Abstract

Wavelength calibration is an important step in charge-coupled device (CCD) spectrometers. In this paper, an accurate calibration method is proposed. A model of a line profile spectrum is built at the beginning, followed by noise reduction, bandwidth correction, and automatic peak-seeking treatment. Experimental tests are conducted on the USB4000 spectrometer with a mercury-argon calibration light source. Compared with the traditional method, the results show that this wavelength calibration procedure obtains higher accuracy and the deviations are within 0.1 nm.

© 2017 Optical Society of America

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References

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  1. H. K. Fu, Y. L. Liu, T. T. Chen, and C. P. Wang, “The study of spectral correction algorithm of charge-coupled device array spectrometer,” IEEE Trans. Electron. Dev. 61, 3796–3802 (2014).
    [Crossref]
  2. K. Liu and F. Yu, “Accurate wavelength calibration method using system parameters for grating spectrometers,” Opt. Eng. 52, 177–182 (2013).
  3. P. Martinsen, B. Jordan, A. Mcglone, P. Gaastra, and T. Laurie, “Accurate and precise wavelength calibration for wide bandwidth array spectrometers,” Appl. Spectrosc. 62, 1008–1012 (2008).
    [Crossref]
  4. X. Li, G. Zhang, L. Xun, P. Xie, J. Hong, and X. Zheng, “Wavelength calibration of shortwave infrared flat spectroradiometer,” Acta Opt. Sinica 28, 902–906 (2008).
    [Crossref]
  5. T. Sundius, “Computer fitting of Voigt profiles to Raman lines,” J. Raman Spectrosc. 1, 471–488 (1973).
    [Crossref]
  6. Y. Chen and L. Dai, “Automated decomposition algorithm for Raman spectra based on a Voigt line profile model,” Appl. Opt. 55, 4085–4094 (2016).
    [Crossref]
  7. J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis for CCD-based ultraviolet and visible spectrophotometry,” Appl. Opt. 54, 8135–8144 (2015).
    [Crossref]
  8. J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis of a CCD based ultra-violet spectrometry system,” in Optical Sensing and Detection II (SPIE, 2012).
  9. G. Zonios, “Noise and stray light characterization of a compact CCD spectrophotometer used in biomedical applications,” Appl. Opt. 49, 163–169 (2010).
    [Crossref]
  10. Y. Chao, Z. He, and W. Yu, “Comparison of public peak detection algorithms for MALDI mass spectrometry data analysis,” BMC Bioinf. 10, 1–13 (2009).
  11. S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [Crossref]
  12. E. R. Woolliams, R. Baribeau, A. Bialek, and M. G. Cox, “Spectrometer bandwidth correction for generalized bandpass functions,” Metrologia 48, 164–172 (2011).
    [Crossref]
  13. J. Dubrovkin, “Identification of peak positions using second-order derivative spectral and Tikhonov deconvolution method: a comparison study,” Int. J. Emerg. Technol. Comput. Appl. Sci. 10, 192–197 (2014).
  14. K. J. Goodman and J. T. Brenna, “Curve fitting for restoration of accuracy for overlapping peaks in gas chromatography/combustion isotope ratio mass spectrometry,” Anal. Chem. 66, 1294–1301 (1994).
    [Crossref]
  15. Y. F. Bi, Y. Li, and R. E. Zheng, “The symmetric zero-area conversion adaptive peak-seeking method research for LIBS/Raman spectra,” Spectrosc. Spectral Anal. 33, 438–443 (2013).
  16. T. O’Haver, “A pragmatic introduction to signal processing with applications in scientific measurement” (2015), https://terpconnect.umd.edu/~toh/spectrum/ .
  17. Z. Lu and Y. Xing-Hai, “The application of an improved wavelet threshold denoising method in heart sound signal,” in Cross Strait Quad-Regional Radio Science and Wireless Technology Conference (CSQRWC) (2011), pp. 1115–1117.
  18. Z. P. Li, Q. J. Fan, L. M. Chang, and X. H. Yang, “Improved wavelet threshold denoising method for MEMS gyroscope,” in IEEE International Conference on Control and Automation (2014), pp. 530–534.
  19. S. G. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process 9, 1522–1531 (2000).
    [Crossref]
  20. F. Luisier, T. Blu, and M. Unser, “A new SURE approach to image denoising: interscale orthonormal wavelet thresholding,” IEEE Trans. Image Process. 16, 593–606 (2007).
    [Crossref]
  21. USB4000 Fiber Optic Spectrometer Installation and Operation Manual.

2016 (1)

2015 (1)

2014 (2)

H. K. Fu, Y. L. Liu, T. T. Chen, and C. P. Wang, “The study of spectral correction algorithm of charge-coupled device array spectrometer,” IEEE Trans. Electron. Dev. 61, 3796–3802 (2014).
[Crossref]

J. Dubrovkin, “Identification of peak positions using second-order derivative spectral and Tikhonov deconvolution method: a comparison study,” Int. J. Emerg. Technol. Comput. Appl. Sci. 10, 192–197 (2014).

2013 (2)

Y. F. Bi, Y. Li, and R. E. Zheng, “The symmetric zero-area conversion adaptive peak-seeking method research for LIBS/Raman spectra,” Spectrosc. Spectral Anal. 33, 438–443 (2013).

K. Liu and F. Yu, “Accurate wavelength calibration method using system parameters for grating spectrometers,” Opt. Eng. 52, 177–182 (2013).

2011 (1)

E. R. Woolliams, R. Baribeau, A. Bialek, and M. G. Cox, “Spectrometer bandwidth correction for generalized bandpass functions,” Metrologia 48, 164–172 (2011).
[Crossref]

2010 (1)

2009 (1)

Y. Chao, Z. He, and W. Yu, “Comparison of public peak detection algorithms for MALDI mass spectrometry data analysis,” BMC Bioinf. 10, 1–13 (2009).

2008 (2)

P. Martinsen, B. Jordan, A. Mcglone, P. Gaastra, and T. Laurie, “Accurate and precise wavelength calibration for wide bandwidth array spectrometers,” Appl. Spectrosc. 62, 1008–1012 (2008).
[Crossref]

X. Li, G. Zhang, L. Xun, P. Xie, J. Hong, and X. Zheng, “Wavelength calibration of shortwave infrared flat spectroradiometer,” Acta Opt. Sinica 28, 902–906 (2008).
[Crossref]

2007 (1)

F. Luisier, T. Blu, and M. Unser, “A new SURE approach to image denoising: interscale orthonormal wavelet thresholding,” IEEE Trans. Image Process. 16, 593–606 (2007).
[Crossref]

2000 (1)

S. G. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process 9, 1522–1531 (2000).
[Crossref]

1994 (1)

K. J. Goodman and J. T. Brenna, “Curve fitting for restoration of accuracy for overlapping peaks in gas chromatography/combustion isotope ratio mass spectrometry,” Anal. Chem. 66, 1294–1301 (1994).
[Crossref]

1989 (1)

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

1973 (1)

T. Sundius, “Computer fitting of Voigt profiles to Raman lines,” J. Raman Spectrosc. 1, 471–488 (1973).
[Crossref]

Baribeau, R.

E. R. Woolliams, R. Baribeau, A. Bialek, and M. G. Cox, “Spectrometer bandwidth correction for generalized bandpass functions,” Metrologia 48, 164–172 (2011).
[Crossref]

Bi, Y. F.

Y. F. Bi, Y. Li, and R. E. Zheng, “The symmetric zero-area conversion adaptive peak-seeking method research for LIBS/Raman spectra,” Spectrosc. Spectral Anal. 33, 438–443 (2013).

Bialek, A.

E. R. Woolliams, R. Baribeau, A. Bialek, and M. G. Cox, “Spectrometer bandwidth correction for generalized bandpass functions,” Metrologia 48, 164–172 (2011).
[Crossref]

Blu, T.

F. Luisier, T. Blu, and M. Unser, “A new SURE approach to image denoising: interscale orthonormal wavelet thresholding,” IEEE Trans. Image Process. 16, 593–606 (2007).
[Crossref]

Brenna, J. T.

K. J. Goodman and J. T. Brenna, “Curve fitting for restoration of accuracy for overlapping peaks in gas chromatography/combustion isotope ratio mass spectrometry,” Anal. Chem. 66, 1294–1301 (1994).
[Crossref]

Chang, L. M.

Z. P. Li, Q. J. Fan, L. M. Chang, and X. H. Yang, “Improved wavelet threshold denoising method for MEMS gyroscope,” in IEEE International Conference on Control and Automation (2014), pp. 530–534.

Chang, S. G.

S. G. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process 9, 1522–1531 (2000).
[Crossref]

Chao, Y.

Y. Chao, Z. He, and W. Yu, “Comparison of public peak detection algorithms for MALDI mass spectrometry data analysis,” BMC Bioinf. 10, 1–13 (2009).

Chen, T. T.

H. K. Fu, Y. L. Liu, T. T. Chen, and C. P. Wang, “The study of spectral correction algorithm of charge-coupled device array spectrometer,” IEEE Trans. Electron. Dev. 61, 3796–3802 (2014).
[Crossref]

Chen, Y.

Cox, M. G.

E. R. Woolliams, R. Baribeau, A. Bialek, and M. G. Cox, “Spectrometer bandwidth correction for generalized bandpass functions,” Metrologia 48, 164–172 (2011).
[Crossref]

Dai, L.

Davenport, J. J.

J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis for CCD-based ultraviolet and visible spectrophotometry,” Appl. Opt. 54, 8135–8144 (2015).
[Crossref]

J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis of a CCD based ultra-violet spectrometry system,” in Optical Sensing and Detection II (SPIE, 2012).

Dubrovkin, J.

J. Dubrovkin, “Identification of peak positions using second-order derivative spectral and Tikhonov deconvolution method: a comparison study,” Int. J. Emerg. Technol. Comput. Appl. Sci. 10, 192–197 (2014).

Fan, Q. J.

Z. P. Li, Q. J. Fan, L. M. Chang, and X. H. Yang, “Improved wavelet threshold denoising method for MEMS gyroscope,” in IEEE International Conference on Control and Automation (2014), pp. 530–534.

Fu, H. K.

H. K. Fu, Y. L. Liu, T. T. Chen, and C. P. Wang, “The study of spectral correction algorithm of charge-coupled device array spectrometer,” IEEE Trans. Electron. Dev. 61, 3796–3802 (2014).
[Crossref]

Gaastra, P.

Goodman, K. J.

K. J. Goodman and J. T. Brenna, “Curve fitting for restoration of accuracy for overlapping peaks in gas chromatography/combustion isotope ratio mass spectrometry,” Anal. Chem. 66, 1294–1301 (1994).
[Crossref]

He, Z.

Y. Chao, Z. He, and W. Yu, “Comparison of public peak detection algorithms for MALDI mass spectrometry data analysis,” BMC Bioinf. 10, 1–13 (2009).

Hodgkinson, J.

J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis for CCD-based ultraviolet and visible spectrophotometry,” Appl. Opt. 54, 8135–8144 (2015).
[Crossref]

J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis of a CCD based ultra-violet spectrometry system,” in Optical Sensing and Detection II (SPIE, 2012).

Hong, J.

X. Li, G. Zhang, L. Xun, P. Xie, J. Hong, and X. Zheng, “Wavelength calibration of shortwave infrared flat spectroradiometer,” Acta Opt. Sinica 28, 902–906 (2008).
[Crossref]

Jordan, B.

Laurie, T.

Li, X.

X. Li, G. Zhang, L. Xun, P. Xie, J. Hong, and X. Zheng, “Wavelength calibration of shortwave infrared flat spectroradiometer,” Acta Opt. Sinica 28, 902–906 (2008).
[Crossref]

Li, Y.

Y. F. Bi, Y. Li, and R. E. Zheng, “The symmetric zero-area conversion adaptive peak-seeking method research for LIBS/Raman spectra,” Spectrosc. Spectral Anal. 33, 438–443 (2013).

Li, Z. P.

Z. P. Li, Q. J. Fan, L. M. Chang, and X. H. Yang, “Improved wavelet threshold denoising method for MEMS gyroscope,” in IEEE International Conference on Control and Automation (2014), pp. 530–534.

Liu, K.

K. Liu and F. Yu, “Accurate wavelength calibration method using system parameters for grating spectrometers,” Opt. Eng. 52, 177–182 (2013).

Liu, Y. L.

H. K. Fu, Y. L. Liu, T. T. Chen, and C. P. Wang, “The study of spectral correction algorithm of charge-coupled device array spectrometer,” IEEE Trans. Electron. Dev. 61, 3796–3802 (2014).
[Crossref]

Lu, Z.

Z. Lu and Y. Xing-Hai, “The application of an improved wavelet threshold denoising method in heart sound signal,” in Cross Strait Quad-Regional Radio Science and Wireless Technology Conference (CSQRWC) (2011), pp. 1115–1117.

Luisier, F.

F. Luisier, T. Blu, and M. Unser, “A new SURE approach to image denoising: interscale orthonormal wavelet thresholding,” IEEE Trans. Image Process. 16, 593–606 (2007).
[Crossref]

Mallat, S. G.

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

Martinsen, P.

Mcglone, A.

Saffell, J. R.

J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis for CCD-based ultraviolet and visible spectrophotometry,” Appl. Opt. 54, 8135–8144 (2015).
[Crossref]

J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis of a CCD based ultra-violet spectrometry system,” in Optical Sensing and Detection II (SPIE, 2012).

Sundius, T.

T. Sundius, “Computer fitting of Voigt profiles to Raman lines,” J. Raman Spectrosc. 1, 471–488 (1973).
[Crossref]

Tatam, R. P.

J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis for CCD-based ultraviolet and visible spectrophotometry,” Appl. Opt. 54, 8135–8144 (2015).
[Crossref]

J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis of a CCD based ultra-violet spectrometry system,” in Optical Sensing and Detection II (SPIE, 2012).

Unser, M.

F. Luisier, T. Blu, and M. Unser, “A new SURE approach to image denoising: interscale orthonormal wavelet thresholding,” IEEE Trans. Image Process. 16, 593–606 (2007).
[Crossref]

Vetterli, M.

S. G. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process 9, 1522–1531 (2000).
[Crossref]

Wang, C. P.

H. K. Fu, Y. L. Liu, T. T. Chen, and C. P. Wang, “The study of spectral correction algorithm of charge-coupled device array spectrometer,” IEEE Trans. Electron. Dev. 61, 3796–3802 (2014).
[Crossref]

Woolliams, E. R.

E. R. Woolliams, R. Baribeau, A. Bialek, and M. G. Cox, “Spectrometer bandwidth correction for generalized bandpass functions,” Metrologia 48, 164–172 (2011).
[Crossref]

Xie, P.

X. Li, G. Zhang, L. Xun, P. Xie, J. Hong, and X. Zheng, “Wavelength calibration of shortwave infrared flat spectroradiometer,” Acta Opt. Sinica 28, 902–906 (2008).
[Crossref]

Xing-Hai, Y.

Z. Lu and Y. Xing-Hai, “The application of an improved wavelet threshold denoising method in heart sound signal,” in Cross Strait Quad-Regional Radio Science and Wireless Technology Conference (CSQRWC) (2011), pp. 1115–1117.

Xun, L.

X. Li, G. Zhang, L. Xun, P. Xie, J. Hong, and X. Zheng, “Wavelength calibration of shortwave infrared flat spectroradiometer,” Acta Opt. Sinica 28, 902–906 (2008).
[Crossref]

Yang, X. H.

Z. P. Li, Q. J. Fan, L. M. Chang, and X. H. Yang, “Improved wavelet threshold denoising method for MEMS gyroscope,” in IEEE International Conference on Control and Automation (2014), pp. 530–534.

Yu, B.

S. G. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process 9, 1522–1531 (2000).
[Crossref]

Yu, F.

K. Liu and F. Yu, “Accurate wavelength calibration method using system parameters for grating spectrometers,” Opt. Eng. 52, 177–182 (2013).

Yu, W.

Y. Chao, Z. He, and W. Yu, “Comparison of public peak detection algorithms for MALDI mass spectrometry data analysis,” BMC Bioinf. 10, 1–13 (2009).

Zhang, G.

X. Li, G. Zhang, L. Xun, P. Xie, J. Hong, and X. Zheng, “Wavelength calibration of shortwave infrared flat spectroradiometer,” Acta Opt. Sinica 28, 902–906 (2008).
[Crossref]

Zheng, R. E.

Y. F. Bi, Y. Li, and R. E. Zheng, “The symmetric zero-area conversion adaptive peak-seeking method research for LIBS/Raman spectra,” Spectrosc. Spectral Anal. 33, 438–443 (2013).

Zheng, X.

X. Li, G. Zhang, L. Xun, P. Xie, J. Hong, and X. Zheng, “Wavelength calibration of shortwave infrared flat spectroradiometer,” Acta Opt. Sinica 28, 902–906 (2008).
[Crossref]

Zonios, G.

Acta Opt. Sinica (1)

X. Li, G. Zhang, L. Xun, P. Xie, J. Hong, and X. Zheng, “Wavelength calibration of shortwave infrared flat spectroradiometer,” Acta Opt. Sinica 28, 902–906 (2008).
[Crossref]

Anal. Chem. (1)

K. J. Goodman and J. T. Brenna, “Curve fitting for restoration of accuracy for overlapping peaks in gas chromatography/combustion isotope ratio mass spectrometry,” Anal. Chem. 66, 1294–1301 (1994).
[Crossref]

Appl. Opt. (3)

Appl. Spectrosc. (1)

BMC Bioinf. (1)

Y. Chao, Z. He, and W. Yu, “Comparison of public peak detection algorithms for MALDI mass spectrometry data analysis,” BMC Bioinf. 10, 1–13 (2009).

IEEE Trans. Electron. Dev. (1)

H. K. Fu, Y. L. Liu, T. T. Chen, and C. P. Wang, “The study of spectral correction algorithm of charge-coupled device array spectrometer,” IEEE Trans. Electron. Dev. 61, 3796–3802 (2014).
[Crossref]

IEEE Trans. Image Process (1)

S. G. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Image Process 9, 1522–1531 (2000).
[Crossref]

IEEE Trans. Image Process. (1)

F. Luisier, T. Blu, and M. Unser, “A new SURE approach to image denoising: interscale orthonormal wavelet thresholding,” IEEE Trans. Image Process. 16, 593–606 (2007).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

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

Int. J. Emerg. Technol. Comput. Appl. Sci. (1)

J. Dubrovkin, “Identification of peak positions using second-order derivative spectral and Tikhonov deconvolution method: a comparison study,” Int. J. Emerg. Technol. Comput. Appl. Sci. 10, 192–197 (2014).

J. Raman Spectrosc. (1)

T. Sundius, “Computer fitting of Voigt profiles to Raman lines,” J. Raman Spectrosc. 1, 471–488 (1973).
[Crossref]

Metrologia (1)

E. R. Woolliams, R. Baribeau, A. Bialek, and M. G. Cox, “Spectrometer bandwidth correction for generalized bandpass functions,” Metrologia 48, 164–172 (2011).
[Crossref]

Opt. Eng. (1)

K. Liu and F. Yu, “Accurate wavelength calibration method using system parameters for grating spectrometers,” Opt. Eng. 52, 177–182 (2013).

Spectrosc. Spectral Anal. (1)

Y. F. Bi, Y. Li, and R. E. Zheng, “The symmetric zero-area conversion adaptive peak-seeking method research for LIBS/Raman spectra,” Spectrosc. Spectral Anal. 33, 438–443 (2013).

Other (5)

T. O’Haver, “A pragmatic introduction to signal processing with applications in scientific measurement” (2015), https://terpconnect.umd.edu/~toh/spectrum/ .

Z. Lu and Y. Xing-Hai, “The application of an improved wavelet threshold denoising method in heart sound signal,” in Cross Strait Quad-Regional Radio Science and Wireless Technology Conference (CSQRWC) (2011), pp. 1115–1117.

Z. P. Li, Q. J. Fan, L. M. Chang, and X. H. Yang, “Improved wavelet threshold denoising method for MEMS gyroscope,” in IEEE International Conference on Control and Automation (2014), pp. 530–534.

J. J. Davenport, J. Hodgkinson, J. R. Saffell, and R. P. Tatam, “Noise analysis of a CCD based ultra-violet spectrometry system,” in Optical Sensing and Detection II (SPIE, 2012).

USB4000 Fiber Optic Spectrometer Installation and Operation Manual.

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Figures (11)

Fig. 1.
Fig. 1.

Schematic diagram of the proposed calibration method.

Fig. 2.
Fig. 2.

Calibration spectrum of mercury-argon calibration light source. The wavelength of characteristic peaks varies from 253.652 to 842.465 nm.

Fig. 3.
Fig. 3.

Result of noise reduction using different denoising methods.

Fig. 4.
Fig. 4.

Simulated model with 27 characteristic peaks. The model mixed with the model of noise is denoised to determine the optimal noise reduction algorithm.

Fig. 5.
Fig. 5.

Corresponding SNR of 21 kinds of wavelet basis and four types of soft thresholding shrinkage functions.

Fig. 6.
Fig. 6.

Normalized response of four spectral lines.

Fig. 7.
Fig. 7.

Measured spectrum and bandwidth corrected spectrum. The corresponding wavelengths of spectral lines are (a) 313.155 nm, (b) 576.960 and 579.066 nm, and (c) 810.369 and 811.531 nm.

Fig. 8.
Fig. 8.

(a) Result of automatic peak seeking without bandwidth correction. The wavelengths of the double peaks are 810.369 and 811.531 nm. (b) Result of automatic peak seeking using three-point differential operator method for bandwidth correction.

Fig. 9.
Fig. 9.

Peaks marked with numbers are found by using automatic peak-seeking algorithm. The small figure at the upper-right side shows the details of the curve fitting process.

Fig. 10.
Fig. 10.

Typical relationship between pixel number and standard wavelength.

Fig. 11.
Fig. 11.

Calibration deviations using proposed calibration and traditional calibration method.

Tables (3)

Tables Icon

Table 1. Pixel Position of the Characteristic Peaks Using the Proposed Calibration Method

Tables Icon

Table 2. Pixel Position of the Characteristic Peaks Using the Traditional Calibration Method

Tables Icon

Table 3. Parameters of Experiment

Equations (17)

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

V(σ)=L(σ)*G(σ).
L(σ)=1πγL[(σcγL)2+1],
G(σ)=12πcex22c2.
G(σ)=ln2πγGeσ2ln2γG2,
V(σ)=G(σ)*L(σ)=+L(σ)G(vσ)dσ=+ln2π32γLγG·1(σcγL)2+1eσ2ln2γG2dσ.
S(σ)=L(σ)*G(σ)+N(σ),
Ne2=Re2+NSe2+NDe2+NFPNe2,
Re=std(Sbias),
NSe=neg,
NDe=std(SD),
y[n]=12k+1i=kkx[ni],
f(t)=kZcJ,kφJ,k(t)+j=1JkZdj,kψj,k(t),
SNR=10log10(i=1Ny12(xiy2)2),
Scorr=[I12δ2+I12δI22δ2]M1+[12I12δ2+I2δ2]M0+[I12δ2I12δI22δ2]M1,
In=λnb(λ)dλ.
pj=f(di,xa,xs,xf),
λ=a3x3+a2x2+a1x+a0.

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