Abstract

Gamma distortion is a dominant error source of phase measuring profilometry. It makes a single frequency for the ideal sinusoidal waveform an infinite width of spectrum. Besides, the defocus of the projector-camera system, like a spatial low-pass filter, attenuates the amplitudes of the high-frequency harmonics. In this paper, a generic distorted fringe model is proposed, which is expressed as a Fourier series. The mathematical model of the harmonic coefficients is derived. Based on the proposed model, a robust gamma calibration method is introduced. It employs the multifrequency phase-shifting method to eliminate the effect of defocus and preserve the influence of gamma distortion. Then, a gamma correction method is proposed to correct the gamma distortion with the calibrated gamma value. The proposed correction method has the advantage of high signal-to-noise ratio. The proposed model is verified through experiments. The results confirm that the phase error is dependent on the defocus and the pitch. The proposed gamma calibration method is compared with the state of the art and proves to be more robust to pitch and defocus variations. After adopting the proposed gamma correction method, the phase precision is much enhanced with higher quality in the measured surfaces.

© 2012 Optical Society of America

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References

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2011

2010

S. Gorthi and P. Rastogi, “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]

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recogn. 43, 2666–2680 (2010).
[CrossRef]

X. Su and Q. Zhang, “Dynamic 3-d shape measurement method: A review,” Opt. Lasers Eng. 48, 191–204 (2010).
[CrossRef]

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

K. Liu, Y. Wang, D. Lau, Q. Hao, and L. Hassebrook, “Gamma model and its analysis for phase measuring profilometry,” J. Opt. Soc. Am. A 27, 553–562 (2010).
[CrossRef]

2009

2008

2007

2006

2004

2003

P. Huang, C. Zhang, and F. Chiang, “High-speed 3-d shape measurement based on digital fringe projection,” Opt. Eng. 42, 163–168 (2003).
[CrossRef]

J. Li, L. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” J. Opt. Soc. Am. A 20, 106–115 (2003).
[CrossRef]

G. H. Notni and G. Notni, “Digital fringe projection in 3d shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[CrossRef]

2002

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

P. Huang, Q. Hu, and F. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41, 4503–4509 (2002).
[CrossRef]

2001

2000

C. Coggrave and J. Huntley, “Optimization of a shape measurement system based on spatial light modulators,” Opt. Eng. 39, 91–98 (2000).
[CrossRef]

1999

1995

1994

T. Judge and P. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

1993

C. Poynton, “Gamma and its disguises: The nonlinear mappings of intensity in perception, crts, film, and video,” SMPTE J. 102, 1099–1108 (1993).
[CrossRef]

1992

X. Su, W. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

1990

Asundi, A.

Baker, M.

Baker, M. J.

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of gamma non-linear luminance effects for digital video projection phase measuring profilometers,” in 4th IEEE International Symposium on Electronic Design, Test and Applications, 2008. DELTA 2008. (IEEE, 2008), pp. 496–501.

Baldi, A.

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

Bertolino, F.

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

Bryanston-Cross, P.

T. Judge and P. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

Chen, M.

Chiang, F.

P. Huang, C. Zhang, and F. Chiang, “High-speed 3-d shape measurement based on digital fringe projection,” Opt. Eng. 42, 163–168 (2003).
[CrossRef]

P. Huang, Q. Hu, and F. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41, 4503–4509 (2002).
[CrossRef]

Chicharo, J.

M. Baker, J. Xi, and J. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233–1243 (2007).
[CrossRef]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678–687 (2006).
[CrossRef]

Y. Hu, J. Xi, J. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in 2006 International Conference on Computer Graphics, Imaging and Visualisation (IEEE, 2006), pp. 161–167.

Chicharo, J. F.

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of gamma non-linear luminance effects for digital video projection phase measuring profilometers,” in 4th IEEE International Symposium on Electronic Design, Test and Applications, 2008. DELTA 2008. (IEEE, 2008), pp. 496–501.

Claxton, C.

Coggrave, C.

C. Coggrave and J. Huntley, “Optimization of a shape measurement system based on spatial light modulators,” Opt. Eng. 39, 91–98 (2000).
[CrossRef]

Dai, J.

Ekstrand, L.

English, C.

Farrant, D.

Fernandez, S.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recogn. 43, 2666–2680 (2010).
[CrossRef]

Freischlad, K.

Ghiglia, D.

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

Ginesu, F.

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

Gorthi, S.

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

Guan, C.

Guo, H.

Hao, Q.

Hassebrook, L.

He, H.

Hibino, K.

Hoang, T.

Hu, Q.

Hu, Y.

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678–687 (2006).
[CrossRef]

Y. Hu, J. Xi, J. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in 2006 International Conference on Computer Graphics, Imaging and Visualisation (IEEE, 2006), pp. 161–167.

Huang, L.

Huang, P.

S. Zhang and P. Huang, “Phase error compensation for a 3-d shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 063601 (2007).
[CrossRef]

P. Huang, C. Zhang, and F. Chiang, “High-speed 3-d shape measurement based on digital fringe projection,” Opt. Eng. 42, 163–168 (2003).
[CrossRef]

P. Huang, Q. Hu, and F. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41, 4503–4509 (2002).
[CrossRef]

Huntley, J.

C. Coggrave and J. Huntley, “Optimization of a shape measurement system based on spatial light modulators,” Opt. Eng. 39, 91–98 (2000).
[CrossRef]

Iwata, K.

Jia, P.

Judge, T.

T. Judge and P. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

Kakunai, S.

Kemao, Q.

Kinell, L.

Kofman, J.

Koliopoulos, C.

Larkin, K.

Lau, D.

Li, E.

Li, J.

Li, Y.

Li, Z.

Liu, K.

Llado, X.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recogn. 43, 2666–2680 (2010).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley-Blackwell, 2007).

Nguyen, D.

Notni, G.

G. H. Notni and G. Notni, “Digital fringe projection in 3d shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[CrossRef]

Notni, G. H.

G. H. Notni and G. Notni, “Digital fringe projection in 3d shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[CrossRef]

Oreb, B.

Pan, B.

Peng, T.

T. Peng, “Algorithms and models for 3-d shape measurement using digital fringe projections,” (University of Maryland, 2006), pp. 221–223.

Poynton, C.

C. Poynton, “Gamma and its disguises: The nonlinear mappings of intensity in perception, crts, film, and video,” SMPTE J. 102, 1099–1108 (1993).
[CrossRef]

Pribanic, T.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recogn. 43, 2666–2680 (2010).
[CrossRef]

Pritt, M.

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

Rastogi, P.

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

Rathjen, C.

Sakamoto, T.

Salvi, J.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recogn. 43, 2666–2680 (2010).
[CrossRef]

Sjödahl, M.

Staunton, R.

Su, X.

X. Su and Q. Zhang, “Dynamic 3-d shape measurement method: A review,” Opt. Lasers Eng. 48, 191–204 (2010).
[CrossRef]

X. Su, W. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Von Bally, G.

X. Su, W. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Vukicevic, D.

X. Su, W. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Wang, Y.

Wang, Z.

Xi, J.

M. Baker, J. Xi, and J. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233–1243 (2007).
[CrossRef]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678–687 (2006).
[CrossRef]

Y. Hu, J. Xi, J. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in 2006 International Conference on Computer Graphics, Imaging and Visualisation (IEEE, 2006), pp. 161–167.

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of gamma non-linear luminance effects for digital video projection phase measuring profilometers,” in 4th IEEE International Symposium on Electronic Design, Test and Applications, 2008. DELTA 2008. (IEEE, 2008), pp. 496–501.

Xu, Y.

Yang, Z.

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678–687 (2006).
[CrossRef]

Y. Hu, J. Xi, J. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in 2006 International Conference on Computer Graphics, Imaging and Visualisation (IEEE, 2006), pp. 161–167.

Yau, S.

Zhang, C.

P. Huang, C. Zhang, and F. Chiang, “High-speed 3-d shape measurement based on digital fringe projection,” Opt. Eng. 42, 163–168 (2003).
[CrossRef]

Zhang, Q.

X. Su and Q. Zhang, “Dynamic 3-d shape measurement method: A review,” Opt. Lasers Eng. 48, 191–204 (2010).
[CrossRef]

Zhang, S.

Y. Xu, L. Ekstrand, J. Dai, and S. Zhang, “Phase error compensation for three-dimensional shape measurement with projector defocusing,” Appl. Opt. 50, 2572–2581 (2011).
[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. Zhang and S. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46, 36–43 (2007).
[CrossRef]

S. Zhang and P. Huang, “Phase error compensation for a 3-d shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 063601 (2007).
[CrossRef]

Zhang, X.

Zhou, W.

X. Su, W. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Zhu, L.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

X. Su, W. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561–573 (1992).
[CrossRef]

Opt. Eng.

S. Zhang and P. Huang, “Phase error compensation for a 3-d shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 063601 (2007).
[CrossRef]

P. Huang, C. Zhang, and F. Chiang, “High-speed 3-d shape measurement based on digital fringe projection,” Opt. Eng. 42, 163–168 (2003).
[CrossRef]

C. Coggrave and J. Huntley, “Optimization of a shape measurement system based on spatial light modulators,” Opt. Eng. 39, 91–98 (2000).
[CrossRef]

Opt. Lasers Eng.

S. Gorthi and P. Rastogi, “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]

X. Su and Q. Zhang, “Dynamic 3-d shape measurement method: A review,” Opt. Lasers Eng. 48, 191–204 (2010).
[CrossRef]

T. Judge and P. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21, 199–239 (1994).
[CrossRef]

A. Baldi, F. Bertolino, and F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

Opt. Lett.

Pattern Recogn.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recogn. 43, 2666–2680 (2010).
[CrossRef]

Proc. SPIE

G. H. Notni and G. Notni, “Digital fringe projection in 3d shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[CrossRef]

SMPTE J.

C. Poynton, “Gamma and its disguises: The nonlinear mappings of intensity in perception, crts, film, and video,” SMPTE J. 102, 1099–1108 (1993).
[CrossRef]

Other

D. Malacara, Optical Shop Testing (Wiley-Blackwell, 2007).

T. Peng, “Algorithms and models for 3-d shape measurement using digital fringe projections,” (University of Maryland, 2006), pp. 221–223.

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

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of gamma non-linear luminance effects for digital video projection phase measuring profilometers,” in 4th IEEE International Symposium on Electronic Design, Test and Applications, 2008. DELTA 2008. (IEEE, 2008), pp. 496–501.

Y. Hu, J. Xi, J. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in 2006 International Conference on Computer Graphics, Imaging and Visualisation (IEEE, 2006), pp. 161–167.

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Figures (13)

Fig. 1.
Fig. 1.

Phase error for three-step phase-shifting method (a) with σ = 0.5 , 1 / f = 120 and different gamma values, (b) with γ = 3.4 , 1 / f = 30 and different PSF, and (c) with γ = 3.4 , σ = 2.5 and different pitches (or the reciprocal of the fundamental frequency).

Fig. 2.
Fig. 2.

Normalized harmonic coefficients.

Fig. 3.
Fig. 3.

Simulated lightness of two gamma correction methods.

Fig. 4.
Fig. 4.

(a) Portable structured light system (a) and the test objects: (b) Box, (c) Marseilles.

Fig. 5.
Fig. 5.

Intensity transfer function of our SLS.

Fig. 6.
Fig. 6.

Different defocus situations of our SLS: (a) defocus 2 , (b) defocus 1 , (c) defocus 0, (d) defocus + 1 , (e) defocus + 2 .

Fig. 7.
Fig. 7.

Phase error for three-step phase-shifting method in different defocus situations: (a) defocus 0, (b) defocus 2 .

Fig. 8.
Fig. 8.

Computed gamma values with Kai’s method.

Fig. 9.
Fig. 9.

Computed gamma values from the proposed method and the traditional gamma calibration method.

Fig. 10.
Fig. 10.

Computed phase errors from (a) the original fringe, (b) the corrected fringes with Eq. (24), and (c) the corrected fringes with Eq. (25). They are normalized with the maximum phase error of the original fringe.

Fig. 11.
Fig. 11.

Phase errors on row 600. STD, standard variance; MAX, maximum value.

Fig. 12.
Fig. 12.

Surface results of the box (a) without gamma correction; (b) with gamma correction in Eq. (24); (c) with gamma correction in Eq. (25); (d) PMP with N = 16 .

Fig. 13.
Fig. 13.

Surface results of Marseilles (a) without gamma correction; (b) with gamma correction in Eq. (24); (c) with gamma correction in Eq. (25); (d) PMP with N = 16 .

Equations (46)

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

I n p = A p + B p cos ( 2 π f x p 2 π n N ) ,
I n c = A c + B c cos ( φ 2 π n N ) ,
φ = arctan [ n = 0 N 1 I n c sin ( 2 π n N ) n = 0 N 1 I n c cos ( 2 π n N ) ] .
I n p = α p ( ( 1 β ) + β cos ( 2 π f x p 2 π n N ) ) ,
I n c = α α p ( ( 1 β ) + β cos ( φ 2 π n N ) ) γ + I 0 ,
I n c = I 0 + B ˜ 0 + i = 1 B ˜ i cos ( i ( φ 2 π n N ) ) ,
( γ + i + 2 ) B ˜ i + 2 = ( γ i ) B ˜ i + ( 2 β 2 ) β ( i + 1 ) B ˜ i + 1 .
B ˜ i + 1 B ˜ i = γ i γ + i + 1 .
G ( x ) = 1 2 π σ exp ( ( x x ¯ ) 2 2 σ 2 ) .
I ˜ n c = I n c * G ( x ) .
B i , f = T ( f ) · B ˜ i ,
I ˜ n c = I 0 + B 0 , f + i = 1 B i , f cos ( i ( φ 2 π n N ) ) .
φ ˜ = arctan ( i = 1 B i , f 2 { n = 0 N 1 sin ( i φ 2 π n ( i 1 ) N ) n = 0 N 1 sin ( i φ 2 π n ( i + 1 ) N ) } i = 1 B i , f 2 { n = 0 N 1 cos ( i φ 2 π n ( i 1 ) N ) + n = 0 N 1 cos ( i φ 2 π n ( i + 1 ) N ) } ) ,
n = 0 N 1 sin ( 2 π n N ) = n = 0 N 1 cos ( 2 π n N ) = 0 , if N > 1.
φ ˜ = arctan ( S N C N ) , where S N = B 1 , f sin ( φ ) + m = 1 ( B m N + 1 , f sin [ ( m N + 1 ) φ ] ) m = 1 ( B m N 1 , f sin [ ( m N 1 ) φ ] ) C N = B 1 , f cos ( φ ) + m = 1 ( B m N + 1 , f cos [ ( m N + 1 , f ) φ ] ) + m = 1 ( B m N 1 , f cos [ ( m N 1 ) φ ] ) .
Δ φ = arctan ( m = 1 ( B m N 1 , f B m N + 1 , f ) sin ( m N φ ) B 1 , f + m = 1 ( B m N 1 , f + B m N + 1 , f ) cos ( m N φ ) ) .
γ = ( k + 2 ) B ˜ k + 2 B ˜ k + k 2 β 2 β ( k + 1 ) B ˜ k + 1 B ˜ k 1 B ˜ k + 2 B ˜ k .
I ^ k , k f ˜ = n = 0 N 1 e i 2 π N n k I ˜ n c k = 0 , 1 , , N 1.
| B k , k f ˜ | = 2 N | I ^ k , k f ˜ | .
Sign ( I ^ k , k f ˜ ) = { 1 , if | angle ( I ^ k , k f ˜ ) k × angle ( I ^ 1 , f ˜ ) | π 2 1 , if | angle ( I ^ k , k f ˜ ) k × angle ( I ^ 1 , f ˜ ) | > π 2 } ,
B k , k f ˜ = Sign ( I ^ k , k f ˜ ) × 2 N | I ^ k , k f ˜ | .
f ˜ 1 = f ˜ 2 / 2 = f ˜ 3 / ,
B 2 , 2 f ˜ 1 B 1 , f ˜ 2 = T ( 2 f ˜ 1 ) B ˜ 2 T ( f ˜ 2 ) B ˜ 1 = B ˜ 2 B ˜ 1 B 3 , 3 f ˜ 1 B 1 , f ˜ 3 = T ( 3 f ˜ 1 ) B ˜ 3 T ( f ˜ 3 ) B ˜ 1 = B ˜ 3 B ˜ 1 .
I ˜ n p = α p ( ( 1 β ) + β cos ( 2 π f x p 2 π n N ) ) 1 / γ ˜ ,
I ˜ n p = α p ( ( 1 β ^ ) + β ^ cos ( 2 π f x p 2 π n N ) ) 1 / γ ˜ , where β ^ = 1 ( 1 2 β ) γ ˜ 2 .
I n c = α α p ( 1 β ) γ ( 1 + β ( 1 β ) cos ( φ 2 π n N ) ) γ + I 0 .
( 1 + x ) t = k = 0 ( t k ) x k ,
I n c = I 0 + α α p ( 1 β ) γ k = 0 ( γ k ) ( β ( 1 β ) ) k ( cos ( φ 2 π n N ) ) k .
cos k ( x ) = 0. 5 k m = 0 k ( k m ) cos ( ( k 2 m ) x ) .
I n c = I 0 + α α p ( 1 β ) γ k = 0 m = 0 k ( γ k ) ( β 2 2 β ) k ( k m ) cos ( ( k 2 m ) ( φ 2 π n N ) ) .
I n c = I 0 + α α p ( 1 β ) γ i = 0 m = 0 ( γ 2 m + i ) ( β 2 2 β ) 2 m + i ( 2 m + i m ) cos ( i ( φ 2 π n N ) ) .
I n c = I 0 + B ˜ 0 + i = 1 B ˜ i cos ( i ( φ 2 π n N ) ) ,
b i + 1 , m b i , m = β ( γ 2 m i ) ( 2 2 β ) ( m + i + 1 ) .
( 2 2 β ) m b i + 1 , m + 2 β m b i , m = β ( γ i ) b i , m + ( 2 β 2 ) ( i + 1 ) b i + 1 , m .
b i , m = ( β ( 2 2 β ) ) 2 m + i γ ( γ 1 ) ( γ 2 m i + 1 ) m ! ( m + i ) ! .
Eq . ( B 2 ) Left = β ( β ( 2 2 β ) ) 2 m + i γ ( γ 1 ) ( γ 2 m i + 1 ) ( γ + i + 2 ) ( m 1 ) ! ( m + i + 1 ) ! .
Eq . ( B 2 ) Left = β ( β ( 2 2 β ) ) 2 z + i + 2 γ ( γ 1 ) ( γ 2 z i 1 ) ( γ + i + 2 ) ( z ) ! ( z + i + 2 ) ! .
Eq . ( B 2 ) Left = β ( γ + i + 2 ) b i + 2 , m .
( γ + i + 2 ) b i + 2 , m = ( γ i ) b i , m + ( 2 β 2 ) β ( i + 1 ) b i + 1 , m .
( γ + i + 2 ) B ˜ i + 2 = ( γ i ) B ˜ i + ( 2 β 2 ) β ( i + 1 ) B ˜ i + 1 .
cos k ( x ) = 0. 5 k m = 0 k ( k m ) cos ( ( k 2 m ) x ) k { 0 , N } .
cos n + 1 ( x ) = cos ( x ) × cos n ( x ) = cos ( x ) × 0.5 n m = 0 n ( n m ) cos ( ( n 2 m ) x ) = 0.5 n m = 0 n ( n m ) cos ( ( n 2 m ) x ) cos ( x ) .
cos n + 1 ( x ) = 0.5 n + 1 { m = 0 n ( n m ) cos ( ( n + 1 2 ( m + 1 ) x ) + m = 0 n ( n m ) cos ( ( n + 1 2 m ) x ) } .
cos n + 1 ( x ) = 0.5 n + 1 { Z = 1 n + 1 ( n z 1 ) cos ( ( n + 1 2 z ) x ) + m = 0 n ( n m ) cos ( ( n + 1 2 m ) x ) } = 0.5 n + 1 { m = 1 n + 1 ( n m 1 ) cos ( ( n + 1 2 m ) x ) + m = 0 n ( n m ) cos ( ( n + 1 2 m ) x ) } = 0.5 n + 1 { m = 1 n ( ( n m 1 ) + ( n m ) ) cos ( ( n + 1 2 m ) x ) + ( n n ) cos ( ( n + 1 ) x ) + ( n 0 ) cos ( ( n + 1 ) x ) } .
( n m 1 ) + ( n m ) = ( n + 1 m ) , ( n 0 ) = ( n + 1 0 ) , ( n n ) = ( n + 1 n + 1 ) .
cos n + 1 ( x ) = 0. 5 n + 1 m = 0 n + 1 ( n + 1 m ) cos ( ( n + 1 2 m ) x ) .

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