Abstract

To increase the accuracy, speed, and robustness of 3D measurements in Fourier transform profilometry (FTP), this paper introduces a cost function according to the intrinsic features of the amplitude and frequency modulated (AF/M) signal and proposes two new algorithms to eliminate the background components of the fringe pattern based on the proposed cost function. Finally, the standard Fourier transform (FT) is used to calculate the phase of the pattern, which no longer contains background components. The two proposed methods are both data-driven and require no parameter adjustments in advance. Theoretical analysis and 80 experimental results show that the proposed cost function is valid. The results of more than 80 experiments with different types of fringe patterns, different carrier frequencies, and different noise variances with frequency overlap and sudden phase variation show that the proposed two methods are more accurate and robust than the 2D Gabor wavelet transform, the 2D Fan wavelet transform, and the 1D complex Morlet wavelet transform profilometry, and they are approximately 70 times faster than the 1D complex Morlet wavelet transform profilometry.

© 2012 Optical Society of America

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  1. S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
    [CrossRef]
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    [CrossRef]
  3. K. Qian, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  4. K. Qian, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
    [CrossRef]
  5. K. Qian, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
    [CrossRef]
  6. G. Busca and E. Zappa, “Sensitivity analysis applied to an improved Fourier-transform profilometry,” Opt. Lasers Eng. 49, 210–221 (2011).
    [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. J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46, 2670–2675 (2007).
    [CrossRef]
  9. H. Li and C. Yang, “Two-dimensional multiscale windowed Fourier transform based on two-dimensional wavelet transform for fringe pattern demodulation,” Opt. Laser Technol. 43, 72–81 (2011).
    [CrossRef]
  10. C. Quan, H. Niu, and C. J. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (2010).
    [CrossRef]
  11. S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
    [CrossRef]
  12. C. Yang, Q. Lu, J. Zhao, and H. Miao, “Window size selection in windowed Fourier transform for phase retrieval,” Opt. Lasers Eng. 48, 1096–1103 (2010).
    [CrossRef]
  13. S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284, 2797–2807 (2011).
    [CrossRef]
  14. 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]
  15. L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303 (2007).
    [CrossRef]
  16. 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]
  17. W. Chen, S. Li, Y. Cai, and Y. Zhao, “Analysis on fringe pattern demodulation by use of 2-D CWT,” Optik 122, 1739–1746 (2011).
    [CrossRef]
  18. A. Z. Abid, Fringe Pattern Analysis using Wavelet Transforms (General Engineering Research Institute (GERI), Liverpool John Moores University , 2008).
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    [CrossRef]
  20. A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120–6126 (2007).
    [CrossRef]
  21. 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]
  22. L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
    [CrossRef]
  23. W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
    [CrossRef]
  24. P. Hlubina, J. Lunek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
    [CrossRef]
  25. 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]
  26. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
    [CrossRef]
  27. S. Li, X. Su, W. Chen, and L. Xiang, “Eliminating the zero spectrum in Fourier transform profilometry using empirical mode decomposition,” J. Opt. Soc. Am. A 26, 1195–1201 (2009).
    [CrossRef]
  28. Q. Zhang, W. Chen, and Y. Tang, “Method of choosing the adaptive level of discrete wavelet decomposition to eliminate zero component,” Opt. Commun. 282, 778–785 (2009).
    [CrossRef]
  29. S. L. Marple, “Computing the discrete-time ‘analytic’signal via FFT,” IEEE Trans. Signal Process. 47, 2600–2603 (1997).
    [CrossRef]
  30. H. Liu, A. N. Cartwright, and C. Basaran, “Moire Interferogram Phase Extraction: A Ridge Detection Algorithm for Continuous Wavelet Transforms,” Appl. Opt. 43, 850–857 (2004).
    [CrossRef]
  31. K. Qian, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420–5428 (2008).
    [CrossRef]

2011 (4)

G. Busca and E. Zappa, “Sensitivity analysis applied to an improved Fourier-transform profilometry,” Opt. Lasers Eng. 49, 210–221 (2011).
[CrossRef]

H. Li and C. Yang, “Two-dimensional multiscale windowed Fourier transform based on two-dimensional wavelet transform for fringe pattern demodulation,” Opt. Laser Technol. 43, 72–81 (2011).
[CrossRef]

S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284, 2797–2807 (2011).
[CrossRef]

W. Chen, S. Li, Y. Cai, and Y. Zhao, “Analysis on fringe pattern demodulation by use of 2-D CWT,” Optik 122, 1739–1746 (2011).
[CrossRef]

2010 (6)

C. Yang, Q. Lu, J. Zhao, and H. Miao, “Window size selection in windowed Fourier transform for phase retrieval,” Opt. Lasers Eng. 48, 1096–1103 (2010).
[CrossRef]

C. Quan, H. Niu, and C. J. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (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. S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (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)

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

Q. Zhang, W. Chen, and Y. Tang, “Method of choosing the adaptive level of discrete wavelet decomposition to eliminate zero component,” Opt. Commun. 282, 778–785 (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]

2008 (2)

P. Hlubina, J. Lunek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[CrossRef]

K. Qian, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420–5428 (2008).
[CrossRef]

2007 (5)

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

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303 (2007).
[CrossRef]

K. Qian, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

K. Qian, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
[CrossRef]

J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46, 2670–2675 (2007).
[CrossRef]

2006 (3)

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (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]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

2005 (1)

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

2004 (3)

1997 (1)

S. L. Marple, “Computing the discrete-time ‘analytic’signal via FFT,” IEEE Trans. Signal Process. 47, 2600–2603 (1997).
[CrossRef]

1983 (1)

Abid, A. Z.

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]

Basaran, C.

Burton, D.

S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284, 2797–2807 (2011).
[CrossRef]

Burton, D. R.

Busca, G.

G. Busca and E. Zappa, “Sensitivity analysis applied to an improved Fourier-transform profilometry,” Opt. Lasers Eng. 49, 210–221 (2011).
[CrossRef]

Cai, Y.

W. Chen, S. Li, Y. Cai, and Y. Zhao, “Analysis on fringe pattern demodulation by use of 2-D CWT,” Optik 122, 1739–1746 (2011).
[CrossRef]

Cao, Y.

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

Cartwright, A. N.

Chen, L.-C.

L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
[CrossRef]

Chen, W.

W. Chen, S. Li, Y. Cai, and Y. Zhao, “Analysis on fringe pattern demodulation by use of 2-D CWT,” Optik 122, 1739–1746 (2011).
[CrossRef]

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

Q. Zhang, W. Chen, and Y. Tang, “Method of choosing the adaptive level of discrete wavelet decomposition to eliminate zero component,” Opt. Commun. 282, 778–785 (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]

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

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
[CrossRef]

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

Chlebus, R.

P. Hlubina, J. Lunek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[CrossRef]

Ciprian, D.

P. Hlubina, J. Lunek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[CrossRef]

Fernandez, S.

S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284, 2797–2807 (2011).
[CrossRef]

Gao, W.

Gdeisat, M. A.

S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284, 2797–2807 (2011).
[CrossRef]

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

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (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]

Gorthi, S. S.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Hlubina, P.

P. Hlubina, J. Lunek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[CrossRef]

Ho, H.-W.

L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
[CrossRef]

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]

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]

Lalor, M. J.

Li, H.

H. Li and C. Yang, “Two-dimensional multiscale windowed Fourier transform based on two-dimensional wavelet transform for fringe pattern demodulation,” Opt. Laser Technol. 43, 72–81 (2011).
[CrossRef]

Li, S.

Lilley, F.

Liu, H.

Lu, Q.

C. Yang, Q. Lu, J. Zhao, and H. Miao, “Window size selection in windowed Fourier transform for phase retrieval,” Opt. Lasers Eng. 48, 1096–1103 (2010).
[CrossRef]

Lunek, J.

P. Hlubina, J. Lunek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[CrossRef]

Marple, S. L.

S. L. Marple, “Computing the discrete-time ‘analytic’signal via FFT,” IEEE Trans. Signal Process. 47, 2600–2603 (1997).
[CrossRef]

Miao, H.

C. Yang, Q. Lu, J. Zhao, and H. Miao, “Window size selection in windowed Fourier transform for phase retrieval,” Opt. Lasers Eng. 48, 1096–1103 (2010).
[CrossRef]

Mutoh, K.

Nguyen, X.-L.

L.-C. Chen, H.-W. Ho, and X.-L. Nguyen, “Fourier transform profilometry (FTP) using an innovative band-pass filter for accurate 3-D surface reconstruction,” Opt. Lasers Eng. 48, 182–190 (2010).
[CrossRef]

Niu, H.

C. Quan, H. Niu, and C. J. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (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]

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]

Qian, K.

K. Qian, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420–5428 (2008).
[CrossRef]

K. Qian, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

K. Qian, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
[CrossRef]

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

C. Quan, H. Niu, and C. J. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (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]

Rastogi, P.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Salvi, J.

S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284, 2797–2807 (2011).
[CrossRef]

Su, X.

S. Li, X. Su, W. Chen, and L. Xiang, “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]

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
[CrossRef]

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

Takeda, M.

Tang, Y.

Q. Zhang, W. Chen, and Y. Tang, “Method of choosing the adaptive level of discrete wavelet decomposition to eliminate zero component,” Opt. Commun. 282, 778–785 (2009).
[CrossRef]

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]

C. Quan, H. Niu, and C. J. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (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]

Wang, H.

Watkins, L. R.

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303 (2007).
[CrossRef]

Weng, J.

Xiang, L.

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

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

Yang, C.

H. Li and C. Yang, “Two-dimensional multiscale windowed Fourier transform based on two-dimensional wavelet transform for fringe pattern demodulation,” Opt. Laser Technol. 43, 72–81 (2011).
[CrossRef]

C. Yang, Q. Lu, J. Zhao, and H. Miao, “Window size selection in windowed Fourier transform for phase retrieval,” Opt. Lasers Eng. 48, 1096–1103 (2010).
[CrossRef]

Zappa, E.

G. Busca and E. Zappa, “Sensitivity analysis applied to an improved Fourier-transform profilometry,” Opt. Lasers Eng. 49, 210–221 (2011).
[CrossRef]

Zeng, H.

Zhang, Q.

Q. Zhang, W. Chen, and Y. Tang, “Method of choosing the adaptive level of discrete wavelet decomposition to eliminate zero component,” Opt. Commun. 282, 778–785 (2009).
[CrossRef]

W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005).
[CrossRef]

Zhao, J.

C. Yang, Q. Lu, J. Zhao, and H. Miao, “Window size selection in windowed Fourier transform for phase retrieval,” Opt. Lasers Eng. 48, 1096–1103 (2010).
[CrossRef]

Zhao, Y.

W. Chen, S. Li, Y. Cai, and Y. Zhao, “Analysis on fringe pattern demodulation by use of 2-D CWT,” Optik 122, 1739–1746 (2011).
[CrossRef]

Zheng, S.

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
[CrossRef]

Zhong, J.

Appl. Opt. (9)

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

H. Liu, A. N. Cartwright, and C. Basaran, “Moire Interferogram Phase Extraction: A Ridge Detection Algorithm for Continuous Wavelet Transforms,” Appl. Opt. 43, 850–857 (2004).
[CrossRef]

K. Qian, “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]

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]

J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46, 2670–2675 (2007).
[CrossRef]

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

K. Qian, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5420–5428 (2008).
[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]

IEEE Trans. Signal Process. (1)

S. L. Marple, “Computing the discrete-time ‘analytic’signal via FFT,” IEEE Trans. Signal Process. 47, 2600–2603 (1997).
[CrossRef]

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

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Opt. Eng. (1)

S. Zheng, W. Chen, and X. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
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H. Li and C. Yang, “Two-dimensional multiscale windowed Fourier transform based on two-dimensional wavelet transform for fringe pattern demodulation,” Opt. Laser Technol. 43, 72–81 (2011).
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C. Quan, H. Niu, and C. J. Tay, “An improved windowed Fourier transform for fringe demodulation,” Opt. Laser Technol. 42, 126–131 (2010).
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C. Yang, Q. Lu, J. Zhao, and H. Miao, “Window size selection in windowed Fourier transform for phase retrieval,” Opt. Lasers Eng. 48, 1096–1103 (2010).
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K. Qian, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
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K. Qian, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).
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C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
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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).
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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).
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Optik (1)

W. Chen, S. Li, Y. Cai, and Y. Zhao, “Analysis on fringe pattern demodulation by use of 2-D CWT,” Optik 122, 1739–1746 (2011).
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Other (1)

A. Z. Abid, Fringe Pattern Analysis using Wavelet Transforms (General Engineering Research Institute (GERI), Liverpool John Moores University , 2008).

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Figures (6)

Fig. 1.
Fig. 1.

The experimentally recorded fringe pattern of dimple, and the carrier frequency is 1/22.5 in this study.

Fig. 2.
Fig. 2.

The experimentally recorded fringe pattern of dummy, and the carrier frequency is 1/17 in this study.

Fig. 3.
Fig. 3.

The experimentally recorded fringe pattern of Cylinder, and the reference fringe pattern.

Fig. 4.
Fig. 4.

The unwrapped phase for “Dimple” pattern using different methods.

Fig. 5.
Fig. 5.

The unwrapped phase for “Dummy” pattern using different methods.

Fig. 6.
Fig. 6.

The unwrapped phase for “Cylinder” pattern using different methods.

Tables (8)

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Table 1. RMSE Comparisons for Different Methods in Simulated Fringe Pattern (1) with Carrier Frequency of 1/8

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Table 2. RMSE Comparisons for Different Methods in Simulated Fringe Pattern (1) with Carrier Frequency of 1/16

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Table 3. RMSE Comparisons for Different Methods in Simulated Fringe Pattern (1) with Carrier Frequency of 1/32

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Table 4. RMSE Comparisons for Different Methods in Simulated Fringe Pattern (2) with Carrier Frequency of 1/8

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Table 5. RMSE Comparisons for Different Methods in Simulated Fringe Pattern (2) with Carrier Frequency of 1/16

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Table 6. Time-Consuming Comparison Among Our Methods and the Others

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Table 7. Comparison of Results between the Level (RMSE) Obtained Using the Proposed First Method and the Optimal Level Jopt (eopt)

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Table 8. Comparison of Results between the Scale (RMSE) Obtained Using the Proposed Second Method and the Optimal Scale Jopt (eopt)

Equations (14)

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

I(x,y)=a(x,y)+b(x,y)cos(2πfox+φ(x,y))+r(x,y),
I(x,k)=a(x,k)+b(x,k)cos(2πfox+φ(x,k))+r(x,k).
I(x)=a(x)+b(x)cos(2πfox+φ(x))+r(x).
Iafm(x)=E(I(x))=A(x)cos(2πfx+ϕ(x)),
Ifm(x)=N(Iafm(x))=cos(2πfx+ϕ(x)),
Ifm(x)=(2πf+ϕ(x))sin(2πfx+ϕ(x)).
z1(x)=IF(Harf(I^fm(x))),
z2(x)=IF(Harf(jwI^fm(x))),
sin2(arctan(Im(z1(x))Re(z1(x))))+sin2(arctan(Im(z2(x))Re(z2(x))))1,x,
sin2(arctan(Im(z1(x1))Re(z1(x1))))+sin2(arctan(Im(z2(x1))Re(z2(x1))))1.
C(I)=i=1N(sin2(arctan(Im(z1(x1))Re(z1(x1))))+sin2(arctan(Im(z2(x1))Re(z2(x1))))1)2,
I(x,y)=0.25ϕ(x,y)+cos(2πf0x+ϕ(x,y))+r(x,y),
ϕ(x,y)=3(1x)2exp(x2(y+1)2)10(x5x3y5)exp(x2y2)13exp((x+1)2y2).
I(x,y)=0.25ϕ(x,y)+cos(2πf0x+ϕ(x,y))+r(x,y),

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