Abstract

What is believed to be a new analysis algorithm to carry out profile measurement with low computational complexity and less noise sensitivity is presented. First, a discrete cosine transform (DCT)-based representation method is introduced to express the height distribution of a 3D surface. Then a novel shift estimation algorithm, called the DCT-based shift estimation (DCT-SE), is presented to reconstruct 3D object surfaces by using the proposed expression and the generalized analysis model. The advantage of DCT-SE is that without loss of measurement precision it provides lower computational complexity to implement 3D reconstruction from nonlinearly distorted fringe patterns and, at the same time, survives the random noise. Simulations and experiments show that the proposed DCT-SE is a fast, accurate, and efficient reconstruction algorithm for digital projection- based fringe pattern profilometry techniques.

© 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. 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. Laser 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. S. Huang, Q. Hu, F. Jin, and F. Chiang, "Color-encoded digitial fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
    [CrossRef]
  16. H. Guo, H. He, and M. Chen, "Gamma correction for digital fringe projection profilometry," Appl. Opt. 43, 2906-2914 (2004).
    [CrossRef] [PubMed]
  17. 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] [PubMed]
  18. F. Berryman, P. Pynsent, and J. Cubillo, "The effect of windowing in Fourier transform profilometry applied to noisy images," Opt. Lasers Eng. 41, 815-825 (2004).
    [CrossRef]
  19. N. Ahmed, T. Natarajan, and K. R. Rao, "Discrete cosine transform," IEEE Trans. Comput. C-23, 90-93 (1974).
    [CrossRef]

2006 (1)

2004 (3)

H. Guo, H. He, and M. Chen, "Gamma correction for digital fringe projection profilometry," Appl. Opt. 43, 2906-2914 (2004).
[CrossRef] [PubMed]

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

F. Berryman, P. Pynsent, and J. Cubillo, "The effect of windowing in Fourier transform profilometry applied to noisy images," Opt. Lasers Eng. 41, 815-825 (2004).
[CrossRef]

2001 (2)

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 (2)

P. S. Huang, Q. Hu, F. Jin, and F. Chiang, "Color-encoded digitial fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

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]

1997 (2)

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]

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

1995 (1)

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

1994 (1)

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

1992 (1)

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 (1)

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 (1)

1984 (1)

1983 (1)

1982 (1)

1974 (1)

N. Ahmed, T. Natarajan, and K. R. Rao, "Discrete cosine transform," IEEE Trans. Comput. C-23, 90-93 (1974).
[CrossRef]

Ahmed, N.

N. Ahmed, T. Natarajan, and K. R. Rao, "Discrete cosine transform," IEEE Trans. Comput. C-23, 90-93 (1974).
[CrossRef]

Berryman, F.

F. Berryman, P. Pynsent, and J. Cubillo, "The effect of windowing in Fourier transform profilometry applied to noisy images," Opt. Lasers Eng. 41, 815-825 (2004).
[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, M.

Chen, W.

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]

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

Chiang, F.

P. S. Huang, Q. Hu, F. Jin, and F. Chiang, "Color-encoded digitial fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Chicharo, J.

Cubillo, J.

F. Berryman, P. Pynsent, and J. Cubillo, "The effect of windowing in Fourier transform profilometry applied to noisy images," Opt. Lasers Eng. 41, 815-825 (2004).
[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]

Guo, H.

Halioua, M.

He, H.

Hu, Q.

P. S. Huang, Q. Hu, F. Jin, and F. Chiang, "Color-encoded digitial fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Hu, Y.

Huang, P. S.

P. S. Huang, Q. Hu, F. Jin, and F. Chiang, "Color-encoded digitial fringe projection technique for high-speed three-dimensional 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.

Iwaasa, Y.

Jin, F.

P. S. Huang, Q. Hu, F. Jin, and F. Chiang, "Color-encoded digitial fringe projection technique for high-speed three-dimensional 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.

Li, E.

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.

Natarajan, T.

N. Ahmed, T. Natarajan, and K. R. Rao, "Discrete cosine transform," IEEE Trans. Comput. C-23, 90-93 (1974).
[CrossRef]

Pynsent, P.

F. Berryman, P. Pynsent, and J. Cubillo, "The effect of windowing in Fourier transform profilometry applied to noisy images," Opt. Lasers Eng. 41, 815-825 (2004).
[CrossRef]

Rao, K. R.

N. Ahmed, T. Natarajan, and K. R. Rao, "Discrete cosine transform," IEEE Trans. Comput. C-23, 90-93 (1974).
[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. Laser Technol. 26, 393-398 (1994).
[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. Laser 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.

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]

Xi, J.

Yang, Z.

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. (5)

IEEE Trans. Comput. (1)

N. Ahmed, T. Natarajan, and K. R. Rao, "Discrete cosine transform," IEEE Trans. Comput. C-23, 90-93 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

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. (2)

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

P. S. Huang, Q. Hu, F. Jin, and F. Chiang, "Color-encoded digitial fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Opt. Laser Technol. (1)

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

Opt. Lasers Eng. (7)

A. Moore and F. Mendoza-Santoyo, "Phase demodulation in the space domain without a fringe carrier," Opt. Lasers Eng. 23, 319-330 (1995).
[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, "The effect of windowing in Fourier transform profilometry applied to noisy images," Opt. Lasers Eng. 41, 815-825 (2004).
[CrossRef]

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

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the FPP system.

Fig. 2
Fig. 2

Simulated fringe patterns and object.

Fig. 3
Fig. 3

Reconstruction results obtained in a noise free situation.

Fig. 4
Fig. 4

Captured fringes corrupted by noise.

Fig. 5
Fig. 5

Reconstruction results with noise.

Fig. 6
Fig. 6

Convergence of the DCT-SE algorithm.

Fig. 7
Fig. 7

Object and fringe patterns that were used in the experiment.

Fig. 8
Fig. 8

Surface reconstructed by FTP.

Fig. 9
Fig. 9

Surface reconstructed by the DCT-SE algorithm.

Tables (1)

Tables Icon

Table 1 Average Number of Iterations and Reconstruction Errors

Equations (30)

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

d ( x ) = s [ x u ( x ) ] ,
h ( x ) = l 0 u ( x ) d 0 + u ( x ) ,
w = L × u ,
l i j = { 1 N ( i = 1 ) 2 N   cos   ( i 1 ) ( 2 j 1 ) π 2 N ( i 1 ) .
u = L 1 × w = L T × w .
p = G × w ,
g i j = { 1, if   w ( j )  is   the   i th   selected   element   of   w 0, others .
C = ( G × L ) T .
u = C × p = ( G × L ) T × ( G × w ) ,
J = E ( e 2 ) = E ( { d ( x ) s [ x u ( x ) ] } 2 ) ,
J = E ( { d ( x ) s [ x u ( x ) ] } 2 ) .
J ( p ) = E ( { d ( x ) s [ x ( C × p ) ( x ) ] } 2 )
= E ( { d ( x ) s [ x C ( x ) × p ] } 2 ) .
p ^ m + 1 = p ^ m η · p J ( p ^ m ) ,
p ^ m + 1 = p ^ m η · ̂ p J ( p ^ m ) ,
̂ p [ J ( p ^ m ) ] = p ( { d ( x ) s [ x C ( x ) × p ^ m ] } 2 ) = p e m 2 = 2 e m p e m = 2 e m p s ( p ^ m ) = 2 e m u s [ u ^ m ( x ) ] p u ^ m ( x ) ,
̂ p [ J ( p ^ m ) ] = 2 e m × s [ x ( u ^ m ( x ) + 1 ) ] s [ x ( u ^ m ( x ) 1 ) ] [ u ^ m ( x ) + 1 ] [ u ^ m ( x ) 1 ] × p u m ( x ) = e m { s [ x u ^ m ( x ) 1 ] s [ x u ^ m ( x ) + 1 ] } × C ( x ) = { d ( x ) s [ x C ( x ) × p ^ m ] } × { s [ x u ^ m ( x ) 1 ] s [ x u ^ m ( x ) + 1 ] } × C ( x ) = { d ( x ) s [ x C ( x ) × p ^ m ] } × { s [ x C ( x ) × p ^ m 1 ] s [ x C ( x ) × p ^ m + 1 ] } C ( x ) .
p ^ m + 1 = p ^ m + η · { d ( x ) s [ x C ( x ) × p ^ m ] } × { s [ x C ( x ) × p ^ m 1 ] s [ x C ( x ) × p ^ m + 1 ] } C ( x ) .
p 1 = G × p FTP ,
k = 1 M | p 1 ( k ) | k = 1 N | p f t p ( k ) | β .
C = ( G × L ) T .
η m = μ · | p m | ,
s ( x ) = 128 + 100   cos ( 2 π f 0 x ) + 10   cos [ 2 π ( 2 f 0 ) x ] ,
ζ ( s ) = 128 tanh [ ( 3 s / 128 ) 3 ] tanh ( 3 ) + 128 ,
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 ) ] ,
d ^ ( x ) = ζ [ d ( x ) ] .
s ^ n ( x ) = s ^ ( x ) + n 1 ( x ) ,
d ^ n ( x ) = d ^ ( x ) + n 2 ( x ) ,
γ = total   number   of   iterations total   number   of   sample   points .

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