Abstract

A two-dimensional continuous wavelet transform employing a real mother wavelet is applied to phase analysis of spatial carrier fringe patterns. In this method, a Hilbert transform is first performed on a carrier fringe pattern to get an analytic signal. Then a two-dimensional wavelet transform is calculated for the signal that is yielded by the first transform. Finally, the height-demodulated phase information can be gotten from the wavelet transform coefficients at the wavelet ridge position. The performance of the proposed method has been evaluated by using computer-generated and real fringe patterns. The result performed better than that of one-dimensional real wavelet transform algorithms in the area with phase discontinuous points and high phase variation, especially when there is much noise in the fringe patterns. Computer simulations and experiments verified the validity of the proposed method.

© 2009 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 3-D object shapes,” Appl. Opt. 22, 3977-3982 (1983).
    [CrossRef]
  2. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263-284 (2001).
  3. J. Li, X. Su, and L. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439-1444 (1990).
  4. S. Li, X. Su, and W. Chen, “Eliminating the zero spectrum in Fourier transform profilometry using empirical mode decomposition,” J. Opt. Soc. Am. A 26, 1195-1201 (2009).
    [CrossRef]
  5. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695-2702 (2004).
    [CrossRef]
  6. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
  7. K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).
  8. J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895-899 (2004).
  9. J. Zhong and J. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt. 43, 4993-4998 (2004).
    [CrossRef]
  10. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30, 2560-2562 (2005).
    [CrossRef]
  11. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722-8732 (2006).
    [CrossRef]
  12. A. Z. Abid, M. A. Gdeisat, and D. R. Burton, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120-6126 (2007).
    [CrossRef]
  13. S. Li, W. Chen, and X. Su, “Reliability-guided phase unwrapping in wavelet-transform profilometry,” Appl. Opt. 47, 3369-3377 (2008).
    [CrossRef]
  14. L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141-148 (2010).
  15. Zh. Xiang and Zh. Hong, “Three-dimensional profilometry based on Mexican hat wavelet transform,” Acta Optica Sinica 29, 197-202 (2009).
  16. R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
    [CrossRef]
  17. J. F. Kirby, “Which wavelet best reproduces the Fourier power spectrum?,” Comput. Geosci. 31, 846-864 (2005).
    [CrossRef]
  18. D. Benitez, P. A. Gaydecki, A. Zaidi, and A. P. Fitzpatrick, “The use of the Hilbert transform in ECG signal analysis,” Comput. Biol. Med. 31, 399-406 (2001).
  19. H. Olkkoned, P. Pesola, J. Olkkonen, and H. Zhou, “Hilbert transform assisted complex wavelet transform for neuroelectric signal analysis,” J. Neurosci. Methods 151, 106-113(2006).
    [CrossRef]
  20. A. A. Nabout and B. Tibken, “Object shape recognition using Mexican hat wavelet descriptors,” in Proceedings of IEEE Conference on Control and Automation (IEEE, 2007), pp. 1313-1318.
  21. S. R. Messer, J. Agzarian, and D. Abbott, “Optimal wavelet denoising for phonocardiograms,” Microelectron. J. 32, 931-941 (2001).
    [CrossRef]
  22. W. L. Anderson and H. Diao, “Two-dimensional wavelet transform and application to holographic particle velocimetry,” Appl. Opt. 34, 249-255 (1995).
    [CrossRef]
  23. S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).
  24. J. Weng, J. Zhong, and C. Hu, “Phase reconstruction of digital holography with the peak of the two-dimensional Gabor wavelet transform,” Appl. Opt. 48, 3308-3316 (2009).
    [CrossRef]
  25. K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
    [CrossRef]
  26. J. P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Proc. 52, 259-281 (1996).
    [CrossRef]
  27. J. P. Antoine and P. Vandergheynst, “Two-dimensional directional wavelets in image processing,” Int. J. Imaging Syst. Technol. 7, 152-165 (1996).
    [CrossRef]
  28. M. Kutter, S. K. Bhattacharjee, and T. Ebrahimi, “Towards second generation watermarking schemes,” in Proceedings of IEEE Conference on Image Processing (IEEE, 1999), pp. 320-323.
  29. L. M. Kaplan and R. Murenzi, “Pose estimation of SAR imagery using the two dimensional continuous wavelet transform,” Patt. Recog. Lett. 24, 2269-2280 (2003).
    [CrossRef]
  30. Yet Another Wavelet Toolbox (YAWTB) home page (accessed in April 2007), http://www.fyma.ucl.ac.be/projects/yawtb/.

2010

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141-148 (2010).

2009

2008

2007

A. Z. Abid, M. A. Gdeisat, and D. R. Burton, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120-6126 (2007).
[CrossRef]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).

2006

H. Olkkoned, P. Pesola, J. Olkkonen, and H. Zhou, “Hilbert transform assisted complex wavelet transform for neuroelectric signal analysis,” J. Neurosci. Methods 151, 106-113(2006).
[CrossRef]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722-8732 (2006).
[CrossRef]

2005

J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30, 2560-2562 (2005).
[CrossRef]

J. F. Kirby, “Which wavelet best reproduces the Fourier power spectrum?,” Comput. Geosci. 31, 846-864 (2005).
[CrossRef]

2004

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695-2702 (2004).
[CrossRef]

J. Zhong and J. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt. 43, 4993-4998 (2004).
[CrossRef]

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).

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

2003

L. M. Kaplan and R. Murenzi, “Pose estimation of SAR imagery using the two dimensional continuous wavelet transform,” Patt. Recog. Lett. 24, 2269-2280 (2003).
[CrossRef]

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

2001

D. Benitez, P. A. Gaydecki, A. Zaidi, and A. P. Fitzpatrick, “The use of the Hilbert transform in ECG signal analysis,” Comput. Biol. Med. 31, 399-406 (2001).

S. R. Messer, J. Agzarian, and D. Abbott, “Optimal wavelet denoising for phonocardiograms,” Microelectron. J. 32, 931-941 (2001).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263-284 (2001).

1997

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).

1996

J. P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Proc. 52, 259-281 (1996).
[CrossRef]

J. P. Antoine and P. Vandergheynst, “Two-dimensional directional wavelets in image processing,” Int. J. Imaging Syst. Technol. 7, 152-165 (1996).
[CrossRef]

1995

1990

J. Li, X. Su, and L. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439-1444 (1990).

1983

Abbott, D.

S. R. Messer, J. Agzarian, and D. Abbott, “Optimal wavelet denoising for phonocardiograms,” Microelectron. J. 32, 931-941 (2001).
[CrossRef]

Abid, A. Z.

Agzarian, J.

S. R. Messer, J. Agzarian, and D. Abbott, “Optimal wavelet denoising for phonocardiograms,” Microelectron. J. 32, 931-941 (2001).
[CrossRef]

Anderson, W. L.

Antoine, J. P.

J. P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Proc. 52, 259-281 (1996).
[CrossRef]

J. P. Antoine and P. Vandergheynst, “Two-dimensional directional wavelets in image processing,” Int. J. Imaging Syst. Technol. 7, 152-165 (1996).
[CrossRef]

Asundi, A. K.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141-148 (2010).

Belaïd, S.

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).

Benitez, D.

D. Benitez, P. A. Gaydecki, A. Zaidi, and A. P. Fitzpatrick, “The use of the Hilbert transform in ECG signal analysis,” Comput. Biol. Med. 31, 399-406 (2001).

Bhattacharjee, S. K.

M. Kutter, S. K. Bhattacharjee, and T. Ebrahimi, “Towards second generation watermarking schemes,” in Proceedings of IEEE Conference on Image Processing (IEEE, 1999), pp. 320-323.

Burton, D. R.

Carmona, R. A.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

Chen, W.

Diao, H.

Ebrahimi, T.

M. Kutter, S. K. Bhattacharjee, and T. Ebrahimi, “Towards second generation watermarking schemes,” in Proceedings of IEEE Conference on Image Processing (IEEE, 1999), pp. 320-323.

Fitzpatrick, A. P.

D. Benitez, P. A. Gaydecki, A. Zaidi, and A. P. Fitzpatrick, “The use of the Hilbert transform in ECG signal analysis,” Comput. Biol. Med. 31, 399-406 (2001).

Gaydecki, P. A.

D. Benitez, P. A. Gaydecki, A. Zaidi, and A. P. Fitzpatrick, “The use of the Hilbert transform in ECG signal analysis,” Comput. Biol. Med. 31, 399-406 (2001).

Gdeisat, M. A.

Guo, L.

J. Li, X. Su, and L. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439-1444 (1990).

Hong, Zh.

Zh. Xiang and Zh. Hong, “Three-dimensional profilometry based on Mexican hat wavelet transform,” Acta Optica Sinica 29, 197-202 (2009).

Hu, C.

Huang, L.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141-148 (2010).

Hwang, W. L.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

Kadooka, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Kaplan, L. M.

L. M. Kaplan and R. Murenzi, “Pose estimation of SAR imagery using the two dimensional continuous wavelet transform,” Patt. Recog. Lett. 24, 2269-2280 (2003).
[CrossRef]

Kemao, Q.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141-148 (2010).

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695-2702 (2004).
[CrossRef]

Kirby, J. F.

J. F. Kirby, “Which wavelet best reproduces the Fourier power spectrum?,” Comput. Geosci. 31, 846-864 (2005).
[CrossRef]

Kunoo, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Kutter, M.

M. Kutter, S. K. Bhattacharjee, and T. Ebrahimi, “Towards second generation watermarking schemes,” in Proceedings of IEEE Conference on Image Processing (IEEE, 1999), pp. 320-323.

Lalor, M. J.

Lebrun, D.

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).

Li, J.

J. Li, X. Su, and L. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439-1444 (1990).

Li, S.

Messer, S. R.

S. R. Messer, J. Agzarian, and D. Abbott, “Optimal wavelet denoising for phonocardiograms,” Microelectron. J. 32, 931-941 (2001).
[CrossRef]

Murenzi, R.

L. M. Kaplan and R. Murenzi, “Pose estimation of SAR imagery using the two dimensional continuous wavelet transform,” Patt. Recog. Lett. 24, 2269-2280 (2003).
[CrossRef]

J. P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Proc. 52, 259-281 (1996).
[CrossRef]

Mutoh, K.

Nabout, A. A.

A. A. Nabout and B. Tibken, “Object shape recognition using Mexican hat wavelet descriptors,” in Proceedings of IEEE Conference on Control and Automation (IEEE, 2007), pp. 1313-1318.

Nagayasu, T.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Olkkoned, H.

H. Olkkoned, P. Pesola, J. Olkkonen, and H. Zhou, “Hilbert transform assisted complex wavelet transform for neuroelectric signal analysis,” J. Neurosci. Methods 151, 106-113(2006).
[CrossRef]

Olkkonen, J.

H. Olkkoned, P. Pesola, J. Olkkonen, and H. Zhou, “Hilbert transform assisted complex wavelet transform for neuroelectric signal analysis,” J. Neurosci. Methods 151, 106-113(2006).
[CrossRef]

Ono, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Özkul, C.

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).

Pan, B.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141-148 (2010).

Pesola, P.

H. Olkkoned, P. Pesola, J. Olkkonen, and H. Zhou, “Hilbert transform assisted complex wavelet transform for neuroelectric signal analysis,” J. Neurosci. Methods 151, 106-113(2006).
[CrossRef]

Qian, K.

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).

Su, X.

S. Li, X. Su, and W. Chen, “Eliminating the zero spectrum in Fourier transform profilometry using empirical mode decomposition,” J. Opt. Soc. Am. A 26, 1195-1201 (2009).
[CrossRef]

S. Li, W. Chen, and X. Su, “Reliability-guided phase unwrapping in wavelet-transform profilometry,” Appl. Opt. 47, 3369-3377 (2008).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263-284 (2001).

J. Li, X. Su, and L. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439-1444 (1990).

Takeda, M.

Tibken, B.

A. A. Nabout and B. Tibken, “Object shape recognition using Mexican hat wavelet descriptors,” in Proceedings of IEEE Conference on Control and Automation (IEEE, 2007), pp. 1313-1318.

Torresani, B.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

Uda, N.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Vandergheynst, P.

J. P. Antoine and P. Vandergheynst, “Two-dimensional directional wavelets in image processing,” Int. J. Imaging Syst. Technol. 7, 152-165 (1996).
[CrossRef]

Weng, J.

Xiang, Zh.

Zh. Xiang and Zh. Hong, “Three-dimensional profilometry based on Mexican hat wavelet transform,” Acta Optica Sinica 29, 197-202 (2009).

Zaidi, A.

D. Benitez, P. A. Gaydecki, A. Zaidi, and A. P. Fitzpatrick, “The use of the Hilbert transform in ECG signal analysis,” Comput. Biol. Med. 31, 399-406 (2001).

Zhong, J.

Zhou, H.

H. Olkkoned, P. Pesola, J. Olkkonen, and H. Zhou, “Hilbert transform assisted complex wavelet transform for neuroelectric signal analysis,” J. Neurosci. Methods 151, 106-113(2006).
[CrossRef]

Acta Optica Sinica

Zh. Xiang and Zh. Hong, “Three-dimensional profilometry based on Mexican hat wavelet transform,” Acta Optica Sinica 29, 197-202 (2009).

Appl. Opt.

Comput. Biol. Med.

D. Benitez, P. A. Gaydecki, A. Zaidi, and A. P. Fitzpatrick, “The use of the Hilbert transform in ECG signal analysis,” Comput. Biol. Med. 31, 399-406 (2001).

Comput. Geosci.

J. F. Kirby, “Which wavelet best reproduces the Fourier power spectrum?,” Comput. Geosci. 31, 846-864 (2005).
[CrossRef]

Exp. Mech.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

IEEE Trans. Signal Process.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

Int. J. Imaging Syst. Technol.

J. P. Antoine and P. Vandergheynst, “Two-dimensional directional wavelets in image processing,” Int. J. Imaging Syst. Technol. 7, 152-165 (1996).
[CrossRef]

J. Neurosci. Methods

H. Olkkoned, P. Pesola, J. Olkkonen, and H. Zhou, “Hilbert transform assisted complex wavelet transform for neuroelectric signal analysis,” J. Neurosci. Methods 151, 106-113(2006).
[CrossRef]

J. Opt. Soc. Am. A

Microelectron. J.

S. R. Messer, J. Agzarian, and D. Abbott, “Optimal wavelet denoising for phonocardiograms,” Microelectron. J. 32, 931-941 (2001).
[CrossRef]

Opt. Eng.

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).

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

J. Li, X. Su, and L. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439-1444 (1990).

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).

Opt. Lasers Eng.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141-148 (2010).

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263-284 (2001).

Opt. Lett.

Patt. Recog. Lett.

L. M. Kaplan and R. Murenzi, “Pose estimation of SAR imagery using the two dimensional continuous wavelet transform,” Patt. Recog. Lett. 24, 2269-2280 (2003).
[CrossRef]

Signal Proc.

J. P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Proc. 52, 259-281 (1996).
[CrossRef]

Other

Yet Another Wavelet Toolbox (YAWTB) home page (accessed in April 2007), http://www.fyma.ucl.ac.be/projects/yawtb/.

M. Kutter, S. K. Bhattacharjee, and T. Ebrahimi, “Towards second generation watermarking schemes,” in Proceedings of IEEE Conference on Image Processing (IEEE, 1999), pp. 320-323.

A. A. Nabout and B. Tibken, “Object shape recognition using Mexican hat wavelet descriptors,” in Proceedings of IEEE Conference on Control and Automation (IEEE, 2007), pp. 1313-1318.

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Figures (13)

Fig. 1
Fig. 1

Optical geometry.

Fig. 2
Fig. 2

Complex Morlet wavelet: (a) real part and imaginary part, (b) spectrum and 1-D Mexican hat wavelet in the spatial domain (c), frequency domain (d).

Fig. 3
Fig. 3

Illustration of a 4-D array produced by the 2-D CWT.

Fig. 4
Fig. 4

Fan wavelet: (a) real part, (b) imaginary part, (c) spectrum and 2-D Mexican hat wavelet in the (d) spatial domain and (e) frequency domain.

Fig. 5
Fig. 5

Simulated phase (a) and the deformed fringe pattern (b).

Fig. 6
Fig. 6

(a), (c), (e), and (g) Wrapped phase obtained by using the complex Morlet wavelet method, fan wavelet method, 1-D Mexican hat wavelet method, and 2-D Mexican hat wavelet method, respectively; (b), (d), (f), and (h) unwrapped phase from the wrapped phase shown in (a), (c), (e) and (g), respectively.

Fig. 7
Fig. 7

256th row of the restored phase (a) and its error distribution (b).

Fig. 8
Fig. 8

Simulated phase.

Fig. 9
Fig. 9

Unwrapped phase (a) and error (b) obtained by using the 2-D Mexican hat method; the unwrapped phase (c) and error (d) obtained by using the 1-D Mexican hat method; the unwrapped phase (e) and error (f) obtained by using the complex Morlet method; and the wrapped phase (g) and unwrapped phase (h) obtained by using the fan wavelet method.

Fig. 10
Fig. 10

Experiment: (a) deformed fringe pattern; (b), (c), (d), and (e) phase retrieved by using the fan method, complex Morlet method, 1-D Mexican hat method, and 2-D Mexican hat method, respectively.

Fig. 11
Fig. 11

(a) Deformed fringe pattern with noise; (b), (c), (d), and (e) phase retrieved by using the 2-D Mexican hat method, 1-D Mexican hat method, complex Morlet method, and fan wavelet method, respectively.

Fig. 12
Fig. 12

Standard deviation curves of the four wavelet methods.

Fig. 13
Fig. 13

Experiment: (a) deformed fringe pattern; (b), (c), (d) and (e) phase retrieved by using the 1-D Mexican hat wavelet method, 2-D-Mexican hat wavelet method, complex Morlet wavelet method, and fan wavelet method, respectively.

Equations (28)

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

g ( x , y ) = C ( x , y ) + B ( x , y ) cos [ 2 π f 0 x + ϕ ( x , y ) ] ,
W f ( s , b ) = < f ( x ) , ψ ¯ s , b ( x ) > = 1 s f ( x ) ψ ¯ ( x b s ) d x .
A ( s , b ) = { Im [ W f ( s , b ) ] } 2 + { Re [ W f ( s , b ) ] } 2 ,
φ ( s , b ) = arctan { Im [ W f ( s , b ) ] / Re [ W f ( s , b ) ] } ,
φ ( b ) = φ ( s r b , b ) .
Δ φ ( x , y ) = φ ( x , y ) φ 0 ( x , y ) .
h ( x , y ) = L 0 2 π f 0 d ϕ ( x , y ) .
ψ ( x ) = π 1 / 4 exp ( i k 0 x ) exp ( x 2 / 2 ) ,
ψ ^ ( s ) = 2 π 1 / 4 e ( s k 0 ) 2 / 2 ,
ψ M ( x , y ) = exp [ i k 0 ( x cos θ + y sin θ ) ] exp ( 1 2 x 2 + y 2 ) ,
ψ ^ M ( s , r ) = exp { 1 2 [ ( r k 0 cos θ ) 2 + ( s k 0 sin θ ) 2 ] } .
ψ F ( x , y ) = j = 0 N θ 1 exp [ i k 0 ( x cos θ j + y sin θ j ) ] exp ( 1 2 x 2 + y 2 ) ,
ψ ^ F ( s , r ) = k = 0 N θ 1 exp { 1 2 [ ( r k 0 cos θ k ) 2 + ( s k 0 sin θ k ) 2 ] } .
H ( f ( x ) ) = 1 π f ( τ ) x τ d τ = 1 π f ( x τ ) τ d τ = f ( x ) * 1 π x .
W [ H ( f ( x ) ) ] = ( f ( x ) * 1 π x ) * ψ ¯ ( x ) = ( f ( x ) * ψ ¯ ( x ) ) * 1 π x = H [ W ( ( x , s ) ) ] .
W [ z ( x ) ] = W [ f ( x ) + i H ( f ( x ) ) ] = W [ f ( x ) ] + W [ i H ( f ( x ) ) ] = W [ f ( x ) ] + i H [ W ( f ( x ) ) ] .
I ^ ( ω ) = i sgn ( ω ) = { i ω 0 i , ω < 0 .
z ^ ( ω ) = { 2 f ^ ( ω ) , ω 0 0 ω < 0 ,
ψ ( x ) = ( 1 x 2 ) e 1 2 x 2 ,
ψ ^ ( ω ) = 2 π ω 2 e 1 2 ω 2 .
W g ( b x , b y , s , θ ) = 1 s g ( x , y ) ψ ( x b x s , y b y s , r θ ) d x d y .
r θ = [ cos θ sin θ sin θ cos θ ]
ψ ( x , y ) = ( 2 ( x 2 + y 2 ) ) e ( x 2 + y 2 ) / 2 ,
ψ ^ ( ω x , ω y ) = 2 π ( ω x 2 + ω y 2 ) e ( ω x 2 + ω y 2 ) / 2 .
ϕ ( x , y ) = { ( π / 80 ) 200 2 ( x 256 ) 2 ( y 256 ) 2 , ( x 256 ) 2 + ( y 256 ) 2 200 2 0 , x 2 + y 2 > 200 2 ,
g ( x , y ) = 1 + cos [ 2 π f 0 x + ϕ ( x , y ) ] .
peaks ( x , y ) = ( 1 x ) 2 exp ( x 2 ( y + 1 ) 2 ) 10 ( x 5 x 3 y 5 ) exp ( x 2 y 2 ) 1 3 exp ( ( x + 1 ) 2 y 2 ) .
g ( x , y ) = 1 + cos [ 2 π f 0 x + ϕ ( x , y ) ] + noise .

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