Abstract

Wavelet ridge techniques utilizing daughter wavelets under two different kinds of definitions in the optical fringe pattern analysis are theoretically clarified. The clarification reveals that the phase of the optical fringe pattern is equal to that of its wavelet transform coefficients on the ridge using both of the two wavelet definitions. The differences between the two definitions in the performance of wavelet transform algorithms are verified in theory. The strict relations between the instantaneous frequency of the fringe pattern and the scale parameter at the wavelet ridge position are also theoretically clarified for the phase gradient method. A simple method for selecting the scale vector is introduced. Computer simulations and experiments reveal the correctness of the clarification and the validity of the proposed method.

© 2010 Optical Society of America

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    [Crossref] [PubMed]
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2010 (3)

X. Su and Q. Zhang, “Dynamic 3-D measurement method: a review,” Opt. Lasers Eng. 48, 191–204 (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]

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]

2009 (4)

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]

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] [PubMed]

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]

2007 (3)

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

W. Chen, J. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt. 54, 2747–2762 (2007).
[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]

2006 (1)

2005 (2)

2004 (2)

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

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]

2002 (1)

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[Crossref]

2001 (1)

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

1999 (1)

1997 (2)

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. (Bellingham) 36, 1947–1951 (1997).
[Crossref]

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]

1995 (1)

1992 (3)

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[Crossref]

S. Mallat and W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).
[Crossref]

B. Telfer and H. H. Szu, “New wavelet transform normalization to remove frequency bias,” Opt. Eng. (Bellingham) 31, 1830–1834 (1992).
[Crossref]

1983 (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]

Afifi, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[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]

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. (Bellingham) 36, 1947–1951 (1997).
[Crossref]

Bian, X.

W. Chen, J. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt. 54, 2747–2762 (2007).
[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] [PubMed]

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, “Eliminating the zero spectrum in Fourier transform profilometry using empirical mode decomposition,” J. Opt. Soc. Am. A 26, 1195–1201 (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] [PubMed]

W. Chen, J. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt. 54, 2747–2762 (2007).
[Crossref]

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

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[Crossref]

Diao, H.

Fassi-Fihri, A.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[Crossref]

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] [PubMed]

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]

S. Mallat and W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).
[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, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[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]

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] [PubMed]

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. (Bellingham) 36, 1947–1951 (1997).
[Crossref]

Li, S.

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]

Mallat, S.

S. Mallat and W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).
[Crossref]

Marjane, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[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]

Mutoh, K.

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]

Nassim, K.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[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. (Bellingham) 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]

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]

Rachafi, S.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[Crossref]

Sidki, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[Crossref]

Su, X.

X. Su and Q. Zhang, “Dynamic 3-D measurement method: a review,” Opt. Lasers Eng. 48, 191–204 (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] [PubMed]

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]

W. Chen, J. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt. 54, 2747–2762 (2007).
[Crossref]

Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13, 3110–3116 (2005).
[Crossref] [PubMed]

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

Sun, J.

W. Chen, J. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt. 54, 2747–2762 (2007).
[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]

Szu, H. H.

B. Telfer and H. H. Szu, “New wavelet transform normalization to remove frequency bias,” Opt. Eng. (Bellingham) 31, 1830–1834 (1992).
[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]

Telfer, B.

B. Telfer and H. H. Szu, “New wavelet transform normalization to remove frequency bias,” Opt. Eng. (Bellingham) 31, 1830–1834 (1992).
[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]

Watkins, L. R.

Weng, J.

Zhang, Q.

X. Su and Q. Zhang, “Dynamic 3-D measurement method: a review,” Opt. Lasers Eng. 48, 191–204 (2010).
[Crossref]

Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13, 3110–3116 (2005).
[Crossref] [PubMed]

Zhong, J.

Appl. Opt. (5)

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. Inf. Theory (1)

S. Mallat and W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).
[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. Mod. Opt. (1)

W. Chen, J. Sun, X. Su, and X. Bian, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt. 54, 2747–2762 (2007).
[Crossref]

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

Opt. Commun. (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]

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[Crossref]

Opt. Eng. (Bellingham) (3)

B. Telfer and H. H. Szu, “New wavelet transform normalization to remove frequency bias,” Opt. Eng. (Bellingham) 31, 1830–1834 (1992).
[Crossref]

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. (Bellingham) 43, 895–899 (2004).
[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. (Bellingham) 36, 1947–1951 (1997).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (7)

X. Su and Q. Zhang, “Dynamic 3-D measurement method: a review,” Opt. Lasers Eng. 48, 191–204 (2010).
[Crossref]

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

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

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]

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]

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

Opt. Lett. (2)

Other (2)

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[Crossref]

http://petercorke.com/Machine_Vision_Toolbox.html.

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Figures (6)

Fig. 1
Fig. 1

Simulated fringe pattern.

Fig. 2
Fig. 2

Modulus analysis: (a) Modulus of the WT coefficient employing the first definition algorithm. (b) Modulus of the WT coefficient employing the second definition algorithm. (c) Comparison of the modulus at the ridge position.

Fig. 3
Fig. 3

Phase extraction method: (a) Extracted modulated phase. (b) Error of the first definition algorithm. (c) Error of the second definition algorithm. (d) Error between the two algorithms.

Fig. 4
Fig. 4

Phase gradient method: (a) Extracted phase gradient. (b) Retrieved phase by using the phase gradient approach. (c) Error of the first definition algorithm. (d) Error of the second definition algorithm.

Fig. 5
Fig. 5

Phase extraction of fringe pattern with noise: (a) Phase extracted by using the phase extraction method. (b) Error of the phase extraction method. (c) Phase extracted by using the phase gradient method. (d) Error of the phase gradient method. (e) Phase extracted by using FT method. (f) Error of FT method.  

Fig. 6
Fig. 6

Experiment: (a) A real deformed fringe pattern. (b) Phase extracted by using the first definition algorithm (phase extraction method). (c) Phase extracted by using the second definition algorithm (phase extraction method). (d) Phase extracted by using the first definition algorithm (phase gradient method). (e) Phase extracted by using the second definition algorithm (phase gradient method). (f) Phase extracted by using FT method.

Equations (26)

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ψ a , b 1 ( x ) = 1 a ψ ( x b a ) ,
ψ a , b 2 ( x ) = 1 a ψ ( x b a ) .
W ( a , b ) = I ( x ) , ψ a , b ( x ) = I ( x ) ψ a , b ( x ) d x ,
ψ ( x ) = 1 π F b exp ( j 2 π F c x ) exp ( 1 F b x 2 ) ,
I ( x ) = A ( x ) + B ( x ) cos   φ ( x ) = A + 1 2 B   exp [ j φ ( x ) ] + 1 2 B   exp [ j φ ( x ) ] ,
φ ( x ) = φ ( b ) + φ ( b ) ( x b ) + φ ( b ) 2 ! ( x b ) 2 + .
W ( a , b ) = I ( x ) ψ a , b ( x ) d x = A ψ a , b ( x ) d x + 1 2 B   exp [ j φ ( x ) ] ψ a , b ( x ) d x + 1 2 B   exp [ j φ ( x ) ] ψ a , b ( x ) d x = W 1 + W 2 + W 3 .
W 1 = A ψ a , b ( x ) d x ,
W 2 = 1 2 a B   exp { { π F c F b [ φ ( b ) a 2 π F c 1 ] } 2 } exp [ j φ ( b ) ] ,
W 3 = 1 2 B a   exp { { π F c F b [ φ ( b ) a 2 π F c + 1 ] } 2 } exp [ j φ ( b ) ] .
W 1 = 1 2 a B   exp { { π F c F b [ φ ( b ) a 2 π F c 1 ] } 2 } exp [ j φ ( b ) ] .
| W 1 ( a , b ) | = | 1 2 a B   exp { { π F c F b [ φ ( b ) a 2 π F c 1 ] } 2 } | .
d | W 1 ( a , b ) | d a = B 1 a exp { { π F c F b [ φ ( b ) a 2 π F c 1 ] } 2 } { 1 4 { π F c F b [ φ ( b ) a 2 π F c 1 ] } φ ( b ) a 2 } ,
a r 1 ( b ) = π F b F c + ( π F b F c ) 2 + F b F b φ ( b ) = π F b F c + ( π F b F c ) 2 + F b 2 π F b 1 f b ,
W 1 ( a r , b ) = 1 2 B a r 1   exp { [ π F c F b ( π F b F c + ( π F b F c ) 2 + F b 2 π F b F c 1 ) ] 2 } exp [ j φ ( b ) ] = 1 2 B f b [ π F b F c + ( π F b F c ) 2 + F b 2 π F b ] 1 / 2 exp { [ π F c F b ( π F b F c + ( π F b F c ) 2 + F b 2 π F b F c 1 ) ] 2 } exp [ j φ ( b ) ] .
W 2 ( a r 2 , b ) = 1 2 B   exp [ j φ ( b ) ] ,
a r 2 = 2 π F c φ ( b ) = F c f b .
φ ( b ) = arctan { Im [ W ( a r , b ) ] Re [ W ( a r , b ) ] } .
φ ( b ) = π F b F c + ( π F b F c ) 2 + F b a r 1 ( b ) F b ,
φ ( b ) = 2 π F c a r 2 .
a min = k 0 / f b _ max = k 0 n min ,
a max = k 0 / f b _ min = k 0 n max ,
W ( a , b ) = f ( x ) ψ a , b ( x ) d x = { IFT [ a F ( ω ) Ψ 1 ( a ω ) ] , IFT [ F ( ω ) Ψ 2 ( a ω ) ] . }
Δ a = k 2 i     ( k = 1 , 2 , 3 , , i = 0 , 1 , 2 , 3 , ) .
ϕ ( x ) = 2 π [ x 13 + ( 8 / 512 ) x x 16 ] .
g ( x , y ) = 1 + cos [ 2 π f 0 x + ϕ ( x , y ) ] .

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