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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 9 — May. 5, 2014
  • pp: 10233–10238
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Accuracy improvement of quantitative analysis in laser-induced breakdown spectroscopy using modified wavelet transform

X. H. Zou, L. B. Guo, M. Shen, X. Y. Li, Z. Q. Hao, Q. D. Zeng, Y. F. Lu, Z. M. Wang, and X. Y. Zeng  »View Author Affiliations


Optics Express, Vol. 22, Issue 9, pp. 10233-10238 (2014)
http://dx.doi.org/10.1364/OE.22.010233


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Abstract

A modified algorithm of background removal based on wavelet transform was developed for spectrum correction in laser-induced breakdown spectroscopy (LIBS). The optimal type of wavelet function, decomposition level and scaling factor γ were determined by the root-mean-square error of calibration (RMSEC) of the univariate regression model of the analysis element, which is considered as the optimization criteria. After background removal by this modified algorithm with RMSEC, the root-mean-square error of cross-validation (RMSECV) and the average relative error (ARE) criteria, the accuracy of quantitative analysis on chromium (Cr), vanadium (V), cuprum (Cu), and manganese (Mn) in the low alloy steel was all improved significantly. The results demonstrated that the algorithm developed is an effective pretreatment method in LIBS to significantly improve the accuracy in the quantitative analysis.

© 2014 Optical Society of America

1. Introduction

As a promising analytical technique, laser-induced breakdown spectroscopy (LIBS) based on laser-material interactions has a wide range of applications, including geological analysis, pharmaceutical industry, and environmental monitoring [1

1. W. B. Lee, J. Y. Wu, Y. I. Lee, and J. Sneddon, “Recent applications of laser-induced breakdown spectrometry: a review of material approaches,” Appl. Spectrosc. Rev. 39(1), 27–97 (2004). [CrossRef]

5

5. L. Dudragne, P. Adam, and J. Amouroux, “Time-resolved laser-induced breakdown spectroscopy: application for qualitative and quantitative detection of fluorine, chlorine, sulfur, and carbon in air,” Appl. Spectrosc. 52(10), 1321–1327 (1998). [CrossRef]

], owing to its advantages of rapid detection, no samples pretreatment, real-time analysis and remote measurement [6

6. J. P. Singh and S. N. Thakur, Laser-Induced Breakdown Spectroscopy (Elsevier Science, 2007).

, 7

7. Z. Wang, Z. Y. Hou, S. L. Lui, D. Jiang, J. M. Liu, and Z. Li, “Utilization of moderate cylindrical confinement for precision improvement of laser-induced breakdown spectroscopy signal,” Opt. Express 20(S6), A1011–A1018 (2012). [CrossRef]

]. In LIBS, elemental concentration is determined by the peak intensities in plasma emission spectra. However, as collected LIBS spectra are often overwhelmed by intense continuum background, which interferes the true intensity of spectra and compromises spectral clarity, and hence reduces the accuracy of quantitative analysis. In LIBS, continuum background is generated by the bremsstrahlung radiation, recombination radiation, as well as stray lights [8

8. T. Fujimoto, Plasma Spectroscopy (Clarendon, 2004).

]. As the delay time increases, isolated characteristic lines go down along with background. The signal-to-background ratio (SBR) of the peak intensities increases initially and then decreases. To obtain a maximum SBR, a time-resolved detector is indispensable. However, for a non-gated detector for applications requiring portability and cost effectiveness, background subtraction is particularly important. Therefore, finding an effective compensation algorithm for accurate estimation and subtraction of background to reduce background interference and enhance spectral SBR, is essential for improving the accuracy of quantitative analysis.

Wavelet transform (WT) is a new signal processing technique which has undergone rapid development in the past two decades. Compared with Fourier transform, WT has the characteristics of time-frequency localization and multi-resolution analyses of signals [9

9. X. G. Shao, A. K. M. Leung, and F. T. Chau, “Wavelet: a new trend in chemistry,” Acc. Chem. Res. 36(4), 276–283 (2003). [CrossRef] [PubMed]

]. Given the superiority for the analysis of non-stationary signals, WT is an extremely powerful tool for correction of spectra, which has been extensively applied in many fields, including Raman spectroscopy [10

10. C. M. Galloway, E. C. Le Ru, and P. G. Etchegoin, “An iterative algorithm for background removal in spectroscopy by wavelet transforms,” Appl. Spectrosc. 63(12), 1370–1376 (2009). [CrossRef] [PubMed]

], inductively coupled plasma atomic emission spectroscopy [11

11. X. G. Ma and Z. X. Zhang, “Application of wavelet transform to background correction in inductively coupled plasma atomic emission spectrometry,” Anal. Chim. Acta 485(2), 233–239 (2003). [CrossRef]

], and near-infrared spectroscopy [12

12. C. X. Ma and X. G. Shao, “Continuous wavelet transform applied to removing the fluctuating background in near-infrared spectra,” J. Chem. Inf. Comput. Sci. 44(3), 907–911 (2004). [CrossRef] [PubMed]

]. Recently, a few researches have explored the wavelet denoising in LIBS. Schlenke et al. adopted stationary WT via adaptive variable threshold for noise reduction, which successfully achieved noise suppression and signal preservation [13

13. J. Schlenke, L. Hildebrand, J. Moros, and J. J. Laserna, “Adaptive approach for variable noise suppression on laser-induced breakdown spectroscopy responses using stationary wavelet transform,” Anal. Chim. Acta 754, 8–19 (2012). [CrossRef] [PubMed]

]. Zhang et al. established a new double threshold optimization model of wavelet denoising, and demonstrated the performance of the method for both synthetic and actual LIBS signals [14

14. B. Zhang, L. X. Sun, H. B. Yu, Y. Xin, and Z. B. Cong, “Wavelet denoising method for laser-induced breakdown spectroscopy,” J. Anal. At. Spectrom. 28(12), 1884–1893 (2013). [CrossRef]

]. However, its application in the background subtraction of LIBS has not been systematically studied. Yuan et al. implemented WT to subtract background for LIBS application by utilizing the wavelet coefficients instead of the spectral information after reconstruction for the first time, and analyzed quantitatively the high-concentration carbon element in coal [15

15. T. B. Yuan, Z. Wang, Z. Li, W. D. Ni, and J. M. Liu, “A partial least squares and wavelet-transform hybrid model to analyze carbon content in coal using Laser-induced breakdown spectroscopy,” Anal. Chim. Acta 807, 29–35 (2014). [CrossRef] [PubMed]

]. Shao et al. proposed a concept of factor of the low-frequency part of the highest level of wavelet decomposition to amend the estimated background in high performance liquid chromatography (HPLC) [9

9. X. G. Shao, A. K. M. Leung, and F. T. Chau, “Wavelet: a new trend in chemistry,” Acc. Chem. Res. 36(4), 276–283 (2003). [CrossRef] [PubMed]

, 16

16. X. G. Shao, W. S. Cai, and Z. X. Pan, “Wavelet transform and its applications in high performance liquid chromatography (HPLC) analysis,” Chemom. Intell. Lab. Syst. 45(1–2), 249–256 (1999). [CrossRef]

]. The value of the factor was just estimated by the operator according to the position of simulated background and the results were judged only by observation of the chromatogram. This method was not suitable for batch processing of spectra data and unable to evaluate the accuracy of the factor.

In this study, a modified algorithm based on WT for automatically subtracting the continuum background was proposed for the purpose of low concentration elements analysis. Through considering the root-mean-square error of calibration (RMSEC) of the univariate regression model as the criterion, the three most important parameters, wavelet function, decomposition level and the scaling factor γ were optimized and discussed in details. To verify the validity, seven low alloy steel samples were used to compare the results with and without using the algorithm, which confirmed the effectiveness of the algorithm to improve the quantitative analyses.

2. Algorithm description

Obviously, the result of background correction depends on the wavelet function, decomposition level and scale factor γ. The Daubechies wavelet family was adopted [15

15. T. B. Yuan, Z. Wang, Z. Li, W. D. Ni, and J. M. Liu, “A partial least squares and wavelet-transform hybrid model to analyze carbon content in coal using Laser-induced breakdown spectroscopy,” Anal. Chim. Acta 807, 29–35 (2014). [CrossRef] [PubMed]

, 19

19. X. G. Shao, L. M. Shao, and G. W. Zhao, “Extraction of extended X-ray absorption fine structure information from the experimental data using the wavelet transform,” Anal. Commun. 35(4), 135–137 (1998). [CrossRef]

21

21. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992).

]. It should be given priority to the selection of the wavelet function and decomposition level, and then determine γ, which can effectively extract background information from the low-frequency component of the optimal wavelet decomposition and then further optimize the result of background estimation. The background-corrected spectra are used to establish the univariate regression model. The RMSEC and the determination coefficient (R2 c) of the calibration curve, the root-mean-square error of cross-validation (RMSECV) and the determination coefficient (R2 v) of the leave-one-out cross-validation curve, as well as the average relative error (ARE) are used as the evaluation parameters to test the performance of the proposed method.

To improve the accuracy of the regression model after background correction, the RMSEC was used as the optimization criterion to obtain the optimal wavelet function, decomposition level and γ. The program used for background subtraction of spectra was written in Matlab R2010b. The algorithm includes the following steps:

  • (1) Select the wavelet function which varies from db1 to db10, and choose the decomposition level which ranges between 4 and 13.
  • (2) Set the approximation coefficients cJ,k to zero, which is equivalent to γini = 1. Then reconstruct the spectra. Extract the intensity information of analysis element to build the calibration regression model, and calculate the RMSEC.
  • (3) Determine the optimum values of the wavelet function (W) and decomposition level (L) which make the RMSEC achieve the minimum.
  • (4) Perform wavelet decomposition again using the wavelet function (W) and decomposition level (L) for the spectra. Then solve the optimum γopt by the least squares method, and the RMSEC is still considered as the optimization goal. That is, γopt satisfies Eq. (2)
    RMSEC(W,L,γopt)=RMSECmin(W,L,γ)
    (2)
  • (5) Multiply cJ,k by (1-γopt) for the spectra and then reconstruct the spectral signal according to Eq. (1). Finally, the background-corrected spectra are obtained.

3. Experimental

The LIBS experimental apparatus used in this study is schematically shown in Fig. 1
Fig. 1 Schematic diagram of the experimental setup.
. A Q-switched Nd:YAG laser (wavelength: 532 nm, repetition rate: 3 Hz, pulse width: 8 ns, pulse energy: 80 mJ) was used to obtain spectral data under air environment. The laser beam was reflected and then focused onto the surface of the low alloy steel sample by a lens with a focal length of 15 cm. The plasma emission was coupled into an optical fiber by a light collector, then collected by an echelle spectrometer (Andor Tech., Mechelle 5000) with a wavelength range of 200–950 nm and a spectral resolution of λ/Δλ=5000. A digital delay generator was adopted to trigger the laser pulses and control the gate delays and widths of the ICCD (Andor Tech., iStar DH-334T). The detector was operated in the gated mode whose delay time and gate width were set to 2 and 10 μs, respectively, to obtain high spectral intensity and SBR. To reduce the effect of the laser energy fluctuation on the spectral intensity, each spectrum accumulated 30 pulses and six spectra were taken for each sample and averaged.

Seven low alloy steel samples with main elements of Cr, V, Cu, and Mn in the matrix element Fe were used in this study. Table 1 lists the certified concentrations of these elements.

Table 1. Composition of Cr, V, Cu, Mn and Fe elements from the low alloy steel samples

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4. Results and discussions

To verify the accuracy of the background correction algorithm, the quantitative analysis was performed for the four elements (namely, Cr, V, Cu, and Mn) in the seven low alloy steel standard samples. According to the Atomic Spectra Database [22

22. A. E. Kramida Y. Ralchenko, J. Reader, and NIST ASD Team, “NIST Atomic Spectra Database (version 5.1)” (National Institute of Standards and Technology, 2013), http://physics.nist.gov/asd.

], together with the observation of actual spectra, the analyte lines of Cr I 425.43 nm, V II 311.07 nm, Cu I 327.40 nm and Mn I 476.64 nm were chosen. Cr was used as an example element to demonstrate how the wavelet function, decomposition level and scaling factor γ influence the RMSEC.

4.1 Optimization of the wavelet function and decomposition level

As described above, the wavelet function and decomposition level should be firstly determined. Theoretically, to obtain the best background estimation, the structure of the wavelet function selected should be similar to a processed spectrum as much as possible. However, it is too difficult to achieve. Practically, the most suitable filter can be quickly determined by the specific optimization goal of the spectral analysis. In this study, RMSEC was used as the evaluation criterion for the optimal wavelet and level. As presented in Fig. 2, the minimal value of RMSEC could be obtained when the wavelet function was db8 and decomposition level was 10.
Fig. 2 RMSEC values at different wavelet functions and decomposition levels.
The background information was certainly stored in the approximation coefficients of the highest level of the wavelet decomposition.

4.2 Optimization of the scaling factor γ

As described in Section 4.1, compared to the wavelet function, the decomposition level is an important parameter. We analyzed the effect of the scaling factor γ on the RMSEC under the condition of optimal decomposition level corresponding to each specific wavelet function, as depicted in Fig. 3(a).
Fig. 3 Influence of the scaling factor γ. (a) The effect of γ on the RMSEC with optimal decomposition level for each wavelet function. (b) Comparison of one original and its corrected spectrum without and with γ at wavelet function was db8 and decomposition level was 10.
Based on the above analysis and performing procedure 4 in the algorithm, the optimum value of the scaling factor γ was acquired. Figure 3(a) shows that all the RMSEC could become smaller by taking the scaling factor of the low-frequency component into consideration. For db8 and 10 decomposition levels, γ = 0.89. Take a spectrum of the sample No. 5 for example, the original and background-corrected spectrum can be seen from Fig. 3(b). It was clear that the background of the spectrum reduced after applying the algorithm and avoided overestimation compared with the traditional method without introducing γ. The traditional method means fully discarding the lowest frequency part of wavelet coefficients.

4.3 Comparison of quantitative analysis

To further verify the effectiveness of the background subtraction, the similar procedure was implemented for V, Cu, and Mn as well. For comparison, the polynomial fitting method proposed by Sun et al. [23

23. L. X. Sun and H. B. Yu, “Automatic estimation of varying continuum background emission in laser-induced breakdown spectroscopy,” Spectrochim. Acta, B At. Spectrosc. 64(3), 278–287 (2009). [CrossRef]

] was also adopted for background subtraction. The detailed data are summarized in Table 2.

Table 2. Influence of different background subtraction methods on the univariate regression model

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In comparison with other methods, the proposed algorithm can further optimize the results. The improvement for Cr, V, Cu and Mn was obtained with lower RMSEC(V), higher R2 and lower ARE. The above analysis revealed that the proposed method can effectively improve the accuracy of the regression model.

5. Conclusions

A modified algorithm based on wavelet transform for background subtraction was developed in this study for quantitative analysis in LIBS spectra of the low alloy steel samples. The method developed requires no prior knowledge of the selection of the proper background points and the mathematical assumption of the background distribution. The results obtained show that for the four elements of Cr, V, Cu, and Mn, their RMSEC(V), R2 and ARE are all significantly improved, successfully proving the validity of the background correction algorithm. This method can effectively improve the quality of the signals and the accuracy of the regression model. Accordingly, the proposed method can be used as a competitive preprocessing tool for LIBS and other spectral analysis.

Acknowledgments

This research has been financially supported by the National Special Fund for the Development of Major Research Equipment and Instruments (No. 2011YQ160017), by the National Natural Science Foundation of China (No. 51128501), and by the Fundamental Research Funds for the Central Universities of China (No. CXY13Q022).

References and links

1.

W. B. Lee, J. Y. Wu, Y. I. Lee, and J. Sneddon, “Recent applications of laser-induced breakdown spectrometry: a review of material approaches,” Appl. Spectrosc. Rev. 39(1), 27–97 (2004). [CrossRef]

2.

Z. Wang, T. B. Yuan, Z. Y. Hou, W. D. Zhou, J. D. Lu, H. B. Ding, and X. Y. Zeng, “Laser-induced breakdown spectroscopy in China,” Front. Phys. (2014).

3.

L. B. Guo, Z. Q. Hao, M. Shen, W. Xiong, X. N. He, Z. Q. Xie, M. Gao, X. Y. Li, X. Y. Zeng, and Y. F. Lu, “Accuracy improvement of quantitative analysis by spatial confinement in laser-induced breakdown spectroscopy,” Opt. Express 21(15), 18188–18195 (2013). [CrossRef] [PubMed]

4.

P. Fichet, P. Mauchien, J. F. Wagner, and C. Moulin, “Quantitative elemental determination in water and oil by laser induced breakdown spectroscopy,” Anal. Chim. Acta 429(2), 269–278 (2001). [CrossRef]

5.

L. Dudragne, P. Adam, and J. Amouroux, “Time-resolved laser-induced breakdown spectroscopy: application for qualitative and quantitative detection of fluorine, chlorine, sulfur, and carbon in air,” Appl. Spectrosc. 52(10), 1321–1327 (1998). [CrossRef]

6.

J. P. Singh and S. N. Thakur, Laser-Induced Breakdown Spectroscopy (Elsevier Science, 2007).

7.

Z. Wang, Z. Y. Hou, S. L. Lui, D. Jiang, J. M. Liu, and Z. Li, “Utilization of moderate cylindrical confinement for precision improvement of laser-induced breakdown spectroscopy signal,” Opt. Express 20(S6), A1011–A1018 (2012). [CrossRef]

8.

T. Fujimoto, Plasma Spectroscopy (Clarendon, 2004).

9.

X. G. Shao, A. K. M. Leung, and F. T. Chau, “Wavelet: a new trend in chemistry,” Acc. Chem. Res. 36(4), 276–283 (2003). [CrossRef] [PubMed]

10.

C. M. Galloway, E. C. Le Ru, and P. G. Etchegoin, “An iterative algorithm for background removal in spectroscopy by wavelet transforms,” Appl. Spectrosc. 63(12), 1370–1376 (2009). [CrossRef] [PubMed]

11.

X. G. Ma and Z. X. Zhang, “Application of wavelet transform to background correction in inductively coupled plasma atomic emission spectrometry,” Anal. Chim. Acta 485(2), 233–239 (2003). [CrossRef]

12.

C. X. Ma and X. G. Shao, “Continuous wavelet transform applied to removing the fluctuating background in near-infrared spectra,” J. Chem. Inf. Comput. Sci. 44(3), 907–911 (2004). [CrossRef] [PubMed]

13.

J. Schlenke, L. Hildebrand, J. Moros, and J. J. Laserna, “Adaptive approach for variable noise suppression on laser-induced breakdown spectroscopy responses using stationary wavelet transform,” Anal. Chim. Acta 754, 8–19 (2012). [CrossRef] [PubMed]

14.

B. Zhang, L. X. Sun, H. B. Yu, Y. Xin, and Z. B. Cong, “Wavelet denoising method for laser-induced breakdown spectroscopy,” J. Anal. At. Spectrom. 28(12), 1884–1893 (2013). [CrossRef]

15.

T. B. Yuan, Z. Wang, Z. Li, W. D. Ni, and J. M. Liu, “A partial least squares and wavelet-transform hybrid model to analyze carbon content in coal using Laser-induced breakdown spectroscopy,” Anal. Chim. Acta 807, 29–35 (2014). [CrossRef] [PubMed]

16.

X. G. Shao, W. S. Cai, and Z. X. Pan, “Wavelet transform and its applications in high performance liquid chromatography (HPLC) analysis,” Chemom. Intell. Lab. Syst. 45(1–2), 249–256 (1999). [CrossRef]

17.

S. G. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. 11(7), 674–693 (1989). [CrossRef]

18.

S. G. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way (Academic, 2008).

19.

X. G. Shao, L. M. Shao, and G. W. Zhao, “Extraction of extended X-ray absorption fine structure information from the experimental data using the wavelet transform,” Anal. Commun. 35(4), 135–137 (1998). [CrossRef]

20.

D. Chen, X. G. Shao, B. Hu, and Q. D. Su, “A background and noise elimination method for quantitative calibration of near infrared spectra,” Anal. Chim. Acta 511(1), 37–45 (2004). [CrossRef]

21.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992).

22.

A. E. Kramida Y. Ralchenko, J. Reader, and NIST ASD Team, “NIST Atomic Spectra Database (version 5.1)” (National Institute of Standards and Technology, 2013), http://physics.nist.gov/asd.

23.

L. X. Sun and H. B. Yu, “Automatic estimation of varying continuum background emission in laser-induced breakdown spectroscopy,” Spectrochim. Acta, B At. Spectrosc. 64(3), 278–287 (2009). [CrossRef]

OCIS Codes
(100.7410) Image processing : Wavelets
(160.3900) Materials : Metals
(300.6365) Spectroscopy : Spectroscopy, laser induced breakdown

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: March 3, 2014
Revised Manuscript: April 11, 2014
Manuscript Accepted: April 14, 2014
Published: April 21, 2014

Citation
X. H. Zou, L. B. Guo, M. Shen, X. Y. Li, Z. Q. Hao, Q. D. Zeng, Y. F. Lu, Z. M. Wang, and X. Y. Zeng, "Accuracy improvement of quantitative analysis in laser-induced breakdown spectroscopy using modified wavelet transform," Opt. Express 22, 10233-10238 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10233


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References

  1. W. B. Lee, J. Y. Wu, Y. I. Lee, J. Sneddon, “Recent applications of laser-induced breakdown spectrometry: a review of material approaches,” Appl. Spectrosc. Rev. 39(1), 27–97 (2004). [CrossRef]
  2. Z. Wang, T. B. Yuan, Z. Y. Hou, W. D. Zhou, J. D. Lu, H. B. Ding, and X. Y. Zeng, “Laser-induced breakdown spectroscopy in China,” Front. Phys. (2014).
  3. L. B. Guo, Z. Q. Hao, M. Shen, W. Xiong, X. N. He, Z. Q. Xie, M. Gao, X. Y. Li, X. Y. Zeng, Y. F. Lu, “Accuracy improvement of quantitative analysis by spatial confinement in laser-induced breakdown spectroscopy,” Opt. Express 21(15), 18188–18195 (2013). [CrossRef] [PubMed]
  4. P. Fichet, P. Mauchien, J. F. Wagner, C. Moulin, “Quantitative elemental determination in water and oil by laser induced breakdown spectroscopy,” Anal. Chim. Acta 429(2), 269–278 (2001). [CrossRef]
  5. L. Dudragne, P. Adam, J. Amouroux, “Time-resolved laser-induced breakdown spectroscopy: application for qualitative and quantitative detection of fluorine, chlorine, sulfur, and carbon in air,” Appl. Spectrosc. 52(10), 1321–1327 (1998). [CrossRef]
  6. J. P. Singh and S. N. Thakur, Laser-Induced Breakdown Spectroscopy (Elsevier Science, 2007).
  7. Z. Wang, Z. Y. Hou, S. L. Lui, D. Jiang, J. M. Liu, Z. Li, “Utilization of moderate cylindrical confinement for precision improvement of laser-induced breakdown spectroscopy signal,” Opt. Express 20(S6), A1011–A1018 (2012). [CrossRef]
  8. T. Fujimoto, Plasma Spectroscopy (Clarendon, 2004).
  9. X. G. Shao, A. K. M. Leung, F. T. Chau, “Wavelet: a new trend in chemistry,” Acc. Chem. Res. 36(4), 276–283 (2003). [CrossRef] [PubMed]
  10. C. M. Galloway, E. C. Le Ru, P. G. Etchegoin, “An iterative algorithm for background removal in spectroscopy by wavelet transforms,” Appl. Spectrosc. 63(12), 1370–1376 (2009). [CrossRef] [PubMed]
  11. X. G. Ma, Z. X. Zhang, “Application of wavelet transform to background correction in inductively coupled plasma atomic emission spectrometry,” Anal. Chim. Acta 485(2), 233–239 (2003). [CrossRef]
  12. C. X. Ma, X. G. Shao, “Continuous wavelet transform applied to removing the fluctuating background in near-infrared spectra,” J. Chem. Inf. Comput. Sci. 44(3), 907–911 (2004). [CrossRef] [PubMed]
  13. J. Schlenke, L. Hildebrand, J. Moros, J. J. Laserna, “Adaptive approach for variable noise suppression on laser-induced breakdown spectroscopy responses using stationary wavelet transform,” Anal. Chim. Acta 754, 8–19 (2012). [CrossRef] [PubMed]
  14. B. Zhang, L. X. Sun, H. B. Yu, Y. Xin, Z. B. Cong, “Wavelet denoising method for laser-induced breakdown spectroscopy,” J. Anal. At. Spectrom. 28(12), 1884–1893 (2013). [CrossRef]
  15. T. B. Yuan, Z. Wang, Z. Li, W. D. Ni, J. M. Liu, “A partial least squares and wavelet-transform hybrid model to analyze carbon content in coal using Laser-induced breakdown spectroscopy,” Anal. Chim. Acta 807, 29–35 (2014). [CrossRef] [PubMed]
  16. X. G. Shao, W. S. Cai, Z. X. Pan, “Wavelet transform and its applications in high performance liquid chromatography (HPLC) analysis,” Chemom. Intell. Lab. Syst. 45(1–2), 249–256 (1999). [CrossRef]
  17. S. G. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. 11(7), 674–693 (1989). [CrossRef]
  18. S. G. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way (Academic, 2008).
  19. X. G. Shao, L. M. Shao, G. W. Zhao, “Extraction of extended X-ray absorption fine structure information from the experimental data using the wavelet transform,” Anal. Commun. 35(4), 135–137 (1998). [CrossRef]
  20. D. Chen, X. G. Shao, B. Hu, Q. D. Su, “A background and noise elimination method for quantitative calibration of near infrared spectra,” Anal. Chim. Acta 511(1), 37–45 (2004). [CrossRef]
  21. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992).
  22. A. E. Kramida Y. Ralchenko, J. Reader, and NIST ASD Team, “NIST Atomic Spectra Database (version 5.1)” (National Institute of Standards and Technology, 2013), http://physics.nist.gov/asd .
  23. L. X. Sun, H. B. Yu, “Automatic estimation of varying continuum background emission in laser-induced breakdown spectroscopy,” Spectrochim. Acta, B At. Spectrosc. 64(3), 278–287 (2009). [CrossRef]

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