Abstract

We present an analysis of a spatial carrier-fringe pattern in three-dimensional (3-D) shape measurement by using the wavelet transform, a tool excelling for its multiresolution in the time- and space-frequency domains. To overcome the limitation of the Fourier transform, we introduce the Gabor wavelet to analyze the phase distributions of the spatial carrier-fringe pattern. The theory of wavelet transform profilometry, an accuracy check by means of a simulation, and an example of 3-D shape measurement are shown.

© 2004 Optical Society of America

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  1. M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
    [CrossRef]
  2. W. Chen, Y. Tan, H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng. 25, 111–120 (1996).
    [CrossRef]
  3. J. Villa, M. Servin, “Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker,” Opt. Lasers Eng. 31, 279–288 (1999).
    [CrossRef]
  4. F. Chen, G. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
    [CrossRef]
  5. J. Zhong, Y. Zhang, “An absolute phase measurement technique based on number theory in multifrequency grating projection profilometry,” Appl. Opt. 40, 492–500 (2001).
    [CrossRef]
  6. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  7. X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Laser Eng. 35, 263–284 (2001).
    [CrossRef]
  8. R. Vander, S. G. Lipson, I. Leizerson, “Fourier fringe analysis with improved spatial resolution,” Appl. Opt. 42, 6830–6837 (2003).
    [CrossRef] [PubMed]
  9. J. Zhong, J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
    [CrossRef]
  10. R.-S. Chang, J.-Y. Sheu, C.-H. Lin, H.-C. Liu, “Analysis of CCD moiré pattern for micro-range measurements using the wavelet transform,” Opt. Laser Technol. 35, 43–47 (2003).
    [CrossRef]
  11. D. P. Casasent, J.-S. Smokelin, A. Ye, “Wavelet and Gabor transforms for detection,” Opt. Eng. 31, 1893–1898 (1992).
    [CrossRef]
  12. M. Lyons, S. Akamatsu, M. Kamachi, J. Gyoba, “Coding facial expressions with Gabor wavelets,” in Proceedings of Third IEEE International Conference on Automatic Face and Gesture Recognition (IEEE Computer Society, Las Alamitos, Calif., 1998), pp. 200–205.
    [CrossRef]
  13. A. Dursun, N. Ecevït, S. Özder, “Application of wavelet and Fourier transforms for the determination of phase and three-dimensional profile,” in Proceedings of International Conference on Signal Processing, Int. J. Comput. Intell. 1, 168–172 (2003).
  14. C. K. Chui, Wavelets: A Tutorial in Theory and Applications (Academic, New York, 1992).
  15. R. A. Carmona, W. L. Hwang, B. Torrésani. “Characterization of signals by the ridges of their wavelet transforms,” IEEE Transactions on Signal Processing, 45, 2586–2590 (1997).
    [CrossRef]
  16. C. A. Sciammarella, T. Kim. “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
    [CrossRef]

2004 (1)

J. Zhong, J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

2003 (4)

R.-S. Chang, J.-Y. Sheu, C.-H. Lin, H.-C. Liu, “Analysis of CCD moiré pattern for micro-range measurements using the wavelet transform,” Opt. Laser Technol. 35, 43–47 (2003).
[CrossRef]

A. Dursun, N. Ecevït, S. Özder, “Application of wavelet and Fourier transforms for the determination of phase and three-dimensional profile,” in Proceedings of International Conference on Signal Processing, Int. J. Comput. Intell. 1, 168–172 (2003).

C. A. Sciammarella, T. Kim. “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

R. Vander, S. G. Lipson, I. Leizerson, “Fourier fringe analysis with improved spatial resolution,” Appl. Opt. 42, 6830–6837 (2003).
[CrossRef] [PubMed]

2001 (2)

2000 (1)

F. Chen, G. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

1999 (1)

J. Villa, M. Servin, “Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker,” Opt. Lasers Eng. 31, 279–288 (1999).
[CrossRef]

1997 (1)

R. A. Carmona, W. L. Hwang, B. Torrésani. “Characterization of signals by the ridges of their wavelet transforms,” IEEE Transactions on Signal Processing, 45, 2586–2590 (1997).
[CrossRef]

1996 (1)

W. Chen, Y. Tan, H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng. 25, 111–120 (1996).
[CrossRef]

1992 (1)

D. P. Casasent, J.-S. Smokelin, A. Ye, “Wavelet and Gabor transforms for detection,” Opt. Eng. 31, 1893–1898 (1992).
[CrossRef]

1990 (1)

M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[CrossRef]

1983 (1)

Akamatsu, S.

M. Lyons, S. Akamatsu, M. Kamachi, J. Gyoba, “Coding facial expressions with Gabor wavelets,” in Proceedings of Third IEEE International Conference on Automatic Face and Gesture Recognition (IEEE Computer Society, Las Alamitos, Calif., 1998), pp. 200–205.
[CrossRef]

Brown, G.

F. Chen, G. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Carmona, R. A.

R. A. Carmona, W. L. Hwang, B. Torrésani. “Characterization of signals by the ridges of their wavelet transforms,” IEEE Transactions on Signal Processing, 45, 2586–2590 (1997).
[CrossRef]

Casasent, D. P.

D. P. Casasent, J.-S. Smokelin, A. Ye, “Wavelet and Gabor transforms for detection,” Opt. Eng. 31, 1893–1898 (1992).
[CrossRef]

Chang, R.-S.

R.-S. Chang, J.-Y. Sheu, C.-H. Lin, H.-C. Liu, “Analysis of CCD moiré pattern for micro-range measurements using the wavelet transform,” Opt. Laser Technol. 35, 43–47 (2003).
[CrossRef]

Chen, F.

F. Chen, G. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Chen, W.

X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Laser Eng. 35, 263–284 (2001).
[CrossRef]

W. Chen, Y. Tan, H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng. 25, 111–120 (1996).
[CrossRef]

Chui, C. K.

C. K. Chui, Wavelets: A Tutorial in Theory and Applications (Academic, New York, 1992).

Dursun, A.

A. Dursun, N. Ecevït, S. Özder, “Application of wavelet and Fourier transforms for the determination of phase and three-dimensional profile,” in Proceedings of International Conference on Signal Processing, Int. J. Comput. Intell. 1, 168–172 (2003).

Ecevït, N.

A. Dursun, N. Ecevït, S. Özder, “Application of wavelet and Fourier transforms for the determination of phase and three-dimensional profile,” in Proceedings of International Conference on Signal Processing, Int. J. Comput. Intell. 1, 168–172 (2003).

Gyoba, J.

M. Lyons, S. Akamatsu, M. Kamachi, J. Gyoba, “Coding facial expressions with Gabor wavelets,” in Proceedings of Third IEEE International Conference on Automatic Face and Gesture Recognition (IEEE Computer Society, Las Alamitos, Calif., 1998), pp. 200–205.
[CrossRef]

Hwang, W. L.

R. A. Carmona, W. L. Hwang, B. Torrésani. “Characterization of signals by the ridges of their wavelet transforms,” IEEE Transactions on Signal Processing, 45, 2586–2590 (1997).
[CrossRef]

Kamachi, M.

M. Lyons, S. Akamatsu, M. Kamachi, J. Gyoba, “Coding facial expressions with Gabor wavelets,” in Proceedings of Third IEEE International Conference on Automatic Face and Gesture Recognition (IEEE Computer Society, Las Alamitos, Calif., 1998), pp. 200–205.
[CrossRef]

Kim, T.

C. A. Sciammarella, T. Kim. “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

Leizerson, I.

Lin, C.-H.

R.-S. Chang, J.-Y. Sheu, C.-H. Lin, H.-C. Liu, “Analysis of CCD moiré pattern for micro-range measurements using the wavelet transform,” Opt. Laser Technol. 35, 43–47 (2003).
[CrossRef]

Lipson, S. G.

Liu, H.-C.

R.-S. Chang, J.-Y. Sheu, C.-H. Lin, H.-C. Liu, “Analysis of CCD moiré pattern for micro-range measurements using the wavelet transform,” Opt. Laser Technol. 35, 43–47 (2003).
[CrossRef]

Lyons, M.

M. Lyons, S. Akamatsu, M. Kamachi, J. Gyoba, “Coding facial expressions with Gabor wavelets,” in Proceedings of Third IEEE International Conference on Automatic Face and Gesture Recognition (IEEE Computer Society, Las Alamitos, Calif., 1998), pp. 200–205.
[CrossRef]

Mutoh, K.

Özder, S.

A. Dursun, N. Ecevït, S. Özder, “Application of wavelet and Fourier transforms for the determination of phase and three-dimensional profile,” in Proceedings of International Conference on Signal Processing, Int. J. Comput. Intell. 1, 168–172 (2003).

Sciammarella, C. A.

C. A. Sciammarella, T. Kim. “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

Servin, M.

J. Villa, M. Servin, “Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker,” Opt. Lasers Eng. 31, 279–288 (1999).
[CrossRef]

Sheu, J.-Y.

R.-S. Chang, J.-Y. Sheu, C.-H. Lin, H.-C. Liu, “Analysis of CCD moiré pattern for micro-range measurements using the wavelet transform,” Opt. Laser Technol. 35, 43–47 (2003).
[CrossRef]

Smokelin, J.-S.

D. P. Casasent, J.-S. Smokelin, A. Ye, “Wavelet and Gabor transforms for detection,” Opt. Eng. 31, 1893–1898 (1992).
[CrossRef]

Song, M.

F. Chen, G. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Su, X.

X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Laser Eng. 35, 263–284 (2001).
[CrossRef]

Takeda, M.

M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[CrossRef]

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

Tan, Y.

W. Chen, Y. Tan, H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng. 25, 111–120 (1996).
[CrossRef]

Torrésani, B.

R. A. Carmona, W. L. Hwang, B. Torrésani. “Characterization of signals by the ridges of their wavelet transforms,” IEEE Transactions on Signal Processing, 45, 2586–2590 (1997).
[CrossRef]

Vander, R.

Villa, J.

J. Villa, M. Servin, “Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker,” Opt. Lasers Eng. 31, 279–288 (1999).
[CrossRef]

Weng, J.

J. Zhong, J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

Ye, A.

D. P. Casasent, J.-S. Smokelin, A. Ye, “Wavelet and Gabor transforms for detection,” Opt. Eng. 31, 1893–1898 (1992).
[CrossRef]

Zhang, Y.

Zhao, H.

W. Chen, Y. Tan, H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng. 25, 111–120 (1996).
[CrossRef]

Zhong, J.

J. Zhong, J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

J. Zhong, Y. Zhang, “An absolute phase measurement technique based on number theory in multifrequency grating projection profilometry,” Appl. Opt. 40, 492–500 (2001).
[CrossRef]

Appl. Opt. (3)

IEEE Transactions on Signal Processing (1)

R. A. Carmona, W. L. Hwang, B. Torrésani. “Characterization of signals by the ridges of their wavelet transforms,” IEEE Transactions on Signal Processing, 45, 2586–2590 (1997).
[CrossRef]

Ind. Metrol. (1)

M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[CrossRef]

Int. J. Comput. Intell. (1)

A. Dursun, N. Ecevït, S. Özder, “Application of wavelet and Fourier transforms for the determination of phase and three-dimensional profile,” in Proceedings of International Conference on Signal Processing, Int. J. Comput. Intell. 1, 168–172 (2003).

Opt. Eng. (4)

C. A. Sciammarella, T. Kim. “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

J. Zhong, J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

D. P. Casasent, J.-S. Smokelin, A. Ye, “Wavelet and Gabor transforms for detection,” Opt. Eng. 31, 1893–1898 (1992).
[CrossRef]

F. Chen, G. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Opt. Laser Eng. (1)

X. Su, W. Chen, “Fourier transform profilometry: a review,” Opt. Laser Eng. 35, 263–284 (2001).
[CrossRef]

Opt. Laser Technol. (1)

R.-S. Chang, J.-Y. Sheu, C.-H. Lin, H.-C. Liu, “Analysis of CCD moiré pattern for micro-range measurements using the wavelet transform,” Opt. Laser Technol. 35, 43–47 (2003).
[CrossRef]

Opt. Lasers Eng. (2)

W. Chen, Y. Tan, H. Zhao, “Automatic analysis technique of spatial carrier-fringe patterns,” Opt. Lasers Eng. 25, 111–120 (1996).
[CrossRef]

J. Villa, M. Servin, “Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker,” Opt. Lasers Eng. 31, 279–288 (1999).
[CrossRef]

Other (2)

M. Lyons, S. Akamatsu, M. Kamachi, J. Gyoba, “Coding facial expressions with Gabor wavelets,” in Proceedings of Third IEEE International Conference on Automatic Face and Gesture Recognition (IEEE Computer Society, Las Alamitos, Calif., 1998), pp. 200–205.
[CrossRef]

C. K. Chui, Wavelets: A Tutorial in Theory and Applications (Academic, New York, 1992).

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Figures (12)

Fig. 1
Fig. 1

(a) Original Ronchi grating pattern and (b) deformed Ronchi grating pattern.

Fig. 2
Fig. 2

Positive spectrum of (a) the original Ronchi grating pattern and (b) the deformed Ronchi grating pattern.

Fig. 3
Fig. 3

(a) Amplitude and (b) phase of the WT components of the deformed Ronchi grating pattern: pi, π.

Fig. 4
Fig. 4

Local phase of the first harmonic wave of the deformed Ronchi grating pattern.

Fig. 5
Fig. 5

Deformed Ronchi grating pattern with (a) white noise and (b) its positive spectrum.

Fig. 6
Fig. 6

(a) Amplitude and (b) phase of the WT components of the deformed Ronchi grating pattern with white noise: pi, π.

Fig. 7
Fig. 7

Local phase of the first harmonic wave of the deformed Ronchi grating pattern with white noise.

Fig. 8
Fig. 8

Phase error of the deformed Ronchi grating pattern with white noise as analyzed by WT.

Fig. 9
Fig. 9

Experimental setup.

Fig. 10
Fig. 10

Phase-modulated grating pattern.

Fig. 11
Fig. 11

Wrapped modulated phase Δϕ as analyzed by WT: pi, π.

Fig. 12
Fig. 12

3-D distribution of unwrapped modulated phase Δϕ as analyzed by WT.

Equations (26)

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

gx, y=rx, yn=- An expi2πnf0x+nϕx, y=rx, yn=- An expiφnx, y,
gox, y= n=- An expi2πnf0x+nϕ0x, y= n=- An expiφonx, y,
g1x, y=A1rx, yexpiφ1x, y,
go1x, y=A1 expiφo1x, y,
φ1x, y=2πf0x+ϕx, y,
φo1x, y=2πf0x+ϕ0x, y.
Δϕx, y=ϕx, y-ϕ0x, y=φ1x, y-φo1x, y=Imlogg1x, yg*ox, y,
|ϕ/x|max< 2πf0/3.
ux=2πx13+8/512x- x16.
ψa,bx= 1a ψx-ba,
Wga, b= - gx, yψa,b*xdx,
ψx= 1π1/42πγ1/2 exp- 2π/γ2x22+i2πx,
φnx, y=φa, b=arctanimagWga, brealWga, b,
φonx, y=φoa, b=arctanimagWgoa, brealWgoa, b.
Aa, b=imagWga, b2+ realWga, b21/2,
Aoa, b= imagWgoa, b2+ realWgoa, b21/2.
Aam, b=maxAa, b,
Aoaom, b=maxAoa, b,
φ1x, y=φam, b,
φo1x, y=φoaom, b
φ1x, y=2πfxx+ϕx, y,
φo1x, y=2πfoxx+ϕ0x, y,
φ1x, y=2πf0x+ϕx, y+ϕ x, y= 2πf0x+ϕx, y,
φo1x, y=2πf0x+ϕ0x, y+ϕ0x, y=2πf0x+ϕ0x, y.
Δϕx, y =ϕx, y-ϕ0x, y =2πf0x+ϕx, y-2πf0x+ϕ0x, y =2πfxx+ϕx, y-2πfoxx+ϕ0x, y =φ1x, y-φo1x, y =φam, b-φoaom, b.
hx, y=l0Δϕx, y/Δϕx, y-2πf0d,

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