Abstract

This paper theoretically discusses modulus of two-dimensional (2D) wavelet transform (WT) coefficients, calculated by using two frequently used 2D daughter wavelet definitions, in an optical fringe pattern analysis. The discussion shows that neither is good enough to represent the reliability of the phase data. The differences between the two frequently used 2D daughter wavelet definitions in the performance of 2D WT also are discussed. We propose a new 2D daughter wavelet definition for reliability-guided phase unwrapping of optical fringe pattern. The modulus of the advanced 2D WT coefficients, obtained by using a daughter wavelet under this new daughter wavelet definition, includes not only modulation information but also local frequency information of the deformed fringe pattern. Therefore, it can be treated as a good parameter that represents the reliability of the retrieved phase data. Computer simulation and experimentation show the validity of the proposed method.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. 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]
  5. S. Li, X. Su, and W. Chen, “Wavelet ridge techniques in optical fringe pattern analysis,” J. Opt. Soc. Am. A 27, 1245–1254 (2010).
    [CrossRef]
  6. 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).
    [CrossRef]
  7. C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
    [CrossRef]
  8. M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
    [CrossRef]
  9. C. J. Tay, C. Quan, and W. Sun, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. J. Weng, and J. Zhong, “Phase reconstruction of digital holography with the peak of the two-dimensional Gabor wavelet transform,” Appl. Opt. 48, 3308–3316 (2009).
    [CrossRef]
  16. 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]
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    [CrossRef]
  18. S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48, 6893–6906 (2009).
    [CrossRef]
  19. H. Niu, C. Quan, and C. J. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
    [CrossRef]
  20. Z. Wang, and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
    [CrossRef]
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  22. S. Zhang, X. Li, and S. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50–57 (2007).
    [CrossRef]
  23. X. Su, and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261(2004).
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  25. X. Su, G. Bally, and D. Vukicevic, “phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
    [CrossRef]
  26. X. Su, A. M. Zarubin, and G. Bally, “Modulation analysis of phase-shifting holographic interferograms,” Opt. Commun. 105, 379–387 (1994).
    [CrossRef]
  27. X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637–643 (2001).
    [CrossRef]
  28. L. Xue and X. Su, “Phase-unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring profilometry method,” Appl. Opt. 40, 1207–1215 (2001).
    [CrossRef]

2010 (3)

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).
[CrossRef]

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

S. Li, X. Su, and W. Chen, “Wavelet ridge techniques in optical fringe pattern analysis,” J. Opt. Soc. Am. A 27, 1245–1254 (2010).
[CrossRef]

2009 (4)

J. Weng, and J. Zhong, “Phase reconstruction of digital holography with the peak of the two-dimensional Gabor wavelet transform,” Appl. Opt. 48, 3308–3316 (2009).
[CrossRef]

S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48, 6893–6906 (2009).
[CrossRef]

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

H. Niu, C. Quan, and C. J. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

2007 (2)

C. J. Tay, C. Quan, and W. Sun, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

S. Zhang, X. Li, and S. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46, 50–57 (2007).
[CrossRef]

2006 (2)

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]

Z. Wang, and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

2005 (1)

2004 (3)

2003 (1)

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 (3)

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

L. Xue and X. Su, “Phase-unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring profilometry method,” Appl. Opt. 40, 1207–1215 (2001).
[CrossRef]

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

1999 (2)

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
[CrossRef]

M. Cherbuliez, P. Jacquot, and X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” Proc. SPIE 3813, 692–702 (1999).
[CrossRef]

1997 (2)

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).
[CrossRef]

1995 (1)

1994 (1)

X. Su, A. M. Zarubin, and G. Bally, “Modulation analysis of phase-shifting holographic interferograms,” Opt. Commun. 105, 379–387 (1994).
[CrossRef]

1993 (1)

X. Su, G. Bally, and D. Vukicevic, “phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

1982 (1)

Abid, A.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

Anderson, W. L.

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).
[CrossRef]

Bally, G.

X. Su, A. M. Zarubin, and G. Bally, “Modulation analysis of phase-shifting holographic interferograms,” Opt. Commun. 105, 379–387 (1994).
[CrossRef]

X. Su, G. Bally, and D. Vukicevic, “phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Barnes, T. H.

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).
[CrossRef]

Burton, D. R.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[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]

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.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

S. Li, X. Su, and W. Chen, “Wavelet ridge techniques in optical fringe pattern analysis,” J. Opt. Soc. Am. A 27, 1245–1254 (2010).
[CrossRef]

S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48, 6893–6906 (2009).
[CrossRef]

X. Su, and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261(2004).

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

Cherbuliez, M.

M. Cherbuliez, P. Jacquot, and X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” Proc. SPIE 3813, 692–702 (1999).
[CrossRef]

de Lega, X.

M. Cherbuliez, P. Jacquot, and X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” Proc. SPIE 3813, 692–702 (1999).
[CrossRef]

Diao, H.

Gdeisat, M. A.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[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]

Ghiglia, D. C.

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

He, G.

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).
[CrossRef]

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]

Ina, H.

Jacquot, P.

M. Cherbuliez, P. Jacquot, and X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” Proc. SPIE 3813, 692–702 (1999).
[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]

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).
[CrossRef]

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

Kobayashi, S.

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]

Lalor, M. J.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[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]

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).
[CrossRef]

Li, S.

Li, X.

Lilley, F.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

Liu, Y.

Lu, Y.

Ma, H.

Z. Wang, and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

Moore, C.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

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]

Niu, H.

H. Niu, C. Quan, and C. J. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[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).
[CrossRef]

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).
[CrossRef]

Pritt, M. D.

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

Quan, C.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

H. Niu, C. Quan, and C. J. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

C. J. Tay, C. Quan, and W. Sun, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

Qudeisat, M.

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

Su, X.

S. Li, X. Su, and W. Chen, “Wavelet ridge techniques in optical fringe pattern analysis,” J. Opt. Soc. Am. A 27, 1245–1254 (2010).
[CrossRef]

S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48, 6893–6906 (2009).
[CrossRef]

X. Su, and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261(2004).

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

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

L. Xue and X. Su, “Phase-unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring profilometry method,” Appl. Opt. 40, 1207–1215 (2001).
[CrossRef]

X. Su, A. M. Zarubin, and G. Bally, “Modulation analysis of phase-shifting holographic interferograms,” Opt. Commun. 105, 379–387 (1994).
[CrossRef]

X. Su, G. Bally, and D. Vukicevic, “phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Sun, W.

C. J. Tay, C. Quan, and W. Sun, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

Takeda, M.

Tan, S. M.

Tay, C. J.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

H. Niu, C. Quan, and C. J. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

C. J. Tay, C. Quan, and W. Sun, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

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]

Vukicevic, D.

X. Su, G. Bally, and D. Vukicevic, “phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Wang, X.

Wang, Z.

Z. Wang, and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[CrossRef]

Watkins, L. R.

Weng, J.

Xue, L.

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

L. Xue and X. Su, “Phase-unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring profilometry method,” Appl. Opt. 40, 1207–1215 (2001).
[CrossRef]

Yau, S.

Zarubin, A. M.

X. Su, A. M. Zarubin, and G. Bally, “Modulation analysis of phase-shifting holographic interferograms,” Opt. Commun. 105, 379–387 (1994).
[CrossRef]

Zhang, S.

Zheng, D.

Zhong, J.

Zhong, X.

Appl. Opt. (7)

Chin. Opt. Lett. (1)

Exp. Mech. (1)

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 (1)

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]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (3)

C. J. Tay, C. Quan, and W. Sun, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327–336 (2007).
[CrossRef]

X. Su, G. Bally, and D. Vukicevic, “phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

X. Su, A. M. Zarubin, and G. Bally, “Modulation analysis of phase-shifting holographic interferograms,” Opt. Commun. 105, 379–387 (1994).
[CrossRef]

Opt. Eng. (3)

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637–643 (2001).
[CrossRef]

Z. Wang, and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45, 045601 (2006).
[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).
[CrossRef]

Opt. Lasers Eng. (6)

H. Niu, C. Quan, and C. J. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

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

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).
[CrossRef]

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

M. A. Gdeisat, A. Abid, D. R. Burton, M. J. Lalor, F. Lilley, C. Moore, and M. Qudeisat, “Spatial and temporal carrier fringe pattern demodulation using the one-dimensional continuous wavelet transform: recent progress, challenges and suggested developments,” Opt. Lasers Eng. 47, 1348–1361 (2009).
[CrossRef]

X. Su, and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245–261(2004).

Opt. Lett. (2)

Proc. SPIE (1)

M. Cherbuliez, P. Jacquot, and X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” Proc. SPIE 3813, 692–702 (1999).
[CrossRef]

Other (1)

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

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Figures (5)

Fig. 1.
Fig. 1.

(a) Reference fringe pattern, (b) modulated phase, and (c) deformed fringe pattern.

Fig. 2.
Fig. 2.

Modulus of the 2D WT coefficients relating to the ridge calculated by (a) using the first daughter wavelet definition, (b) the second daughter wavelet definition, and (c) the proposed daughter wavelet definition.

Fig. 3.
Fig. 3.

(a) Real deformed fringe pattern, and (b) the wrapped phase extracted by using 2D WT method employing the second daughter wavelet definition.

Fig. 4.
Fig. 4.

Modulus of the 2D WT coefficients relating to the ridge calculated by (a) using the first daughter wavelet definition, (b) the second daughter wavelet definition and (c) the proposed daughter wavelet definition, respectively; (d), (e), (f) are the unwrapped phases guided by (a), (b) and (c), respectively.

Fig. 5.
Fig. 5.

(a) Residues of the wrapped phase, (b) unwrapped phase by using the branch—cut method, and (c) unwrapped phase by row scanning method.

Equations (18)

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g(x,y)=A(x,y)+B(x,y)cosφ(x,y)=A+12Bexp[jφ(x,y)]+12Bexp[jφ(x,y)],
W(bx,by,a,θ)=g(x,y),ψbx,by,a,θ(x,y)=g(x,y)ψ¯(xbxa,xbya,rθ)dxdy,
ϕ(x,y)=arctan{imag[Wr(bx,by,ar,θ)]real[Wr(bx,by,ar,θ)]},
ψ(x,y)=0orΨ(0,0)=0,
ψ1(x,y)=a1ψ[a1rθ(xbx,yby)].
ψ2(x,y)=a2ψ[a1rθ(xbx,yby)].
W1(bx,by,a,θ)=g(x,y)ψ¯1(x,y)dxdy=g(x,y)a1ψ¯[a1rθ(xbx,yby)]dxdy=g(x,y)a1ψ¯[a1rθ(bxx,byy)]dxdy=g(x,y)*a1ψ¯[a1rθ(x,y)]=aIFT{G(ωx,ωy)Ψ¯[arθ(ωx,ωy)]},
W2(bx,by,a,θ)=IFT{G(ωx,ωy)Ψ¯[arθ(ωx,ωy)]}.
G(ωx,ωy)=Q0+Q+1(ωxω0x,ωyω0y)+Q1(ωx+ω0x,ωy+ω0y),
W1r(bx,by,ar,θ)=arIFT{G(ωx,ωy)Ψ¯[arrθ(ωx,ωy)]}=arIFT{[Q0+Q+1(ωxω0x,ωyω0y)+Q1(ωx+ω0x,ωy+ω0y)]Ψ¯[arrθ(ωx,ωy)]}=arIFT{Q+1(ωxω0x,ωyω0y)}=12CarBexp[jϕ(x,y)].
W2r(bx,by,ar,θ)=12CBexp[jϕ(x,y)],
Ψ(ωx,ωy)=exp{12ε2[(ωxk0)2+ωy2]},
|W1r(bx,by,ar,θ)|=12arB,
|W2r(bx,by,ar,θ)|=12B.
ψ3(x,y)=a2exp[(aa0)2]ψ[a1rθ(xbx,yby)].
W3(bx,by,a,θ)=exp[(aa0)2]IFT{G(ωx,ωy)Ψ¯[arθ(ωx,ωy)]}.
W3r(bx,by,ar,θ)=12CBexp[(ara0)2]exp[jϕ(x,y)].
|W3r(bx,by,ar,θ)|=12Bexp[(ara0)2].

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