Abstract

This paper presents a new approach to fringe pattern profilometry. In this paper, a generalized model describing the relationship between the projected fringe pattern and the deformed fringe pattern is derived, in which the projected fringe pattern can be arbitrary rather than being limited to being sinusoidal, as are those for the conventional approaches. Based on this model, what is believed to be a new approach is proposed to reconstruct the three-dimensional object surface by estimating the shift between the projected and deformed fringe patterns. Additionally, theoretical analysis, computer simulation, and experimental results are presented, which show how the proposed approach can significantly improve the measurement accuracy, especially when the fringe patterns are distorted by unknown factors.

© 2006 Optical Society of America

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References

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  1. M. Takeda and K. Mutoh, "Fourier transform profilometry for the automatic measurement of 3-D object shapes," Appl. Opt. 22, 3977-3982 (1983).
    [CrossRef] [PubMed]
  2. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. A 72, 156-160 (1982).
    [CrossRef]
  3. R. Green, J. Walker, and D. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
    [CrossRef]
  4. X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
    [CrossRef]
  5. J. Yi and S. Huang, "Modified Fourier transform profilometry for the measurement of 3-D steep shapes," Opt. Lasers Eng. 27, 493-505 (1997).
    [CrossRef]
  6. X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3-D shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
    [CrossRef]
  7. V. Srinivasan, H. Liu, and M. Halioua, "Automated phase-measuring profilometry of 3-D diffuse objects," Appl. Opt. 23, 3105-3108 (1984).
    [CrossRef] [PubMed]
  8. X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of Ronchi grating," Opt. Commun. 94, 561-573 (1992).
    [CrossRef]
  9. H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. 36, 1799-1805 (1997).
    [CrossRef]
  10. S. Toyooka and Y. Iwaasa, "Automatic profilometry of 3-D diffuse objects by spatial phase detection," Appl. Opt. 25, 1630-1633 (1986).
    [CrossRef] [PubMed]
  11. R. Rodriguez-Vera and M. Servin, "Phase locked loop profilometry," Opt. Lasers Technol. 26, 393-398 (1994).
    [CrossRef]
  12. A. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
    [CrossRef]
  13. J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
    [CrossRef]
  14. L. Kinell, "Multichannel method for absolute shape measurement using projected fringes," Opt. Lasers Eng. 41, 57-71 (2004).
    [CrossRef]
  15. P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digital fringe projection technique for high-speed 3D surface contouring," Opt. Eng. 38, 1065-1071 (1999).
    [CrossRef]
  16. L. Biancardi, G. Sansoni, and F. Docchio, "Adaptive whole-field optical profilometry: A study of the systematic errors," IEEE Trans. Instrum. Meas. 44, 36-41 (1995).
    [CrossRef]
  17. F. Berryman, P. Pynsent, and J. Cubillo, "A theoretical comparison of three fringe analysis methods for determining the three dimensional shape of an object in the presence of noise," Opt. Lasers Eng. 39, 35-50 (2003).
    [CrossRef]

2004

L. Kinell, "Multichannel method for absolute shape measurement using projected fringes," Opt. Lasers Eng. 41, 57-71 (2004).
[CrossRef]

2003

F. Berryman, P. Pynsent, and J. Cubillo, "A theoretical comparison of three fringe analysis methods for determining the three dimensional shape of an object in the presence of noise," Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

2001

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3-D shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

1999

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[CrossRef]

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digital fringe projection technique for high-speed 3D surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

1997

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. 36, 1799-1805 (1997).
[CrossRef]

J. Yi and S. Huang, "Modified Fourier transform profilometry for the measurement of 3-D steep shapes," Opt. Lasers Eng. 27, 493-505 (1997).
[CrossRef]

1995

A. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

L. Biancardi, G. Sansoni, and F. Docchio, "Adaptive whole-field optical profilometry: A study of the systematic errors," IEEE Trans. Instrum. Meas. 44, 36-41 (1995).
[CrossRef]

1994

R. Rodriguez-Vera and M. Servin, "Phase locked loop profilometry," Opt. Lasers Technol. 26, 393-398 (1994).
[CrossRef]

1992

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of Ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

1988

R. Green, J. Walker, and D. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

1986

1984

1983

1982

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. A 72, 156-160 (1982).
[CrossRef]

Berryman, F.

F. Berryman, P. Pynsent, and J. Cubillo, "A theoretical comparison of three fringe analysis methods for determining the three dimensional shape of an object in the presence of noise," Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

Biancardi, L.

L. Biancardi, G. Sansoni, and F. Docchio, "Adaptive whole-field optical profilometry: A study of the systematic errors," IEEE Trans. Instrum. Meas. 44, 36-41 (1995).
[CrossRef]

Castillo, L.

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[CrossRef]

Chao, Y.

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3-D shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Chen, W.

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3-D shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Chiang, F.

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digital fringe projection technique for high-speed 3D surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Cubillo, J.

F. Berryman, P. Pynsent, and J. Cubillo, "A theoretical comparison of three fringe analysis methods for determining the three dimensional shape of an object in the presence of noise," Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

Docchio, F.

L. Biancardi, G. Sansoni, and F. Docchio, "Adaptive whole-field optical profilometry: A study of the systematic errors," IEEE Trans. Instrum. Meas. 44, 36-41 (1995).
[CrossRef]

Green, R.

R. Green, J. Walker, and D. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

Halioua, M.

Ho, Q.

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digital fringe projection technique for high-speed 3D surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Huang, P.

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digital fringe projection technique for high-speed 3D surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Huang, S.

J. Yi and S. Huang, "Modified Fourier transform profilometry for the measurement of 3-D steep shapes," Opt. Lasers Eng. 27, 493-505 (1997).
[CrossRef]

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. A 72, 156-160 (1982).
[CrossRef]

Iwaasa, Y.

Jin, F.

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digital fringe projection technique for high-speed 3D surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Kinell, L.

L. Kinell, "Multichannel method for absolute shape measurement using projected fringes," Opt. Lasers Eng. 41, 57-71 (2004).
[CrossRef]

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. A 72, 156-160 (1982).
[CrossRef]

Li, J.

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. 36, 1799-1805 (1997).
[CrossRef]

Liu, H.

Mendoza-Santoyo, F.

A. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

Moore, A.

A. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

Mutoh, K.

Pynsent, P.

F. Berryman, P. Pynsent, and J. Cubillo, "A theoretical comparison of three fringe analysis methods for determining the three dimensional shape of an object in the presence of noise," Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

Robinson, D.

R. Green, J. Walker, and D. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

Rodriguez-Vera, R.

R. Rodriguez-Vera and M. Servin, "Phase locked loop profilometry," Opt. Lasers Technol. 26, 393-398 (1994).
[CrossRef]

Sansoni, G.

L. Biancardi, G. Sansoni, and F. Docchio, "Adaptive whole-field optical profilometry: A study of the systematic errors," IEEE Trans. Instrum. Meas. 44, 36-41 (1995).
[CrossRef]

Servin, M.

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[CrossRef]

R. Rodriguez-Vera and M. Servin, "Phase locked loop profilometry," Opt. Lasers Technol. 26, 393-398 (1994).
[CrossRef]

Srinivasan, V.

Su, H.

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. 36, 1799-1805 (1997).
[CrossRef]

Su, X.

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3-D shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. 36, 1799-1805 (1997).
[CrossRef]

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of Ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Takeda, M.

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

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. A 72, 156-160 (1982).
[CrossRef]

Toyooka, S.

Villa, J.

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[CrossRef]

von Bally, G.

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of 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 Ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Walker, J.

R. Green, J. Walker, and D. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

Yi, J.

J. Yi and S. Huang, "Modified Fourier transform profilometry for the measurement of 3-D steep shapes," Opt. Lasers Eng. 27, 493-505 (1997).
[CrossRef]

Zhang, Q.

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3-D shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Zhou, W.

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of Ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Appl. Opt.

IEEE Trans. Instrum. Meas.

L. Biancardi, G. Sansoni, and F. Docchio, "Adaptive whole-field optical profilometry: A study of the systematic errors," IEEE Trans. Instrum. Meas. 44, 36-41 (1995).
[CrossRef]

J. Opt. Soc. Am. A

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. A 72, 156-160 (1982).
[CrossRef]

Opt. Commun.

J. Villa, M. Servin, and L. Castillo, "Profilometry for the measurement of 3-D object shapes based on regularized filters," Opt. Commun. 161, 13-18 (1999).
[CrossRef]

X. Su, W. Zhou, G. von Bally, and D. Vukicevic, "Automated phase-measuring profilometry using defocused projection of Ronchi grating," Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Opt. Eng.

H. Su, J. Li, and X. Su, "Phase algorithm without the influence of carrier frequency," Opt. Eng. 36, 1799-1805 (1997).
[CrossRef]

P. Huang, Q. Ho, F. Jin, and F. Chiang, "Colour-enhanced digital fringe projection technique for high-speed 3D surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Opt. Lasers Eng.

A. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
[CrossRef]

L. Kinell, "Multichannel method for absolute shape measurement using projected fringes," Opt. Lasers Eng. 41, 57-71 (2004).
[CrossRef]

R. Green, J. Walker, and D. Robinson, "Investigation of the Fourier-transform method of fringe pattern analysis," Opt. Lasers Eng. 8, 29-44 (1988).
[CrossRef]

X. Su and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

J. Yi and S. Huang, "Modified Fourier transform profilometry for the measurement of 3-D steep shapes," Opt. Lasers Eng. 27, 493-505 (1997).
[CrossRef]

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3-D shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

F. Berryman, P. Pynsent, and J. Cubillo, "A theoretical comparison of three fringe analysis methods for determining the three dimensional shape of an object in the presence of noise," Opt. Lasers Eng. 39, 35-50 (2003).
[CrossRef]

Opt. Lasers Technol.

R. Rodriguez-Vera and M. Servin, "Phase locked loop profilometry," Opt. Lasers Technol. 26, 393-398 (1994).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of FPP system.

Fig. 2
Fig. 2

Maximum absolute error of measuring height distribution.

Fig. 3
Fig. 3

Simulated fringe patterns and object.

Fig. 4
Fig. 4

Reconstructed 3D surfaces in simulation.

Fig. 5
Fig. 5

Reconstruction results and errors.

Fig. 6
Fig. 6

Object and fringe patterns in experiment.

Fig. 7
Fig. 7

Surface reconstructed by PSP.

Fig. 8
Fig. 8

Surface reconstructed by GSE.

Tables (1)

Tables Icon

Table 1 Absolute Error and Standard Deviation

Equations (237)

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

s ( x )
d ( x )
s ^ ( x ) , d ^ ( x )
s ( x )
d ( x )
h ( x )
s ^ ( x ) = s ( x )
d ^ ( x ) = d ( x )
s ( x )
d ( x )
s ( x )
d ( x )
s ( x )
d ( x )
d ( x d ) = s ( x c ) ,
x c
x d
u = x d - x c .
d ( x d ) = s ( x d - u ) .
x c - x d - h ( x h ) = d 0 l 0 - h ( x h ) ,
x h
l 0
d 0
x h = x d
x c - x d - h ( x d ) = d 0 l 0 - h ( x d ) .
- u - h ( x d ) = d 0 l 0 - h ( x d ) .
h ( x )
x d
x d
u ( x d ) = d 0 h ( x d ) l 0 - h ( x d ) .
h ( x d ) = l 0 u ( x d ) d 0 + u ( x d ) .
d ( x d ) = s [ x d - u ( x d ) ] ,
u ( x d )
x d
x d = x
d ( x ) = s [ x - u ( x ) ] ,
h ( x ) = l 0 u ( x ) d 0 + u ( x ) .
d ( x )
s ( x )
u ( x )
s ^ ( x ) = ζ ( s ( x ) ) ,
d ^ ( x ) = ζ ( d ( x ) ) ,
ζ ( )
s ^ ( x ) , d ^ ( x )
ζ ( )
s ( x )
d ( x )
u ( x )
u ( x )
ζ ( )
s ( x )
d ( x )
s ^ ( x )
d ^ ( x )
s ( x )
d ( x )
s ^ ( x )
d ^ ( x )
s ( x )
d ( x )
u ( x )
s ^ ( x )
d ^ ( x )
u ( x )
d ^ ( x ) = s ^ [ x - u ( x ) ]
ζ ( d ( x ) ) = ζ ( s [ x - u ( x ) ] ) .
ζ ( )
d ( x ) = s [ x - u ( x ) ] ,
u ( x )
s ^ ( x )
d ^ ( x )
s ^ ( x )
d ^ ( x )
u ( x )
s ( x )
s ( x ) = A cos ( 2 π f 0 x )
f 0
d ( x )
d ( x ) = s [ x - u ( x ) ] = A cos [ 2 πf 0 x - 2 π f 0 u ( x ) ] = A cos [ 2 π f 0 x + ϕ ( x ) ] ,
ϕ ( x ) = - 2 π f 0 u ( x )
u ( x ) = - ϕ ( x ) / 2 π f 0
h ( x ) = l 0 ϕ ( x ) ϕ ( x ) - 2 π f 0 d 0 .
ϕ ( x ) = 2 π f 0 d 0 h ( x ) h ( x ) - l 0 .
f 0
u ( x )
h ( x )
u ( x )
u ( x )
s ( x )
d ( x )
u ^ ( x )
u ( x )
e 2 [ u ^ ( x ) ] = { d ( x ) - s [ x - u ^ ( x ) ] } 2 .
e 2
u ^ ( x )
u ^ m + 1 = u ^ m - η d e 2 d u ^ | u ̂ = u ̂ m ,
d e 2 d u ^ | u ̂ = u ̂ m = 2 e d e d u ^ | u ̂ = u ̂ m = - 2 e d s d u ^ | u ̂ = u ̂ m = - 2 e s [ x - ( u ^ m + 1 ) ] - s [ x - ( u ^ m - 1 ) ] ( u ^ m + 1 ) - ( u ^ m - 1 ) = - e [ s ( x - u ^ m - 1 ) - s ( x - u ^ m + 1 ) ] = - [ d ( x ) - s ( x - u ^ m ) ] × [ s ( x - u ^ m - 1 ) - s ( x - u ^ m + 1 ) ] .
u ( x )
u ^ m + 1 ( x ) = u ^ m ( x ) + η { d ( x ) - s [ x - u ^ m ( x ) ] } × { s [ x - u ^ m ( x ) - 1 ] - s [ x - u ^ m ( x ) + 1 ] } .
| u ^ m + 1 - u ^ m |
u ( x )
u ^ ( x ) = u ^ m ( x )
u ^ ( x )
x + 1
u ^ 1 ( x + 1 ) = u ^ ( x )
x + 1
β = l 0 π f 0 d 0 arctan ( p 1 - p ) ,
l 0
5 m
d 0
2 m
0.1 m
f 0 = 10 / m
160 mm
160 mm
s ( x ) = 128 + 100 cos ( 2 π f 0 x ) + 10 cos [ 2 π ( 2 f 0 ) x ] ,
f 0
10 / m
100 mm
l 0
d 0
2 m
1 pixel / mm
- 20 db
s ( x )
ζ ( s ) = 128 tanh ( 3 s 128 - 3 ) tanh ( 3 ) + 128 ,
s ( x )
s ^ ( x ) = ζ ( s ( x ) ) .
d ( x ) = 128 + 100 cos [ 2 π f 0 x + ϕ ( x ) ] + 10 cos [ 2 π ( 2 f 0 ) x + 2 ϕ ( x ) ] ,
ϕ ( x )
d ^ ( x ) = ζ ( d ( x ) ) .
s ^ ( x ) , d ^ ( x )
ε a
ε s
ε a
ε s
2.4966 mm
8.2116 mm
0.5113 mm
2 m
81 cm
22.8 mm
99 mm
16 mm
1392 × 1039 pixels
260 × 194 mm 2
0.1868 mm / pixel
25.7 mm
f 0 = 38.9 / m
f 0
ϕ ( x )
g n ( x ) = a ( x ) + b ( x ) cos [ 2 π f 0 x + ϕ ( x ) + 2 π n / N ]   for   n = 0 , 1 , 2 , , N - 1 ,
ϕ ( x )
g n ( x )
a ( x )
b ( x )
f 0
ϕ ( x )
l 0 h ( x )
ϕ ( x )
ϕ ( x ) 2 π f 0 d 0 h ( x ) - l 0 .
h ( x ) - l 0 2 π f 0 d 0 ϕ ( x ) .
ϕ ( x )
tan [ 2 π f 0 x + ϕ ( x ) ] = - n = 0 N - 1 g n ( x ) sin ( 2 π n / N ) n = 0 N - 1 g n ( x ) cos ( 2 π n / N ) .
g ˜ n ( x ) = a ( x ) + b ( x ) cos [ 2 π f 0 x + ϕ ( x ) + 2 π n / N ] + c m ( x ) cos { m [ 2 π f 0 x + ϕ ( x ) + 2 π n / N ] }   for   n = 0 , 1 , 2 , , N - 1.
ϕ ( x )
ϕ ( x )
2 π f 0 x + ϕ ( x )
tan ( θ + δ ) = - n = 0 N - 1 g ˜ n ( x ) sin ( 2 π n / N ) n = 0 N - 1 g ˜ n ( x ) cos ( 2 π n / N ) = - ( n = 0 N - 1 b cos ( θ + 2 πn / N ) sin ( 2 πn / N ) + n = 0 N - 1 c m cos [ m ( θ + 2 πn / N ) ] sin ( 2 πn / N ) ) / ( n = 0 N - 1 b cos ( θ + 2 πn / N ) cos ( 2 πn / N ) + n = 0 N - 1 c m cos [ m ( θ + 2 πn / N ) ] cos ( 2 πn / N ) ) .
n = 0 N - 1  c m cos [ m ( θ + 2 π n / N ) ] sin ( 2 π n / N )
n = 0 N - 1  c m cos [ m ( θ + 2 π n / N ) ] cos ( 2 π n / N )
m = 3 , N = 3
m = 2 , N = 5
m = 2
N = 3
N = 3
m = 2
c 2
tan ( θ + δ ) = b sin ( θ ) - c sin ( 2 θ ) b cos ( θ ) + c cos ( 2 θ ) .
δ ( θ )
ϕ ^ 0
ϕ ( x ) 0
tan ( ϕ ^ 0 ) = b sin ( 2 π f 0 x ) - c sin ( 2 × 2 π f 0 x ) b cos ( 2 π f 0 x ) + c cos ( 2 × 2 π f 0 x ) .
θ = 2 π f 0 x + ϕ ( x )
ϕ ^ 0 = arctan ( b sin ( θ - ϕ ) - c sin [ 2 ( θ - ϕ ) ] b cos ( θ - ϕ ) + c cos [ 2 ( θ - ϕ ) ] ) .
ϕ ^ = arctan ( b sin ( θ ) - c sin ( 2 θ ) b cos ( θ ) + c cos ( 2 θ ) ) - ϕ ^ 0 = arctan ( b sin ( θ ) - c sin ( 2 θ ) b cos ( θ ) + c cos ( 2 θ ) ) - arctan ( b sin ( θ - ϕ ) - c sin [ 2 ( θ - ϕ ) ] b cos ( θ - ϕ + c cos [ 2 ( θ - ϕ ) ] ) = [ θ + δ ( θ ) ] - [ ( θ - ϕ ) + δ ( θ - ϕ ) ] = ϕ + δ ( θ ) - δ ( θ - ϕ ) .
ε = ϕ ^ - ϕ = δ ( θ ) - δ ( θ - ϕ ) .
ε max = δ max ( θ ) - δ min ( θ ) ,
δ max ( θ )
δ min ( θ )
δ ( θ )
δ ( θ )
tan ( θ + δ ) = sin ( θ ) cos ( δ ) + cos ( θ ) sin ( δ ) cos ( θ ) cos ( δ ) - sin ( θ ) sin ( δ ) .
sin ( δ ) = - p sin ( 3 θ + δ ) ,
p = c 2 / b 2
δ = - arctan ( p sin ( 3 θ ) 1 + p cos ( 3 θ ) ) .
/ = 0
δ ( θ )
δ max = arctan ( p 1 - p )
δ min = - arctan ( p 1 - p ) .
ε max = 2 arctan ( p 1 - p ) .
β = | - l 0 2 π f 0 d 0 ε max | = l 0 2 π f 0 d 0 ε max .
β = l 0 π f 0 d 0 arctan ( p 1 - p )
g ( x ) = b cos ( 2 π f 0 x ) + c m cos ( m 2 π f 0 x )
g ˜ ( x ) = b cos [ 2 π f 0 x + ϕ ( x ) ] + c m cos { m [ 2 π f 0 x + ϕ ( x ) ] } .
g ( x )
g ˜ ( x )
G ( x ) = b exp [ i ξ ( x ) ] + c m exp { i [ m ξ ( x ) ] } ,
ξ ( x ) = 2 π f 0 x
G ˜ ( x ) = b exp { i [ 2 π f 0 x + ϕ ( x ) ] } + c m exp { i [ 2 π f 0 x + ϕ ( x ) ] m } = b exp { i [ ξ ( x ) + ϕ ( x ) ] } + c m exp { i [ ξ ( x ) + ϕ ( x ) ] m }
ϕ ( x )
ϕ ^ = Im { log [ G ^ ( x ) G * ( x ) ] } ,
log ( )
Im ( )
p = c m 2 / b 2
ϕ ^ = Im ( log { b exp [ i ( ξ + ϕ ) ] + c m exp [ i ( ξ + ϕ ) m ] } + log { b exp ( - i ξ ) + c m exp [ - i ( m ξ ) ] } ) = Im ( log { exp [ i ( ξ + ϕ ) ] + p    exp [ i ( ξ + ϕ ) m ] } + log { exp ( - i ξ ) + p exp [ - i ( m ξ ) ] } ) = Im ( i ( ξ + ϕ ) + log { 1 + p exp [ i ( ξ + ϕ ) ( m - 1 ) ] } - i ξ + log { 1 + p exp [ - i ξ ( m - 1 ) ] } ) = Im ( i ϕ + log { 1 + p exp [ i ( ξ + ϕ ) ( m - 1 ) ] } + log { 1 + p exp [ - i ξ ( m - 1 ) ] } ) = ϕ + Im ( log { 1 + p exp [ i ( ξ + ϕ ) ( m - 1 ) ] } + log { 1 + p exp [ - i ξ ( m - 1 ) ] } ) .
ε = ϕ ^ - ϕ = Im ( log { 1 + p exp [ i ( ξ + ϕ ) ( m - 1 ) ] } + log { 1 + p exp [ - i ξ ( m - 1 ) ] } ) = Im ( log { 1 + p exp [ i ( ξ + ϕ ) ( m - 1 ) ] } ) + Im ( log { 1 + p exp [ - i ξ ( m - 1 ) ] } ) .
Im [ log ( ) ]
Λ ( )
Λ ( α ) = Im ( log { 1 + p exp [ i α ( m - 1 ) ] } ) = arctan { p sin [ α ( m - 1 ) ] 1 + p cos [ α ( m - 1 ) ] } .
ε = Λ ( ξ + ϕ ) + Λ ( - ξ ) = Λ ( ξ + ϕ ) - Λ ( ξ ) .
ε max = Λ max ( ξ ) - Λ min ( ξ ) .
/ = 0
Λ max = arctan ( p 1 - p )
Λ min = - arctan ( p 1 - p ) .
ε max = 2 arctan ( p 1 - p ) .
β = l 0 π f 0 d 0 arctan ( p 1 - p ) .
ε a
ε s
6.7768 mm
8.2116 mm
2.0240 mm
2.4966 mm
3.8216 mm
4.9457 mm
0.2094 mm
0.5113 mm

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