Abstract

Based on empirical mode decomposition (EMD), the background removal and de-noising procedures of the data taken by polarization interference imaging interferometer (PIIS) are implemented. Through numerical simulation, it is discovered that the data processing methods are effective. The assumption that the noise mostly exists in the first intrinsic mode function is verified, and the parameters in the EMD thresholding de-noising methods is determined. In comparison, the wavelet and windowed Fourier transform based thresholding de-noising methods are introduced. The de-noised results are evaluated by the SNR, spectral resolution and peak value of the de-noised spectrums. All the methods are used to suppress the effect from the Gaussian and Poisson noise. The de-noising efficiency is higher for the spectrum contaminated by Gaussian noise. The interferogram obtained by the PIIS is processed by the proposed methods. Both the interferogram without background and noise free spectrum are obtained effectively. The adaptive and robust EMD based methods are effective to the background removal and de-noising in PIIS.

© 2013 OSA

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Errata

Wenyi Ren, Chunmin Zhang, Tingkui Mu, Lili Fu, and Chenling Jia, "Empirical mode decomposition based background removal and de-noising in polarization interference imaging spectrometer: erratum," Opt. Express 21, 10207-10207 (2013)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-21-8-10207

References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012

2011

A. Moghtaderi, P. Borgnat, and P. Flandrin, “Trend filtering: empirical mode decompositions Versus l1 and Hodrick-Prescott,” Adv. Adapt. Data Anal.3, 41–61 (2011).
[CrossRef]

2010

2008

M. Blanco-Velasco, B. Weng, and K. E. Barner, “ECG signal de-noising and baseline wander correction based on the empirical mode decomposition,” Comput. Biol. Med.38(1), 1–13 (2008).
[CrossRef] [PubMed]

Q. Gao, C. Duan, H. Fan, and Q. Meng, “Rotating machine fault diagnosis using empirical mode decomposition,” Mech. Syst. Signal Process.22(5), 1072–1081 (2008).
[CrossRef]

Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol.40(8), 1091–1098 (2008).
[CrossRef]

2007

L. Wu, C. Zhang, and B. Zhao, “Analysis of the lateral displacement and optical path difference in Wide-field-of –view polarization interference imaging spectrometer,” Opt. Commun.273(1), 67–73 (2007).
[CrossRef]

P. D. Spanos, A. Giaralis, and N. P. Politis, “Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition,” Soil. Dyn. Earthquake Eng.27(7), 675–689 (2007).
[CrossRef]

A. O. Boudraa and J. C. Cexus, “EMD-based signal filtering,” IEEE Trans. Instrum. Meas.56(6), 2196–2202 (2007).
[CrossRef]

2005

H. Liang, Q. H. Lin, and J. D. Z. Chen, “Application of the empirical mode decomposition to the analysis of esophageal manometric data in gastroesophageal reflux disease,” IEEE Trans. Biomed. Eng.52(10), 1692–1701 (2005).
[CrossRef] [PubMed]

2004

A. O. Boudraa, J. C. Cexus, and Z. Saidi, “EMD-based signal noise reduction,” Int. J. Signal Process.1, 33–37 (2004).

Z. Wu and N. E. Huang, “A study of the characteristcs of white noise using the Empirical Mode Decomposition method,” Proc. R. Soc. Lond. A460(2046), 1597–1611 (2004).
[CrossRef]

P. Flandrin, G. Rilling, and P. Gonçalves, “Empirical mode decomposition as a filter bank,” IEEE Signal Process. Lett.11(2), 112–114 (2004).
[CrossRef]

2000

H. Liang, Z. Lin, and R. W. McCallum, “Artifact reduction in electrogastrogram based on empirical mode decomposition method,” Med. Biol. Eng. Comput.38(1), 35–41 (2000).
[CrossRef] [PubMed]

C. Zhang, X. Bin, and B. Zhao, “Static polarization interference imaging spectrometer (SPIIS),” Proc. SPIE4087, 957–961 (2000).
[CrossRef]

1999

N. E. Huang, Z. Shen, and S. R. Long, “A new view of nonlinear water waves: The Hilbert spectrum,” Annu. Rev. Fluid Mech.31(1), 417–457 (1999).
[CrossRef]

1998

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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]

1995

L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory41(3), 613–627 (1995).
[CrossRef]

Barner, K. E.

M. Blanco-Velasco, B. Weng, and K. E. Barner, “ECG signal de-noising and baseline wander correction based on the empirical mode decomposition,” Comput. Biol. Med.38(1), 1–13 (2008).
[CrossRef] [PubMed]

Bin, X.

C. Zhang, X. Bin, and B. Zhao, “Static polarization interference imaging spectrometer (SPIIS),” Proc. SPIE4087, 957–961 (2000).
[CrossRef]

Blanco-Velasco, M.

M. Blanco-Velasco, B. Weng, and K. E. Barner, “ECG signal de-noising and baseline wander correction based on the empirical mode decomposition,” Comput. Biol. Med.38(1), 1–13 (2008).
[CrossRef] [PubMed]

Borgnat, P.

A. Moghtaderi, P. Borgnat, and P. Flandrin, “Trend filtering: empirical mode decompositions Versus l1 and Hodrick-Prescott,” Adv. Adapt. Data Anal.3, 41–61 (2011).
[CrossRef]

Boudraa, A. O.

A. O. Boudraa and J. C. Cexus, “EMD-based signal filtering,” IEEE Trans. Instrum. Meas.56(6), 2196–2202 (2007).
[CrossRef]

A. O. Boudraa, J. C. Cexus, and Z. Saidi, “EMD-based signal noise reduction,” Int. J. Signal Process.1, 33–37 (2004).

Cexus, J. C.

A. O. Boudraa and J. C. Cexus, “EMD-based signal filtering,” IEEE Trans. Instrum. Meas.56(6), 2196–2202 (2007).
[CrossRef]

A. O. Boudraa, J. C. Cexus, and Z. Saidi, “EMD-based signal noise reduction,” Int. J. Signal Process.1, 33–37 (2004).

Chen, J. D. Z.

H. Liang, Q. H. Lin, and J. D. Z. Chen, “Application of the empirical mode decomposition to the analysis of esophageal manometric data in gastroesophageal reflux disease,” IEEE Trans. Biomed. Eng.52(10), 1692–1701 (2005).
[CrossRef] [PubMed]

Dai, H.

Donoho, L.

L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory41(3), 613–627 (1995).
[CrossRef]

Duan, C.

Q. Gao, C. Duan, H. Fan, and Q. Meng, “Rotating machine fault diagnosis using empirical mode decomposition,” Mech. Syst. Signal Process.22(5), 1072–1081 (2008).
[CrossRef]

Fan, H.

Q. Gao, C. Duan, H. Fan, and Q. Meng, “Rotating machine fault diagnosis using empirical mode decomposition,” Mech. Syst. Signal Process.22(5), 1072–1081 (2008).
[CrossRef]

Flandrin, P.

A. Moghtaderi, P. Borgnat, and P. Flandrin, “Trend filtering: empirical mode decompositions Versus l1 and Hodrick-Prescott,” Adv. Adapt. Data Anal.3, 41–61 (2011).
[CrossRef]

P. Flandrin, G. Rilling, and P. Gonçalves, “Empirical mode decomposition as a filter bank,” IEEE Signal Process. Lett.11(2), 112–114 (2004).
[CrossRef]

Gao, Q.

Q. Gao, C. Duan, H. Fan, and Q. Meng, “Rotating machine fault diagnosis using empirical mode decomposition,” Mech. Syst. Signal Process.22(5), 1072–1081 (2008).
[CrossRef]

Giaralis, A.

P. D. Spanos, A. Giaralis, and N. P. Politis, “Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition,” Soil. Dyn. Earthquake Eng.27(7), 675–689 (2007).
[CrossRef]

Gonçalves, P.

P. Flandrin, G. Rilling, and P. Gonçalves, “Empirical mode decomposition as a filter bank,” IEEE Signal Process. Lett.11(2), 112–114 (2004).
[CrossRef]

Huang, N. E.

Z. Wu and N. E. Huang, “A study of the characteristcs of white noise using the Empirical Mode Decomposition method,” Proc. R. Soc. Lond. A460(2046), 1597–1611 (2004).
[CrossRef]

N. E. Huang, Z. Shen, and S. R. Long, “A new view of nonlinear water waves: The Hilbert spectrum,” Annu. Rev. Fluid Mech.31(1), 417–457 (1999).
[CrossRef]

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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]

Jia, C.

Jian, X.

Kemao, Q.

Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol.40(8), 1091–1098 (2008).
[CrossRef]

Liang, H.

H. Liang, Q. H. Lin, and J. D. Z. Chen, “Application of the empirical mode decomposition to the analysis of esophageal manometric data in gastroesophageal reflux disease,” IEEE Trans. Biomed. Eng.52(10), 1692–1701 (2005).
[CrossRef] [PubMed]

H. Liang, Z. Lin, and R. W. McCallum, “Artifact reduction in electrogastrogram based on empirical mode decomposition method,” Med. Biol. Eng. Comput.38(1), 35–41 (2000).
[CrossRef] [PubMed]

Lin, Q. H.

H. Liang, Q. H. Lin, and J. D. Z. Chen, “Application of the empirical mode decomposition to the analysis of esophageal manometric data in gastroesophageal reflux disease,” IEEE Trans. Biomed. Eng.52(10), 1692–1701 (2005).
[CrossRef] [PubMed]

Lin, Z.

H. Liang, Z. Lin, and R. W. McCallum, “Artifact reduction in electrogastrogram based on empirical mode decomposition method,” Med. Biol. Eng. Comput.38(1), 35–41 (2000).
[CrossRef] [PubMed]

Liu, H. H.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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, and S. R. Long, “A new view of nonlinear water waves: The Hilbert spectrum,” Annu. Rev. Fluid Mech.31(1), 417–457 (1999).
[CrossRef]

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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]

McCallum, R. W.

H. Liang, Z. Lin, and R. W. McCallum, “Artifact reduction in electrogastrogram based on empirical mode decomposition method,” Med. Biol. Eng. Comput.38(1), 35–41 (2000).
[CrossRef] [PubMed]

Meng, Q.

Q. Gao, C. Duan, H. Fan, and Q. Meng, “Rotating machine fault diagnosis using empirical mode decomposition,” Mech. Syst. Signal Process.22(5), 1072–1081 (2008).
[CrossRef]

Moghtaderi, A.

A. Moghtaderi, P. Borgnat, and P. Flandrin, “Trend filtering: empirical mode decompositions Versus l1 and Hodrick-Prescott,” Adv. Adapt. Data Anal.3, 41–61 (2011).
[CrossRef]

Mu, T.

Politis, N. P.

P. D. Spanos, A. Giaralis, and N. P. Politis, “Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition,” Soil. Dyn. Earthquake Eng.27(7), 675–689 (2007).
[CrossRef]

Ren, W.

Rilling, G.

P. Flandrin, G. Rilling, and P. Gonçalves, “Empirical mode decomposition as a filter bank,” IEEE Signal Process. Lett.11(2), 112–114 (2004).
[CrossRef]

Saidi, Z.

A. O. Boudraa, J. C. Cexus, and Z. Saidi, “EMD-based signal noise reduction,” Int. J. Signal Process.1, 33–37 (2004).

Shen, Z.

N. E. Huang, Z. Shen, and S. R. Long, “A new view of nonlinear water waves: The Hilbert spectrum,” Annu. Rev. Fluid Mech.31(1), 417–457 (1999).
[CrossRef]

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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. 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]

Spanos, P. D.

P. D. Spanos, A. Giaralis, and N. P. Politis, “Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition,” Soil. Dyn. Earthquake Eng.27(7), 675–689 (2007).
[CrossRef]

Tung, C. C.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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]

Weng, B.

M. Blanco-Velasco, B. Weng, and K. E. Barner, “ECG signal de-noising and baseline wander correction based on the empirical mode decomposition,” Comput. Biol. Med.38(1), 1–13 (2008).
[CrossRef] [PubMed]

Wu, L.

L. Wu, C. Zhang, and B. Zhao, “Analysis of the lateral displacement and optical path difference in Wide-field-of –view polarization interference imaging spectrometer,” Opt. Commun.273(1), 67–73 (2007).
[CrossRef]

Wu, M. C.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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, “A study of the characteristcs of white noise using the Empirical Mode Decomposition method,” Proc. R. Soc. Lond. A460(2046), 1597–1611 (2004).
[CrossRef]

Yen, N.-C.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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]

Zhang, C.

Zhang, L.

Zhao, B.

X. Jian, C. Zhang, L. Zhang, and B. Zhao, “The data processing of the temporarily and spatially mixed modulated polarization interference imaging spectrometer,” Opt. Express18(6), 5674–5680 (2010).
[CrossRef] [PubMed]

L. Wu, C. Zhang, and B. Zhao, “Analysis of the lateral displacement and optical path difference in Wide-field-of –view polarization interference imaging spectrometer,” Opt. Commun.273(1), 67–73 (2007).
[CrossRef]

C. Zhang, X. Bin, and B. Zhao, “Static polarization interference imaging spectrometer (SPIIS),” Proc. SPIE4087, 957–961 (2000).
[CrossRef]

Zheng, Q.

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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]

Adv. Adapt. Data Anal.

A. Moghtaderi, P. Borgnat, and P. Flandrin, “Trend filtering: empirical mode decompositions Versus l1 and Hodrick-Prescott,” Adv. Adapt. Data Anal.3, 41–61 (2011).
[CrossRef]

Annu. Rev. Fluid Mech.

N. E. Huang, Z. Shen, and S. R. Long, “A new view of nonlinear water waves: The Hilbert spectrum,” Annu. Rev. Fluid Mech.31(1), 417–457 (1999).
[CrossRef]

Comput. Biol. Med.

M. Blanco-Velasco, B. Weng, and K. E. Barner, “ECG signal de-noising and baseline wander correction based on the empirical mode decomposition,” Comput. Biol. Med.38(1), 1–13 (2008).
[CrossRef] [PubMed]

IEEE Signal Process. Lett.

P. Flandrin, G. Rilling, and P. Gonçalves, “Empirical mode decomposition as a filter bank,” IEEE Signal Process. Lett.11(2), 112–114 (2004).
[CrossRef]

IEEE Trans. Biomed. Eng.

H. Liang, Q. H. Lin, and J. D. Z. Chen, “Application of the empirical mode decomposition to the analysis of esophageal manometric data in gastroesophageal reflux disease,” IEEE Trans. Biomed. Eng.52(10), 1692–1701 (2005).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory41(3), 613–627 (1995).
[CrossRef]

IEEE Trans. Instrum. Meas.

A. O. Boudraa and J. C. Cexus, “EMD-based signal filtering,” IEEE Trans. Instrum. Meas.56(6), 2196–2202 (2007).
[CrossRef]

Int. J. Signal Process.

A. O. Boudraa, J. C. Cexus, and Z. Saidi, “EMD-based signal noise reduction,” Int. J. Signal Process.1, 33–37 (2004).

Mech. Syst. Signal Process.

Q. Gao, C. Duan, H. Fan, and Q. Meng, “Rotating machine fault diagnosis using empirical mode decomposition,” Mech. Syst. Signal Process.22(5), 1072–1081 (2008).
[CrossRef]

Med. Biol. Eng. Comput.

H. Liang, Z. Lin, and R. W. McCallum, “Artifact reduction in electrogastrogram based on empirical mode decomposition method,” Med. Biol. Eng. Comput.38(1), 35–41 (2000).
[CrossRef] [PubMed]

Opt. Commun.

L. Wu, C. Zhang, and B. Zhao, “Analysis of the lateral displacement and optical path difference in Wide-field-of –view polarization interference imaging spectrometer,” Opt. Commun.273(1), 67–73 (2007).
[CrossRef]

Opt. Express

Opt. Laser Technol.

Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol.40(8), 1091–1098 (2008).
[CrossRef]

Opt. Lett.

Proc. R. Soc. Lond. A

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. 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]

Z. Wu and N. E. Huang, “A study of the characteristcs of white noise using the Empirical Mode Decomposition method,” Proc. R. Soc. Lond. A460(2046), 1597–1611 (2004).
[CrossRef]

Proc. SPIE

C. Zhang, X. Bin, and B. Zhao, “Static polarization interference imaging spectrometer (SPIIS),” Proc. SPIE4087, 957–961 (2000).
[CrossRef]

Soil. Dyn. Earthquake Eng.

P. D. Spanos, A. Giaralis, and N. P. Politis, “Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition,” Soil. Dyn. Earthquake Eng.27(7), 675–689 (2007).
[CrossRef]

Other

R. J. Bell, Introductory Fourier Transform Spectroscopy (Academic Press & London, 1972), Chap. 3.

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Figures (17)

Fig. 1
Fig. 1

The simulated noise free interferogram (the up) and contaminated interferogram (the down).

Fig. 2
Fig. 2

The simulated noise free interferogram is decomposed into 16 IMFs (including the residual mode).

Fig. 3
Fig. 3

Background estimation: (a) The absolute value of the summation of the first d IMFS; (b) The assumed background (the solid black curve) and the estimated background (The dotted red curve) obtained from the partial reconstruction with IMFs 11 to 15 and the residual.

Fig. 4
Fig. 4

The de-noised spectrums, whose SNR are 20.26, 13.02 and 20.72 dB, obtained by the EMDT (the upper), WFT (the middle) and WT (the lower) de-noising methods. Herein, the SNR of the contaminated interferogram is 5 dB. The red dashed lines marked with purple circles are the noisy spectrums; the solid black lines are the original noise free spectrums; the dashed blue lines are the de-noised spectrums.

Fig. 5
Fig. 5

The SNR of the de-noised spectrums v. s. SNR of the contaminated interferograms.

Fig. 6
Fig. 6

The de-noised spectrums obtained by the EMDT (the upper), WFT (the middle) and WT (the lower) de-noising methods. δλ is the spectral resolution of the de-noised spectrum. The SNR and resolution of the de-noised spectrums, respectively, are 6.02, 10.12 and 5.62 dB and 7.62, 6.76 and 8.07nm. Herein, the SNR of the contaminated interferogram is 5 dB. The dashed lines marked with purple circles, solid black lines and dashed blue lines denote the noisy spectrums, the original noise free spectrums and the de-noised spectrums.

Fig. 7
Fig. 7

The de-noising results: The SNR (a), spectral resolution δλ (b), peak value (c) of the de-noised spectrums B Peak v. s. SNR Ic .

Fig. 8
Fig. 8

The de-noised results obtained via the WFT de-noising method with different width of Hamming window functions

Fig. 9
Fig. 9

The de-noised spectrums obtained by the EMDT (the upper), WFT (the middle) and WT (the lower) de-noising methods. δλ is the spectral resolution of the de-noised spectrum. The SNR and resolution of the de-noised spectrums, respectively, are −9.59, −2.66 and −6.26 dB and 8.41, 12.54 and 10.52nm.The dashed lines marked with purple circles, solid black lines and dashed blue lines denote the noisy spectrums, the original noise free spectrums and the de-noised spectrums.

Fig. 10
Fig. 10

The de-noising results with respective to the different noise assumptions.

Fig. 11
Fig. 11

The de-noising results with respective to the different values of γ.

Fig. 12
Fig. 12

The SNR of the de-noised polychromatic spectrums obtained by the WFT de-noising method with different width of windows and thresholds.

Fig. 13
Fig. 13

The SNR (a) and spectral resolution (b) of the de-noised polychromatic spectrums obtained by the WFT de-noising method with different width of windows and thresholds.

Fig. 14
Fig. 14

The measured interferogram.

Fig. 15
Fig. 15

The measured interferogram is decomposed into 9 IMFs (including the residual mode).

Fig. 16
Fig. 16

One row of the measured interferogram I m , estimated background I be and interferogram without background I p .

Fig. 17
Fig. 17

The noise free spectrum obtained by the proposed methods.

Equations (10)

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I c (x)= j=1 j=N B j cos[2π σ j Δ j (x)] +b(x)+n(x),
s(x)= i=1 i=K c i (x) +r(x),
I c (x)= i=1 i=K c i (x) +r(x).
I p (x)= i=D+1 i=K c i (x), b e (x)= i=D+1 i=K c i (x) +r(x),
S(d)=| i=1 i=d x c i (x) |,d=1,,K.
B c (σ)= i=1 i=M B ic (σ)+ B rc (σ),
B(σ)= i= K 1 i= K 2 B i (σ)+ i= K 2 +1 i=M B ic (σ)+ B rc (σ),
B(σ)= B 1 (σ)+ i=2 i=M B ic (σ)+ B rc (σ),
B 1 (σ)=sgn[ B 1c (σ)]max(| B 1c (σ) | τ 1 ,0),
τ 1 =γ κ 1 2log(T) , κ 1 = MAD 1 /0.6745, MAD 1 =Median{| B 1c (σ)Median{ B 1c (σ)} |},

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