. Xing and H. Guo, “Iterative calibration method for measurement system having lens distortions in fringe projection profilometry,” Opt. Express 28(2), 1177–1195 (2020).
[Crossref]

S. Xing and H. Guo, “Directly recognizing and removing the projector nonliearity errors from a phase map in phase-shifting fringe projection profilometry,” Opt. Commun. 435, 212–220 (2019).
[Crossref]

S. Xing and H. Guo, “Correction of projector nonlinearity in multi-frequency phase-shifting fringe projection profilometry,” Opt. Express 26(13), 16277–16291 (2018).
[Crossref]

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).
[Crossref]

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).
[Crossref]

R. Zhang and H. Guo, “Depth recovering method immune to projector errors in fringe projection profilometry by use of cross-ratio invariance,” Optics Express. 25(23), 29272–29286 (2017).
[Crossref]

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).
[Crossref]

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).
[Crossref]

H. Guo and B. Lü, “Phase-shifting algorithm by use of Hough transform,” Opt. Express 20(23), 26037–26049 (2012).
[Crossref]

H. Guo, “A simple algorithm for fitting a Gaussian function,” IEEE Signal Process. Mag. 28(5), 134–137 (2011).
[Crossref]

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48(2), 166–171 (2010).
[Crossref]

H. Guo and M. Chen, “Least-squares algorithm for phase-stepping interferometry with an unknown relative step,” Appl. Opt. 44(23), 4854–4859 (2005).
[Crossref]

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44(3), 033603 (2005).
[Crossref]

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

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

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[Crossref]

Halioua, M.

Hausler, G.

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring Deflectometry: a new approach to measuring specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[Crossref]

He, H.

H. Guo, H. He, Y. Yu, and M. Chen, “Least-squares calibration method for fringe projection profilometry,” Opt. Eng. 44(3), 033603 (2005).
[Crossref]

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

Herriott, D. R.

Hoang, T.

Huang, C.

K. Yan, Y. Yu, C. Huang, L. Sui, K. Qian, and A. Asundi, “Fringe pattern denoising based on deep learning,” Opt. Commun. 437, 148–152 (2019).
[Crossref]

Huang, K.

Z.-W. Li, Y.-S. Shi, C.-J. Wang, D.-H. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (2009).
[Crossref]

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]

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(4), 416–418 (2009).
[Crossref]

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(9), 1573–1581 (1999).
[Crossref]

Hyun, J. S.

Iwata, K.

Jones, J. D. C.

Kakunai, S.

Kaminski, J.

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring Deflectometry: a new approach to measuring specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[Crossref]

Kemao, Q.

Kim, S.-W.

Y. B. Choi and S.-W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37(3), 1005–1010 (1998).
[Crossref]

Knauer, M. C.

M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring Deflectometry: a new approach to measuring specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).
[Crossref]

Krishnamuthy, R. S.

Li, B.

B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014).
[Crossref]

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(5), 533–538 (2012).
[Crossref]

Li, Y.

Li, Z.

Li, Z.-W.

Z.-W. Li, Y.-S. Shi, C.-J. Wang, D.-H. Qin, and K. Huang, “Complex object 3D measurement based on phase-shifting and a neural network,” Opt. Commun. 282(14), 2699–2706 (2009).
[Crossref]

Liang, R.

Liu, H. C.

Liu, H.-C.

Lohry, W.

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Figures (9)

Fig. 1.
Fig. 1. (a) Measurement system. (b) The transfer of fringe patterns in the measurement system.
Fig. 2.
Fig. 2. The transfer of a 1D fringe pattern. (a) A standard sinusoidal fringe pattern generated using Eq. (1). (b) The projector nonlinearity of the brightness versus the gray level. (c) The output fringes of the projector. (d) The reflected brightness by the object surface. (e) The nonlinear response curve of the camera. (f) The captured fringe pattern by the camera.
Fig. 3.
Fig. 3. Simulation of phase measuring and error correcting. (a) A simulated 1D phase curve without carrier. (b) One of the 1D fringe patterns of (a). (c) The phases calculated using the method in Section 3 with the carrier subtracted out. (d) The errors in (c) obtained by subtracting (a). (e) The phase correcting result using the iterative least-squares algorithm in Section 4.1. (f) The residual phase errors in (e). (g) The phase correcting result with its pitfall points being processed using the interpolation-based method in Section 4.3. (h) The residual errors in (g).
Fig. 4.
Fig. 4. Numerical simulation results of phase error correction when using three-step phase-shifting method. (a) A fringe pattern having a phase shift of 0. (b) The wrapped phase map of (a). (c) and (d) are similar to (a) and (b), but have a different fringe frequency. (e) The unwrapped phase map without carrier, calculated using the method in Section 3. (f) The residual errors in (e). (g) The phase map with its errors corrected using the technique in [32]. (h) The residual errors in (g). (i) The phase map with its errors induced by fringe harmonics corrected using the newly proposed technique. (j) The residual errors in (i). The colorbars for the phases and phase errors have a unit of radian.
Fig. 5.
Fig. 5. Using the proposed technique, RMS phase errors decrease with the number of iterations under different noise conditions.
Fig. 6.
Fig. 6. Measurement results of a circular object by using three-step phase-shifting method. (a) A fringe pattern having a phase shift of 0. (b) The wrapped phase map of (a) calculated by using the method in Section 3.1. (c) and (d) are similar to (a) and (b) but have different frequencies. (e) The unwrapped phase map calculated from (b) and (d) using the method in Section 3.2. (f) The same phase map as in (e) with its carrier removed. (g) The phase map with the harmonics-induced errors being corrected using the method in [32]. (h) The same phase map as in (g) with its carrier removed. (i) The phase map with the harmonics-induced errors having been corrected using the proposed technique. (j) The same phase map as in (i) but its carrier has been removed. The colorbars have a unit of radian.
Fig. 7.
Fig. 7. Comparison of measurement results for a circular object. The rows, (a), (b), and (c), show the results by using three-, four- and five-step phase-shifting method, respectively. The leftmost column shows the depth maps in millimeters reconstructed using the conventional technique without correcting the harmonics-induced errors. The second column shows their cross-sections along a pixel column within the region of the plane outside the circular object. The third column gives the results with the harmonics-induced errors being corrected by using our proposed method. The rightmost column shows their cross-sections.
Fig. 8.
Fig. 8. With the proposed technique, RMS depth errors decrease with the number of iterations when using different number of phase shifts.
Fig. 9.
Fig. 9. Comparison of measurement results for an object having a relatively large height and a complex shape. The layout is the same as that of Fig. 7.

Tables (2)

Tables Icon

Table 1. Phase errors (rad) under different noise conditions.

Tables Icon

Table 2. RMS errors (mm) in the reconstructed depth maps.

Equations (33)

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

g n ( u , v ) = a 0 + a 1 cos ( 2 π u / u p p + 2 π n / n N N ) ,
h n ( u , v ) = k = 0 c k g   n k ( u , v ) ,
f n ( x , y ) = R ( x , y ) h n ( x , y ) + B ( x , y ) ,
I n ( x , y ) = k = 0 d k f   n k ( x , y ) ,
I n ( x , y ) = k = 0  +  b k ( x , y ) cos { k [ ϕ ( x , y ) + 2 π n / N ] } ,
ϕ ^ ( x , y ) tan 1 n = 0 N 1 I n ( x , y ) sin ( 2 π n / N ) n = 0 N 1 I n ( x , y ) cos ( 2 π n / N ) ,
b ^ 0 ( x , y ) 1 N n = 0 N 1 I n ( x , y ) ,
b ^ 1 ( x , y ) 2 N { [ n = 0 N 1 I n ( x , y ) sin ( 2 π n / N ) ] 2 + [ n = 0 N 1 I n ( x , y ) cos ( 2 π n / N ) ] 2 } 1 2 ,
λ = p I / p I p II p II .
p eq = p I p II / p I p II ( p II p I ) ( p II p I ) ,
ϕ eq ( x , y ) = [ ϕ I ( x , y ) ϕ II ( x , y ) ]  mod  2 π ,
Φ I ( x , y ) = round [ p eq ϕ eq ( x , y ) p I ϕ I ( x , y ) 2 π p I ] 2 π + ϕ I ( x , y ) ,
Φ II ( x , y ) = λ Φ I ( x , y ) = p I Φ I ( x , y ) / p I Φ I ( x , y ) p II p II
{ I I n = b 0 + b 1 cos ( Φ I + 2 π n / N ) + b N 1 cos [ ( N 1 ) ( Φ I + 2 π n / N ) ] I II n = b 0 + b 1 cos ( λ Φ I + 2 π n / N ) + b N 1 cos [ ( N 1 ) ( λ Φ I + 2 π n / N ) ]
δ  = [ δ b 0 , δ b 1 , δ b N 1 , δ Φ I ] T = [ b ^ 0 , b ^ 1 , b ^ N 1 , Φ ^ I ] T [ b 0 , b 1 , b N 1 , Φ I ] T
{ I ^ I n = b ^ 0 + b ^ 1 cos ( Φ ^ I + 2 π n / N ) + b ^ N 1 cos [ ( N 1 ) ( Φ ^ I + 2 π n / N ) ] I ^ II n = b ^ 0 + b ^ 1 cos ( λ Φ ^ I + 2 π n / N ) + b ^ N 1 cos [ ( N 1 ) ( λ Φ ^ I + 2 π n / N ) ]
Δ = [ I ^ I 0 , , I ^ I( N 1 ) , I ^ II 0 , I ^ II( N 1 ) ] T [ I I 0 , , I I( N 1 ) , I II 0 , I II( N 1 ) ] T .
{ I ^ I n = I I n  +  I I n b 0 δ b 0  +  I I n b 1 δ b 1  +  I I n b N  - 1 δ b N  - 1  +  I I n Φ I δ Φ I I ^ II n = I II n  +  I II n b 0 δ b 0  +  I II n b 1 δ b 1  +  I II n b N  - 1 δ b N  - 1  +  I II n Φ I δ Φ I
{ I I n / I I n b 0 b 0 = I II n / I II n b 0 b 0 = 1 I I n / I I n b 1 b 1 = cos ( Φ I + 2 π n / N ) I I n / I I n b N 1 b N 1 = cos [ ( N 1 ) ( Φ I + 2 π n / N ) ] I I n / I I n Φ I Φ I = b 1 sin ( Φ I + 2 π n / N ) b N 1 ( N 1 ) sin [ ( N 1 ) ( Φ I + 2 π n / N ) ] I II n / I II n b 1 b 1 = cos ( λ Φ I + 2 π n / N ) I II n / I II n b N 1 b N 1 = cos [ ( N 1 ) ( λ Φ I + 2 π n / N ) ] I II n / I II n Φ I Φ I = b 1 λ sin ( λ Φ I + 2 π n / N ) b N 1 ( N 1 ) λ sin [ ( N 1 ) ( λ Φ I + 2 π n / N ) ]
Ω δ = Δ ,
Ω = [ I I 0 / I I 0 b 0 b 0 I I 0 / I I 0 b 1 b 1 I I 0 / I I 0 b N 1 b N 1 I I 0 / I I 0 Φ I Φ I I I( N 1 ) / I I( N 1 ) b 0 b 0 I I( N 1 ) / I I( N 1 ) b 1 b 1 I I( N 1 ) / I I( N 1 ) b N 1 b N 1 I I( N 1 ) / I I( N 1 ) Φ I Φ I I II 0 / I II 0 b 0 b 0 I II 0 / I II 0 b 1 b 1 I II 0 / I II 0 b N 1 b N 1 I II 0 / I II 0 Φ I Φ I I II( N 1 ) / I II( N 1 ) b 0 b 0 I II( N 1 ) / I II( N 1 ) b 1 b 1 I II( N 1 ) / I II( N 1 ) b N 1 b N 1 I II ( N 1 ) / I II ( N 1 ) Φ I Φ I   ] .
Ω T Ω δ = Ω T Δ
[ Ω ( i ) ] T Ω ( i ) δ ( i + 1 ) = [ Ω ( i ) ] T Δ ( i )
[ b 0 ( i + 1 ) , b 1 ( i + 1 ) , b N 1 ( i + 1 ) , Φ I ( i + 1 ) ] = [ b 0 ( i ) , b 1 ( i ) , b N 1 ( i ) , Φ I ( i ) ] [ δ b 0 ( i + 1 ) , δ b 1 ( i + 1 ) , δ b N 1 ( i + 1 ) , δ Φ I ( i + 1 ) ] .
Ω T Ω = [ 2 N 0 0 0 0 N ω 1 ω 2 0 ω 1 N ω 3 0 ω 2 ω 3 ω 4 ] ,
{ ω 1 = N [ cos ( N Φ I ) + cos ( λ N Φ I ) ] / N [ cos ( N Φ I ) + cos ( λ N Φ I ) ] 2 2 ω 2  =  N ( N 1 ) b N 1 [ sin ( N Φ I ) + λ sin ( λ N Φ I ) ] / N ( N 1 ) b N 1 [ sin ( N Φ I ) + λ sin ( λ N Φ I ) ] 2 2 ω 3  =  b 1 N [ sin ( N Φ I ) + λ sin ( λ N Φ I ) ] / b 1 N [ sin ( N Φ I ) + λ sin ( λ N Φ I ) ] 2 2 ω 4 = N ( 1 + λ 2 ) [ b 1 2 + ( N b N 1 b N 1 ) 2 ] / N ( 1 + λ 2 ) [ b 1 2 + ( N b N 1 b N 1 ) 2 ] 2 2 N ( N 1 ) b 1 b N 1 [ cos ( N Φ I ) + λ 2 cos ( λ N Φ I ) ]
cos ( N Φ I ) = cos ( λ N Φ I ) = ± 1
sin ( N Φ I ) = sin ( λ N Φ I ) = 0 ,
Φ I = 2 k π p e q / 2 k π p e q N p I N p I ,
| cos ( N Φ I ) + cos ( λ N Φ I ) | 2 T ,
γ ( x , y ) = b N 1 ( x , y ) / b N 1 ( x , y ) b 1 ( x , y ) b 1 ( x , y ) ,
δ  = [ δ b 0   δ b 1   δ Φ I ] T
Ω = [ I I 0 / I I 0 b 0 b 0 I I 0 / I I 0 b 1 b 1  +  γ I I 0 / I I 0 b N 1 b N 1 I I 0 / I I 0 Φ I Φ I I I( N 1 ) / I I( N 1 ) b 0 b 0 I I( N 1 ) / I I( N 1 ) b 1 b 1  +  γ I I( N 1 ) / I I( N 1 ) b N 1 b N 1 I I( N 1 ) / I I( N 1 ) Φ I Φ I I II 0 / I II 0 b 0 b 0 I II 0 / I II 0 b 1 b 1  +  γ I II 0 / I II 0 b N 1 b N 1 I II 0 / I II 0 Φ I Φ I I II( N 1 ) / I II( N 1 ) b 0 b 0 I II( N 1 ) / I II( N 1 ) b 1 b 1 + γ I II( N 1 ) / I II( N 1 ) b N 1 b N 1 I II ( N 1 ) / I II ( N 1 ) Φ I Φ I   ] .

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