Abstract

Preprocessing of spectral data is a key part of infrared spectroscopy and is an important basis for building robust models. Therefore, the measured signal needs to be preprocessed to achieve accurate and reliable measurement results. After sketching out the basic principles and basic methods of the wavelet transform, a new modified double-threshold denoising method combined with the proposed threshold method is presented in the paper. Two sets of comparative simulation experiments are also done to demonstrate the performance of the new denoising method. Block signals with a signal length of 2000 and the sinusoidal signal with a signal length of 1000 and the measured spectra are used for denoising with traditional schemes and the proposed method. The results of simulation data have demonstrated that the proposed method outperforms the traditional thresholding schemes for increasing the signal-to-noise ratio (SNR) without distorting the signal.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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  1. J. Zhu, W. Liu, Y. Lu, M. Gao, X. Zhao, T. Zhang, and L. Xu, “Research on analyzing atmospheric trasmittance based on infrared radiation measurements,” Proc. SPIE 5832, 83–90 (2005).
    [Crossref]
  2. R.J. Berry, B.K. Hart, R.L. Richardson, and P.R. Griffiths, “A low-resolution spectrometer for open-path Fourier-transform infrared spectrometry,” Field Anal. Chem. Technol. 3(2), 131–138 (1999).
    [Crossref]
  3. J.L. Hopkins, Introduction to Spectroscopy (Saunders College, 2001), pp. 1–19.
  4. W.K. Surewicz, H.H. Mantsch, and D. Chapman, “Determination of protein secondary structure by Fourier transform infrared spectroscopy: a critical assessment,” Biochemistry 32(2), 389–394 (1993).
    [Crossref]
  5. P.S. Snowball, “Spectral analysis of signals,” Leber Magen Darm 13(2), 57–63 (2005).
  6. S.L Marple, “Digital spectral analysis: with applications,” J. Acoust. Soc. Am. 86(5), 2043 (1989).
    [Crossref]
  7. Y. Xu, J.B. Weaver, D.M. Healy, and J. Lu, “Wavelet transform domain filters: a spatially selective noise filtration technique,” IEEE Trans. on Image Process. 3(6), 747–758 (1994).
    [Crossref]
  8. P. Li, Y. Zheng, C. Han, L.J. Song, and B.F. Cao, “Wavelet threshold methods used in lightning transient electrical signals denoising,” in International Conference on Fuzzy Systems and Knowledge Discovery. 2012.
  9. D. Griffin and J.S. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. 32(2), 236–243 (1984).
    [Crossref]
  10. S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust., Speech, Signal Process. 31(4), 986–998 (1983).
    [Crossref]
  11. P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
    [Crossref]
  12. J. Bertoin and A. R. Watson, “A probabilistic approach to spectral analysis of growth-fragmentation equations,” J. Funct. Anal. 274(8), 2163–2204 (2018).
    [Crossref]
  13. A. R. F. da Silva, “Wavelet denoising with evolutionary algorithms,” IEEE Trans. Circuits Syst. 15(4), 382–399 (2005).
    [Crossref]
  14. I. K. Fodor and C. Kamath, “Denoising through wavelet shrinkage: An empirical study,” J. Electron. Imaging 12(1), 151–160 (2003).
    [Crossref]
  15. M. Safta, P. Svasta, and M.O. Dima, “Wavelet signal denoising applied on electromagnetic traces,” in Design and Technology in Electronic Packaging. 2018.
  16. D.L Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41(3), 613–627 (1995).
    [Crossref]
  17. G. Deng, D. B. H. Tay, and S. Marusic, “A signal denoising algorithm based on overcomplete wavelet representations and Gaussian models,” Signal Processing 87(5), 866–876 (2007).
    [Crossref]
  18. H. Zang, Z. Wang, and Y. Zheng, “Analysis of signal de-noising method based on an improved wavelet thresholding,” in International Conference on Electronic Measurement & Instruments. 2009.
  19. J.F. Wang, “A wavelet denoising method based on the improved threshold function,” in International Conference on Wavelet Analysis and Pattern Recognition. 2014.
  20. H. Cui, R. Zhao, and Y. Hou, “Improved Threshold Denoising Method Based on Wavelet Transform,” in International Conference on Intelligent Information Technology Application. 2016.
  21. H. Liu, W. Wang, C. Xiang, L. Han, and H. Nie, “A de-noising method using the improved wavelet threshold function based on noise variance estimation,” Mech. Syst. Signal Process. 99, 30–46 (2018).
    [Crossref]
  22. G. Chen, Y. Zeng, and L.I. Zhe, “Research of self-adaptive double threshold method for ECG signal detection,” Journal of Jinan University, 2018.
  23. J. Zhang, Z. Qiang, J. Zhang, S. Lin, and J. Wang, “A novel algorithm for threshold image denoising based on wavelet construction,” Cluster Computing, 2018: 1–8.
  24. Z. Di, J. Zhang, and C. Jia, “An Improved Wavelet Threshold Denoising Algorithm,” in International Conference on Intelligent System Design & Engineering Applications. 2013.
  25. M. Srivastava, C.L. Anderson, and J.H. Freed, “A New Wavelet Denoising Method for Selecting Decomposition Levels and Noise Thresholds,” IEEE Access 4, 3862–3877 (2016).
    [Crossref]
  26. G. Y. Chen and T. D. Bui, “Multiwavelets denoising using neighboring coefficients,” IEEE Signal Process. Lett. 10(7), 211–214 (2003).
    [Crossref]

2018 (2)

J. Bertoin and A. R. Watson, “A probabilistic approach to spectral analysis of growth-fragmentation equations,” J. Funct. Anal. 274(8), 2163–2204 (2018).
[Crossref]

H. Liu, W. Wang, C. Xiang, L. Han, and H. Nie, “A de-noising method using the improved wavelet threshold function based on noise variance estimation,” Mech. Syst. Signal Process. 99, 30–46 (2018).
[Crossref]

2016 (1)

M. Srivastava, C.L. Anderson, and J.H. Freed, “A New Wavelet Denoising Method for Selecting Decomposition Levels and Noise Thresholds,” IEEE Access 4, 3862–3877 (2016).
[Crossref]

2010 (1)

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

2007 (1)

G. Deng, D. B. H. Tay, and S. Marusic, “A signal denoising algorithm based on overcomplete wavelet representations and Gaussian models,” Signal Processing 87(5), 866–876 (2007).
[Crossref]

2005 (3)

A. R. F. da Silva, “Wavelet denoising with evolutionary algorithms,” IEEE Trans. Circuits Syst. 15(4), 382–399 (2005).
[Crossref]

J. Zhu, W. Liu, Y. Lu, M. Gao, X. Zhao, T. Zhang, and L. Xu, “Research on analyzing atmospheric trasmittance based on infrared radiation measurements,” Proc. SPIE 5832, 83–90 (2005).
[Crossref]

P.S. Snowball, “Spectral analysis of signals,” Leber Magen Darm 13(2), 57–63 (2005).

2003 (2)

I. K. Fodor and C. Kamath, “Denoising through wavelet shrinkage: An empirical study,” J. Electron. Imaging 12(1), 151–160 (2003).
[Crossref]

G. Y. Chen and T. D. Bui, “Multiwavelets denoising using neighboring coefficients,” IEEE Signal Process. Lett. 10(7), 211–214 (2003).
[Crossref]

1999 (1)

R.J. Berry, B.K. Hart, R.L. Richardson, and P.R. Griffiths, “A low-resolution spectrometer for open-path Fourier-transform infrared spectrometry,” Field Anal. Chem. Technol. 3(2), 131–138 (1999).
[Crossref]

1995 (1)

D.L Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41(3), 613–627 (1995).
[Crossref]

1994 (1)

Y. Xu, J.B. Weaver, D.M. Healy, and J. Lu, “Wavelet transform domain filters: a spatially selective noise filtration technique,” IEEE Trans. on Image Process. 3(6), 747–758 (1994).
[Crossref]

1993 (1)

W.K. Surewicz, H.H. Mantsch, and D. Chapman, “Determination of protein secondary structure by Fourier transform infrared spectroscopy: a critical assessment,” Biochemistry 32(2), 389–394 (1993).
[Crossref]

1989 (1)

S.L Marple, “Digital spectral analysis: with applications,” J. Acoust. Soc. Am. 86(5), 2043 (1989).
[Crossref]

1984 (1)

D. Griffin and J.S. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. 32(2), 236–243 (1984).
[Crossref]

1983 (1)

S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust., Speech, Signal Process. 31(4), 986–998 (1983).
[Crossref]

Anderson, C.L.

M. Srivastava, C.L. Anderson, and J.H. Freed, “A New Wavelet Denoising Method for Selecting Decomposition Levels and Noise Thresholds,” IEEE Access 4, 3862–3877 (2016).
[Crossref]

Bassan, P.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Berry, R.J.

R.J. Berry, B.K. Hart, R.L. Richardson, and P.R. Griffiths, “A low-resolution spectrometer for open-path Fourier-transform infrared spectrometry,” Field Anal. Chem. Technol. 3(2), 131–138 (1999).
[Crossref]

Bertoin, J.

J. Bertoin and A. R. Watson, “A probabilistic approach to spectral analysis of growth-fragmentation equations,” J. Funct. Anal. 274(8), 2163–2204 (2018).
[Crossref]

Brown, M.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Bui, T. D.

G. Y. Chen and T. D. Bui, “Multiwavelets denoising using neighboring coefficients,” IEEE Signal Process. Lett. 10(7), 211–214 (2003).
[Crossref]

Byrne, H.J.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Cao, B.F.

P. Li, Y. Zheng, C. Han, L.J. Song, and B.F. Cao, “Wavelet threshold methods used in lightning transient electrical signals denoising,” in International Conference on Fuzzy Systems and Knowledge Discovery. 2012.

Chapman, D.

W.K. Surewicz, H.H. Mantsch, and D. Chapman, “Determination of protein secondary structure by Fourier transform infrared spectroscopy: a critical assessment,” Biochemistry 32(2), 389–394 (1993).
[Crossref]

Chen, G.

G. Chen, Y. Zeng, and L.I. Zhe, “Research of self-adaptive double threshold method for ECG signal detection,” Journal of Jinan University, 2018.

Chen, G. Y.

G. Y. Chen and T. D. Bui, “Multiwavelets denoising using neighboring coefficients,” IEEE Signal Process. Lett. 10(7), 211–214 (2003).
[Crossref]

Clarke, N.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Cui, H.

H. Cui, R. Zhao, and Y. Hou, “Improved Threshold Denoising Method Based on Wavelet Transform,” in International Conference on Intelligent Information Technology Application. 2016.

da Silva, A. R. F.

A. R. F. da Silva, “Wavelet denoising with evolutionary algorithms,” IEEE Trans. Circuits Syst. 15(4), 382–399 (2005).
[Crossref]

Deng, G.

G. Deng, D. B. H. Tay, and S. Marusic, “A signal denoising algorithm based on overcomplete wavelet representations and Gaussian models,” Signal Processing 87(5), 866–876 (2007).
[Crossref]

Di, Z.

Z. Di, J. Zhang, and C. Jia, “An Improved Wavelet Threshold Denoising Algorithm,” in International Conference on Intelligent System Design & Engineering Applications. 2013.

Dima, M.O.

M. Safta, P. Svasta, and M.O. Dima, “Wavelet signal denoising applied on electromagnetic traces,” in Design and Technology in Electronic Packaging. 2018.

Donoho, D.L

D.L Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41(3), 613–627 (1995).
[Crossref]

Dumas, P.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Fodor, I. K.

I. K. Fodor and C. Kamath, “Denoising through wavelet shrinkage: An empirical study,” J. Electron. Imaging 12(1), 151–160 (2003).
[Crossref]

Freed, J.H.

M. Srivastava, C.L. Anderson, and J.H. Freed, “A New Wavelet Denoising Method for Selecting Decomposition Levels and Noise Thresholds,” IEEE Access 4, 3862–3877 (2016).
[Crossref]

Gao, M.

J. Zhu, W. Liu, Y. Lu, M. Gao, X. Zhao, T. Zhang, and L. Xu, “Research on analyzing atmospheric trasmittance based on infrared radiation measurements,” Proc. SPIE 5832, 83–90 (2005).
[Crossref]

Gardner, P.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Gazi, E.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Griffin, D.

D. Griffin and J.S. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. 32(2), 236–243 (1984).
[Crossref]

Griffiths, P.R.

R.J. Berry, B.K. Hart, R.L. Richardson, and P.R. Griffiths, “A low-resolution spectrometer for open-path Fourier-transform infrared spectrometry,” Field Anal. Chem. Technol. 3(2), 131–138 (1999).
[Crossref]

Han, C.

P. Li, Y. Zheng, C. Han, L.J. Song, and B.F. Cao, “Wavelet threshold methods used in lightning transient electrical signals denoising,” in International Conference on Fuzzy Systems and Knowledge Discovery. 2012.

Han, L.

H. Liu, W. Wang, C. Xiang, L. Han, and H. Nie, “A de-noising method using the improved wavelet threshold function based on noise variance estimation,” Mech. Syst. Signal Process. 99, 30–46 (2018).
[Crossref]

Hart, B.K.

R.J. Berry, B.K. Hart, R.L. Richardson, and P.R. Griffiths, “A low-resolution spectrometer for open-path Fourier-transform infrared spectrometry,” Field Anal. Chem. Technol. 3(2), 131–138 (1999).
[Crossref]

Healy, D.M.

Y. Xu, J.B. Weaver, D.M. Healy, and J. Lu, “Wavelet transform domain filters: a spatially selective noise filtration technique,” IEEE Trans. on Image Process. 3(6), 747–758 (1994).
[Crossref]

Hopkins, J.L.

J.L. Hopkins, Introduction to Spectroscopy (Saunders College, 2001), pp. 1–19.

Hou, Y.

H. Cui, R. Zhao, and Y. Hou, “Improved Threshold Denoising Method Based on Wavelet Transform,” in International Conference on Intelligent Information Technology Application. 2016.

Jia, C.

Z. Di, J. Zhang, and C. Jia, “An Improved Wavelet Threshold Denoising Algorithm,” in International Conference on Intelligent System Design & Engineering Applications. 2013.

Kamath, C.

I. K. Fodor and C. Kamath, “Denoising through wavelet shrinkage: An empirical study,” J. Electron. Imaging 12(1), 151–160 (2003).
[Crossref]

Kohler, A.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Lee, J.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Li, P.

P. Li, Y. Zheng, C. Han, L.J. Song, and B.F. Cao, “Wavelet threshold methods used in lightning transient electrical signals denoising,” in International Conference on Fuzzy Systems and Knowledge Discovery. 2012.

Lim, J. S.

S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust., Speech, Signal Process. 31(4), 986–998 (1983).
[Crossref]

Lim, J.S.

D. Griffin and J.S. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. 32(2), 236–243 (1984).
[Crossref]

Lin, S.

J. Zhang, Z. Qiang, J. Zhang, S. Lin, and J. Wang, “A novel algorithm for threshold image denoising based on wavelet construction,” Cluster Computing, 2018: 1–8.

Liu, H.

H. Liu, W. Wang, C. Xiang, L. Han, and H. Nie, “A de-noising method using the improved wavelet threshold function based on noise variance estimation,” Mech. Syst. Signal Process. 99, 30–46 (2018).
[Crossref]

Liu, W.

J. Zhu, W. Liu, Y. Lu, M. Gao, X. Zhao, T. Zhang, and L. Xu, “Research on analyzing atmospheric trasmittance based on infrared radiation measurements,” Proc. SPIE 5832, 83–90 (2005).
[Crossref]

Lu, J.

Y. Xu, J.B. Weaver, D.M. Healy, and J. Lu, “Wavelet transform domain filters: a spatially selective noise filtration technique,” IEEE Trans. on Image Process. 3(6), 747–758 (1994).
[Crossref]

Lu, Y.

J. Zhu, W. Liu, Y. Lu, M. Gao, X. Zhao, T. Zhang, and L. Xu, “Research on analyzing atmospheric trasmittance based on infrared radiation measurements,” Proc. SPIE 5832, 83–90 (2005).
[Crossref]

Mantsch, H.H.

W.K. Surewicz, H.H. Mantsch, and D. Chapman, “Determination of protein secondary structure by Fourier transform infrared spectroscopy: a critical assessment,” Biochemistry 32(2), 389–394 (1993).
[Crossref]

Marple, S.L

S.L Marple, “Digital spectral analysis: with applications,” J. Acoust. Soc. Am. 86(5), 2043 (1989).
[Crossref]

Martens, H.

P. Bassan, A. Kohler, H. Martens, J. Lee, H.J. Byrne, P. Dumas, E. Gazi, M. Brown, N. Clarke, and P. Gardner, “Resonant Mie scattering (RMieS) correction of infrared spectra from highly scattering biological samples,” Analyst 135(2), 268–277 (2010).
[Crossref]

Marusic, S.

G. Deng, D. B. H. Tay, and S. Marusic, “A signal denoising algorithm based on overcomplete wavelet representations and Gaussian models,” Signal Processing 87(5), 866–876 (2007).
[Crossref]

Nawab, S. H.

S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust., Speech, Signal Process. 31(4), 986–998 (1983).
[Crossref]

Nie, H.

H. Liu, W. Wang, C. Xiang, L. Han, and H. Nie, “A de-noising method using the improved wavelet threshold function based on noise variance estimation,” Mech. Syst. Signal Process. 99, 30–46 (2018).
[Crossref]

Qiang, Z.

J. Zhang, Z. Qiang, J. Zhang, S. Lin, and J. Wang, “A novel algorithm for threshold image denoising based on wavelet construction,” Cluster Computing, 2018: 1–8.

Quatieri, T. F.

S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust., Speech, Signal Process. 31(4), 986–998 (1983).
[Crossref]

Richardson, R.L.

R.J. Berry, B.K. Hart, R.L. Richardson, and P.R. Griffiths, “A low-resolution spectrometer for open-path Fourier-transform infrared spectrometry,” Field Anal. Chem. Technol. 3(2), 131–138 (1999).
[Crossref]

Safta, M.

M. Safta, P. Svasta, and M.O. Dima, “Wavelet signal denoising applied on electromagnetic traces,” in Design and Technology in Electronic Packaging. 2018.

Snowball, P.S.

P.S. Snowball, “Spectral analysis of signals,” Leber Magen Darm 13(2), 57–63 (2005).

Song, L.J.

P. Li, Y. Zheng, C. Han, L.J. Song, and B.F. Cao, “Wavelet threshold methods used in lightning transient electrical signals denoising,” in International Conference on Fuzzy Systems and Knowledge Discovery. 2012.

Srivastava, M.

M. Srivastava, C.L. Anderson, and J.H. Freed, “A New Wavelet Denoising Method for Selecting Decomposition Levels and Noise Thresholds,” IEEE Access 4, 3862–3877 (2016).
[Crossref]

Surewicz, W.K.

W.K. Surewicz, H.H. Mantsch, and D. Chapman, “Determination of protein secondary structure by Fourier transform infrared spectroscopy: a critical assessment,” Biochemistry 32(2), 389–394 (1993).
[Crossref]

Svasta, P.

M. Safta, P. Svasta, and M.O. Dima, “Wavelet signal denoising applied on electromagnetic traces,” in Design and Technology in Electronic Packaging. 2018.

Tay, D. B. H.

G. Deng, D. B. H. Tay, and S. Marusic, “A signal denoising algorithm based on overcomplete wavelet representations and Gaussian models,” Signal Processing 87(5), 866–876 (2007).
[Crossref]

Wang, J.

J. Zhang, Z. Qiang, J. Zhang, S. Lin, and J. Wang, “A novel algorithm for threshold image denoising based on wavelet construction,” Cluster Computing, 2018: 1–8.

Wang, J.F.

J.F. Wang, “A wavelet denoising method based on the improved threshold function,” in International Conference on Wavelet Analysis and Pattern Recognition. 2014.

Wang, W.

H. Liu, W. Wang, C. Xiang, L. Han, and H. Nie, “A de-noising method using the improved wavelet threshold function based on noise variance estimation,” Mech. Syst. Signal Process. 99, 30–46 (2018).
[Crossref]

Wang, Z.

H. Zang, Z. Wang, and Y. Zheng, “Analysis of signal de-noising method based on an improved wavelet thresholding,” in International Conference on Electronic Measurement & Instruments. 2009.

Watson, A. R.

J. Bertoin and A. R. Watson, “A probabilistic approach to spectral analysis of growth-fragmentation equations,” J. Funct. Anal. 274(8), 2163–2204 (2018).
[Crossref]

Weaver, J.B.

Y. Xu, J.B. Weaver, D.M. Healy, and J. Lu, “Wavelet transform domain filters: a spatially selective noise filtration technique,” IEEE Trans. on Image Process. 3(6), 747–758 (1994).
[Crossref]

Xiang, C.

H. Liu, W. Wang, C. Xiang, L. Han, and H. Nie, “A de-noising method using the improved wavelet threshold function based on noise variance estimation,” Mech. Syst. Signal Process. 99, 30–46 (2018).
[Crossref]

Xu, L.

J. Zhu, W. Liu, Y. Lu, M. Gao, X. Zhao, T. Zhang, and L. Xu, “Research on analyzing atmospheric trasmittance based on infrared radiation measurements,” Proc. SPIE 5832, 83–90 (2005).
[Crossref]

Xu, Y.

Y. Xu, J.B. Weaver, D.M. Healy, and J. Lu, “Wavelet transform domain filters: a spatially selective noise filtration technique,” IEEE Trans. on Image Process. 3(6), 747–758 (1994).
[Crossref]

Zang, H.

H. Zang, Z. Wang, and Y. Zheng, “Analysis of signal de-noising method based on an improved wavelet thresholding,” in International Conference on Electronic Measurement & Instruments. 2009.

Zeng, Y.

G. Chen, Y. Zeng, and L.I. Zhe, “Research of self-adaptive double threshold method for ECG signal detection,” Journal of Jinan University, 2018.

Zhang, J.

J. Zhang, Z. Qiang, J. Zhang, S. Lin, and J. Wang, “A novel algorithm for threshold image denoising based on wavelet construction,” Cluster Computing, 2018: 1–8.

Z. Di, J. Zhang, and C. Jia, “An Improved Wavelet Threshold Denoising Algorithm,” in International Conference on Intelligent System Design & Engineering Applications. 2013.

J. Zhang, Z. Qiang, J. Zhang, S. Lin, and J. Wang, “A novel algorithm for threshold image denoising based on wavelet construction,” Cluster Computing, 2018: 1–8.

Zhang, T.

J. Zhu, W. Liu, Y. Lu, M. Gao, X. Zhao, T. Zhang, and L. Xu, “Research on analyzing atmospheric trasmittance based on infrared radiation measurements,” Proc. SPIE 5832, 83–90 (2005).
[Crossref]

Zhao, R.

H. Cui, R. Zhao, and Y. Hou, “Improved Threshold Denoising Method Based on Wavelet Transform,” in International Conference on Intelligent Information Technology Application. 2016.

Zhao, X.

J. Zhu, W. Liu, Y. Lu, M. Gao, X. Zhao, T. Zhang, and L. Xu, “Research on analyzing atmospheric trasmittance based on infrared radiation measurements,” Proc. SPIE 5832, 83–90 (2005).
[Crossref]

Zhe, L.I.

G. Chen, Y. Zeng, and L.I. Zhe, “Research of self-adaptive double threshold method for ECG signal detection,” Journal of Jinan University, 2018.

Zheng, Y.

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Figures (11)

Fig. 1.
Fig. 1. Comparison of soft and hard thresholds and proposed thresholds.
Fig. 2.
Fig. 2. Original signal s and its low and high frequency parts.
Fig. 3.
Fig. 3. (a) Fourth layer low frequency coefficient; (b) First layer high frequency coefficient; (c) Second layer high frequency coefficient; (d) Third layer high frequency coefficient; (e) Fourth layer high frequency coefficient.
Fig. 4.
Fig. 4. (a) Original Signal; (b) Level 4 approximation coefficient.
Fig. 5.
Fig. 5. (a) Original pure signal; (b) Noisy signal; (c) Hard threshold de-noised signal; (d) Soft threshold de-noised signal; (e) Literature [21] threshold de-noised signal; (f) Improved threshold de-noised signal.
Fig. 6.
Fig. 6. (A) The differential spectrum between(a) and (c); (B) The differential spectrum between (a) and (d); (C) The differential spectrum between (a) and (e); (D) The differential spectrum between (a) and (f).Diagram (a), (c), (d), (e)and (f) are based on Fig. 3.
Fig. 7.
Fig. 7. (a) Original pure signal; (b) Noisy signal; (c) Hard threshold de-noised signal; (d) Soft threshold de-noised signal; (e) Literature [21] threshold de-noised signal; (f) Improved threshold de-noised signal.
Fig. 8.
Fig. 8. (a) Fourth layer low frequency coefficient; (b) First layer high frequency coefficient; (c) Second layer high frequency coefficient; (d) Third layer high frequency coefficient; (e) Fourth layer high frequency coefficient.
Fig. 9.
Fig. 9. (a) Original Signal; (b) Level 4 approximation coefficient.
Fig. 10.
Fig. 10. (a) Measured signal; (b) Hard threshold de-noised signal; (c) Soft threshold de-noised signal; (d) Literature [21] threshold de-noised signal; (e) Improved threshold de-noised signal.
Fig. 11.
Fig. 11. (A) The differential spectrum between(a) and (b); (B) The differential spectrum between (a) and (c); (C) The differential spectrum between (a) and (d); (D) The differential spectrum between (a) and (e). Diagram (b), (c), (d) and(e) are based on Fig. 7.

Tables (2)

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Table 1. The SNR and MSE values after denoising of different threshold method based on block signal

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Table 2. The SNR and MSE values after denoising of different threshold method based on sinusoidal signal

Equations (15)

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s ( t ) = f ( t ) + σ e ( t )
W s ( j , k ) = W f ( j , k ) + W e ( j , k )
T i = σ 2 logN
σ = median ( | w j , k | ) 0.6745
k ( u , v ) = corr ( u , v ) P w ( u ) / P corr ( v ) w ( u , v )
P w ( u ) = v w ( u , v ) 2
P corr ( u ) = v corr ( u , v ) 2
h ( u , v ) = 1 ln | k ( u , v ) |
T ( u , v ) = σ 2 logN h ( u , v )
w ^ j , k = { w j , k | w j , k | λ 0 | w j , k | < λ
w ^ j , k = { s i g n ( w j , k ) ( | w j , k | λ ) | w j , k | λ 0 | w j , k | < λ
w ^ j , k = { s g n ( w j , k ) [ | w j , k | λ 1 e ( λ 1 λ 2 ) ] | w j , k | λ 2 s g n ( w j , k ) [ | w j , k | λ 1 e ( λ 1 | w j , k | ) ] λ 1 < | w j , k | < λ 2 0 | w j , k | < λ 1
w ^ j , k = { s g n ( w j , k ) [ | w j , k | λ 1 e ( λ 1 λ 2 | w j , k | ) ] , | w j , k | λ 2 s g n ( w j , k ) [ | w j , k | λ 1 e ( λ 1 | w j , k | ) ] , λ 1 < | w j , k | < λ 2 0 , | w j , k | < λ 1
SNR = 10 lg [ n = 1 N X ¯ 2 ( n ) 1 N n = 1 N [ S ( n ) X ( n ) ] 2 ]
RMSE = 1 N n = 1 N [ X ( n ) X ¯ ( n ) ] 2

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