Abstract

A multiscale windowed Fourier transform for phase extraction of fringe patterns is presented. A local stationary length of signal is used to control the window width of a windowed Fourier transform automatically, which is measured by an instantaneous frequency gradient. The instantaneous frequency of the fringe pattern is obtained by detecting the ridge of the wavelet transform. The numerical simulation and experiment have proved the validity of this method. The combination of the windowed Fourier transform and the wavelet transform makes the extracted phase more precise than other methods.

© 2007 Optical Society of America

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    [CrossRef]
  4. L. R. Watkins, S. M. Tan, and T. H. Barnes, "Determination of interferometer phase distributions by use of wavelets," Opt. Lett. 24, 905-907 (1997).
    [CrossRef]
  5. 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] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  11. X. Su and W. Chen, "Reliability-guided phase unwrapping algorithm: a review," Opt. Lasers Eng. 42, 245-261 (2004).
    [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]

1997 (2)

A. René, L. Wen, and B. Torrésani, "Characterization of signals by the ridges of their wavelet transforms," IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, "Determination of interferometer phase distributions by use of wavelets," Opt. Lett. 24, 905-907 (1997).
[CrossRef]

1996 (1)

R. G. Stockwell, L. Mansinha, and R. P. Lowe, "Localization of the complex spectrum: the S transform," IEEE Trans. Signal Process. 44, 998-1001 (1996).
[CrossRef]

1992 (1)

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[CrossRef]

1983 (1)

Barnes, T. H.

Chen, W.

X. Su and W. Chen, "Reliability-guided phase unwrapping algorithm: a review," Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

Delprat, N.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[CrossRef]

Escudié, B.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[CrossRef]

Fu, Y.

Guillemain, P.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[CrossRef]

Huang, Y.

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]

Kronland-Martinet, R.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[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]

Lowe, R. P.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, "Localization of the complex spectrum: the S transform," IEEE Trans. Signal Process. 44, 998-1001 (1996).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).

Mansinha, L.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, "Localization of the complex spectrum: the S transform," IEEE Trans. Signal Process. 44, 998-1001 (1996).
[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]

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]

Qian, K.

Quan, C.

René, A.

A. René, L. Wen, and B. Torrésani, "Characterization of signals by the ridges of their wavelet transforms," IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

Stockwell, R. G.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, "Localization of the complex spectrum: the S transform," IEEE Trans. Signal Process. 44, 998-1001 (1996).
[CrossRef]

Su, X.

X. Su and W. Chen, "Reliability-guided phase unwrapping algorithm: a review," Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

Takeda, M.

Tan, S. M.

Tay, C. J.

Tchamitchian, P.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[CrossRef]

Torrésani, B.

A. René, L. Wen, and B. Torrésani, "Characterization of signals by the ridges of their wavelet transforms," IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[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.

Wen, L.

A. René, L. Wen, and B. Torrésani, "Characterization of signals by the ridges of their wavelet transforms," IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

Weng, J.

Zhong, J.

Appl. Opt. (3)

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)

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani, "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies," IEEE Trans. Inf. Theory 38, 644-664 (1992).
[CrossRef]

IEEE Trans. Signal Process. (2)

A. René, L. Wen, and B. Torrésani, "Characterization of signals by the ridges of their wavelet transforms," IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

R. G. Stockwell, L. Mansinha, and R. P. Lowe, "Localization of the complex spectrum: the S transform," IEEE Trans. Signal Process. 44, 998-1001 (1996).
[CrossRef]

Opt. Lasers Eng. (1)

X. Su and W. Chen, "Reliability-guided phase unwrapping algorithm: a review," Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

Opt. Lett. (2)

Other (1)

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).

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Figures (11)

Fig. 1
Fig. 1

Conventional windowed Fourier transform. (a) Invariable Gaussian window, (b) Heisenberg box.

Fig. 2
Fig. 2

Multiscale windowed Fourier transform. (a) Variable Gaussian window, (b) Heisenberg box.

Fig. 3
Fig. 3

Nonstationary signal with a local stationary signal in local AB.

Fig. 4
Fig. 4

Simulated fringe pattern. (a) Base grating pattern, (b) deformed grating pattern.

Fig. 5
Fig. 5

Instantaneous frequency distributions obtained by wavelet transform.

Fig. 6
Fig. 6

Modulated phase distributions extracted by MWFT.

Fig. 7
Fig. 7

Phase distributions analyzed by WFT with window widths of 4, 42, and 100 pixels.

Fig. 8
Fig. 8

Comparison of the phase error distributions obtained by MWFT and WFT with a window width of 42 pixels.

Fig. 9
Fig. 9

Comparison of the phase error distributions obtained by MWFT and WT.

Fig. 10
Fig. 10

Real deformed grating pattern.

Fig. 11
Fig. 11

Phase distributions obtained by MWFT.

Equations (19)

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W F δ b ( ω ) = [ f ( x ) exp ( j ω x ) ] G δ b ( x ) d x ,
G δ b ( x ) = | δ | 1 g ( x b δ ) ( δ > 0 ) ,
g ( x ) = 1 2 π exp ( x 2 2 ) .
Δ x = 2 δ 2 ln 2 .
Δ x Δ f 1 / 4 π ,
W ( a , b ) = f ( x ) M a , b * ( x ) d x ,
M a , b ( x ) = 1 a M ( x b a ) .
a = 1 f i n s .
| x i x i + m i T f i n s ( x ) d x | σ ( i = 1 , 2 , 3 , … )
L i = m i T ,
F ( ω ) = f ( x ) exp ( j ω x ) d x .
G δ b ( x ) d x = | δ | 1 g ( x b δ ) d x = 1
W F δ b ( ω ) d b = f ( x ) exp ( j ω x ) G δ b ( x ) d x d b = f ( x ) exp ( j ω x ) [ G δ b ( x ) d b ] d x = F ( ω ) .
I ( x ) = r ( x ) n = A n exp { j [ n ω 0 x + n ϕ ( x ) ] } , = r ( x ) n = A n exp { j [ n φ ( x ) ] } .
F ( ω 0 ) = W F δ b ( ω 0 ) d b .
F ( ω 0 ) exp ( j ω x ) d ω = r ( x ) A 1 ( x ) exp [ j φ ( x ) ] = I 1 ( x ) .
φ ( x ) = arctan { i m a g [ I 1 ( x ) ] r e a l [ I 1 ( x ) ] } .
Δ ϕ ( x ) = 140 π 16 ( 1 sin 2 π x 512 ) .
M ( x ) = 1 π 4 2 π γ exp [ ( 2 π / γ ) 2 2 + j 2 π x ] ,

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