Abstract

The phase demodulation method of adaptive windowed Fourier transform (AWFT) is proposed based on Hilbert-Huang transform (HHT). HHT is analyzed and performed on fringe pattern to obtain instantaneous frequencies firstly. These instantaneous frequencies are further analyzed based on the condition of AWFT to locate local stationary areas where the fundamental spectrum will not be interfered by high-order spectrum. Within each local stationary area, the fundamental spectrum can be extracted accurately and adaptively by using AWFT with the background, which has been determined previously with the presented criterion during HHT, being eliminated to remove the zero-spectrum. This method is adaptive and unconstrained by any precondition for the measured phase. Experiments demonstrate its robustness and effectiveness for measuring the object with discontinuities or complex surface.

© 2012 OSA

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2012

M. Zhong, W. Chen, and M. Jiang, “Application of S-transform profilometry in eliminating nonlinearity in fringe pattern,” Appl. Opt.51(5), 577–587 (2012).
[CrossRef] [PubMed]

Z. Wang, J. Ma, and M. Vo, “Recent progress in two-dimensional continuous wavelet transform technique for fringe pattern analysis,” Opt. Lasers Eng.50(8), 1052–1058 (2012).
[CrossRef]

2011

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(12), 2797–2807 (2011).
[CrossRef]

W. Gao and Q. Kemao, “Statistical analysis for windowed Fourier ridge algorithm in fringe pattern analysis,” Appl. Opt.51(3), 328–337 (2011).
[CrossRef] [PubMed]

2010

2009

2008

G. Rilling and P. Flandrin, “One or two frequencies? The empirical mode decomposition answers,” IEEE Trans. Signal Process.56(1), 85–95 (2008).
[CrossRef]

2007

2006

S. Zheng, W. Chen, and X. Su, “Adaptive Windowed Fourier transform in 3-D shape measurement,” Opt. Eng.45(6), 063601 (2006).
[CrossRef]

2005

2004

1998

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

1983

1982

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(12), 2797–2807 (2011).
[CrossRef]

Chen, W.

Coskun, E.

Da, F.

S. Gai and F. Da, “A novel phase-shifting method based on strip marker,” Opt. Lasers Eng.48(2), 205–211 (2010).
[CrossRef]

S. Gai and F. Da, “Fringe image analysis based on the amplitude modulation method,” Opt. Express18(10), 10704–10719 (2010).
[CrossRef] [PubMed]

Equis, S.

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(12), 2797–2807 (2011).
[CrossRef]

Flandrin, P.

G. Rilling and P. Flandrin, “One or two frequencies? The empirical mode decomposition answers,” IEEE Trans. Signal Process.56(1), 85–95 (2008).
[CrossRef]

Gai, S.

S. Gai and F. Da, “Fringe image analysis based on the amplitude modulation method,” Opt. Express18(10), 10704–10719 (2010).
[CrossRef] [PubMed]

S. Gai and F. Da, “A novel phase-shifting method based on strip marker,” Opt. Lasers Eng.48(2), 205–211 (2010).
[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(12), 2797–2807 (2011).
[CrossRef]

Göktas, H.

Huang, N. E.

Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise assisted data analysis method,” Adv. Adapt. Data Anal.1(1), 1–41 (2009).
[CrossRef]

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Ina, H.

Jacquot, P.

Jiang, M.

Jiang, T.

Kemao, Q.

Kobayashi, S.

Kocahan, Ö.

Li, S.

Li, X.

Liu, H. H.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Long, S. R.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Ma, J.

Z. Wang, J. Ma, and M. Vo, “Recent progress in two-dimensional continuous wavelet transform technique for fringe pattern analysis,” Opt. Lasers Eng.50(8), 1052–1058 (2012).
[CrossRef]

Mutoh, K.

Özder, S.

Rilling, G.

G. Rilling and P. Flandrin, “One or two frequencies? The empirical mode decomposition answers,” IEEE Trans. Signal Process.56(1), 85–95 (2008).
[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(12), 2797–2807 (2011).
[CrossRef]

Shen, Z.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Shih, H. H.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Su, X.

Takeda, M.

Tung, C. C.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Vo, M.

Z. Wang, J. Ma, and M. Vo, “Recent progress in two-dimensional continuous wavelet transform technique for fringe pattern analysis,” Opt. Lasers Eng.50(8), 1052–1058 (2012).
[CrossRef]

Wang, Z.

Z. Wang, J. Ma, and M. Vo, “Recent progress in two-dimensional continuous wavelet transform technique for fringe pattern analysis,” Opt. Lasers Eng.50(8), 1052–1058 (2012).
[CrossRef]

Weng, J.

Wu, M. C.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Wu, Z.

Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise assisted data analysis method,” Adv. Adapt. Data Anal.1(1), 1–41 (2009).
[CrossRef]

Xiang, L.

Yau, S. T.

Yen, N. C.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Zeng, H.

Zhang, S.

Zhao, H.

Zheng, Q. N.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Zheng, S.

S. Zheng, W. Chen, and X. Su, “Adaptive Windowed Fourier transform in 3-D shape measurement,” Opt. Eng.45(6), 063601 (2006).
[CrossRef]

Zhong, J.

Zhong, M.

Zhou, X.

Adv. Adapt. Data Anal.

Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise assisted data analysis method,” Adv. Adapt. Data Anal.1(1), 1–41 (2009).
[CrossRef]

Appl. Opt.

IEEE Trans. Signal Process.

G. Rilling and P. Flandrin, “One or two frequencies? The empirical mode decomposition answers,” IEEE Trans. Signal Process.56(1), 85–95 (2008).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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(12), 2797–2807 (2011).
[CrossRef]

Opt. Eng.

S. Zheng, W. Chen, and X. Su, “Adaptive Windowed Fourier transform in 3-D shape measurement,” Opt. Eng.45(6), 063601 (2006).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

Z. Wang, J. Ma, and M. Vo, “Recent progress in two-dimensional continuous wavelet transform technique for fringe pattern analysis,” Opt. Lasers Eng.50(8), 1052–1058 (2012).
[CrossRef]

S. Gai and F. Da, “A novel phase-shifting method based on strip marker,” Opt. Lasers Eng.48(2), 205–211 (2010).
[CrossRef]

Opt. Lett.

Proc. R. Soc. Lond. A

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

Other

G. Rilling, P. Flandrin, and P. Goncalves, “On empirical mode decomposition and its algorithms,” in IEEE-EURASIP Workshop on Nonlinear signal and Image Processing, NSTP-03, GRADO (I) (2003).

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Figures (17)

Fig. 1
Fig. 1

Condition for extracting the fundamental spectrum completely.

Fig. 2
Fig. 2

(a) Simulated signal S(x). (b) The IMFs m1(x) ~m7(x) and the residual r(x) after EMD. (c) Instantaneous frequencies Insf1(x) ~Insf7(x) for each IMF. (d) Marginal spectra h1(f) ~h7(f) for each IMF.

Fig. 3
Fig. 3

(a) The minima in Insf1(x). (b) The maxima in Insf2(x).

Fig. 4
Fig. 4

Schematic diagram for Step3~Step5.

Fig. 5
Fig. 5

The located local stationary areas for S(x) in Fig. 2(a) and any Gaussian window centered at different pixel.

Fig. 6
Fig. 6

(a)The result got by eliminating the background by the presented method. (b) The standard value of simulation without background. (c) The error of the presented method compared with the standard value.

Fig. 7
Fig. 7

The flow chart of the proposed method and the method of locating local stationary areas.

Fig. 8
Fig. 8

Simulated fringe pattern.

Fig. 9
Fig. 9

The located local stationary areas of S(x) got by: (a) the RWT method; (b) our method.

Fig. 10
Fig. 10

The respective maps of scale factors got by: (a) the RWT method; (b) our method.

Fig. 11
Fig. 11

The wrapped phase of the local area got by: (a) the FT method, (b) the RWT method, (c) the RWT-MWFT method, (d) our method.

Fig. 12
Fig. 12

The comparison between the true phase and the retrieved phase for S(x) got by the FT method, the RWT method, the RWT-MWFT method and our method.

Fig. 13
Fig. 13

Errors for phase demodulation of S(x) got by: (a) the FT method and our method; (b) the RWT method and our method; (c) the RWT-MWFT method and our method.

Fig. 14
Fig. 14

Comparison of errors got by FT, RWT, RWT-MWFT and our method: (a)~(d): simulated fringe pattern with different level of deformation and noise; (a1)~(a4): errors for (a) with four methods respectively; (b1)~(b4): errors for (b) respectively; (c1)~(c4): errors for (c) respectively; (d1)~(d4): errors for (d) respectively.

Fig. 15
Fig. 15

(a) The measured object. (b) The obtained fringe pattern.

Fig. 16
Fig. 16

The retrieved phase for the 380th line of fringe pattern got by using five methods, respectively.

Fig. 17
Fig. 17

Restored 3D phase map got by: (a) FT; (b) RWT; (c) RWT-MWFT; (d) our method; (e) phase-shift method.

Tables (3)

Tables Icon

Table 1 Corresponding γ for each IMF

Tables Icon

Table 2 The mean values of Insf1(x) ~Insf7(x) in Fig. 2(c)

Tables Icon

Table 3 Error statistics for different methods

Equations (6)

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

I( x )=A( x )+B( x )cos[2π f 0 x+f(x)]+η(x),
f 0 < ( f 1 ) max < ( f m ) min , (m=2,3,...)
v(x)= n=1 N m n (x)+r(x) , (1nN)
v(x)= n=1 N λ n (x)exp{j2π [Ins f n (x)]dx } .
h n (f)= H(f,x)dx = | λ n (x)exp{j2π [Ins f n (x)]dx }|dx ,
ϕ(x,y) =α{3(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 ]},

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