Abstract

An FTIR spectrometer often suffers from common problems of band overlap and Poisson noises. In this paper, we show that the issue of infrared (IR) spectrum degradation can be considered as a maximum a posterior (MAP) problem and solved by minimized a cost function that includes a likelihood term and two prior terms. In the MAP framework, the likelihood probability density function (PDF) is constructed based on the observed Poisson noise model. A fitted distribution of curvelet transform coefficient is used as spectral prior PDF, and the instrument response function (IRF) prior is described based on a Gauss-Markov function. Moreover, the split Bregman iteration method is employed to solve the resulting minimization problem, which highly reduces the computational load. As a result, the Poisson noises are perfectly removed, while the spectral structure information is well preserved. The novelty of the proposed method lies in its ability to estimate the IRF and latent spectrum in a joint framework, thus eliminating the degradation effects to a large extent. The reconstructed IR spectrum is more convenient for extracting the spectral feature and interpreting the unknown chemical or biological materials.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (3)

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

T. Liu, H. Liu, Z. Chen, and A. M. Lesgold, "Fast Blind Instrument Function Estimation Method for Industrial Infrared Spectrometers," IEEE Trans. Ind. Inf. 279, 4449 (2018).
[Crossref]

H. Zhu, L. Deng, G. Xu, Y. Chen, and Y. Li, “Spectral semi-blind deconvolution methods based on modified φHS regularizations,” Opt. Laser Technol. 1, 46 (2018).
[Crossref]

2017 (4)

S. Hugelier, P. H. C. Eilers, O. Devos, and C. Ruckebusch, “Improved superresolution microscopy imaging by sparse deconvolution with an interframe penalty,” J. Chemom.  31, e2847 (2017).

Q. Han, Q. Xie, S. Peng, and B. Guo, “Simultaneous spectrum fitting and baseline correction using sparse representation,” Analyst (Lond.) 142(13), 2460–2468 (2017).
[Crossref] [PubMed]

S. Chen, G. Wang, X. Cui, and Q. Liu, “Stepwise method based on Wiener estimation for spectral reconstruction in spectroscopic Raman imaging,” Opt. Express 25(2), 1005–1018 (2017).
[Crossref] [PubMed]

N. P. Ayerden and R. F. Wolffenbuttel, “The Miniaturization of an Optical Absorption Spectrometer for Smart Sensing of Natural Gas,” IEEE Trans. Ind. Electron. 64(12), 9666–9674 (2017), doi:.
[Crossref]

2015 (3)

2014 (1)

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

2013 (1)

F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, and S. Katsura, “Industrial Applications of the Kalman Filter: A Review,” IEEE Trans. Ind. Electron. 60(12), 5458–5471 (2013).
[Crossref]

2012 (2)

L. Yan, H. Liu, S. Zhong, and H. Fang, “Semi-blind spectral deconvolution with adaptive Tikhonov regularization,” Appl. Spectrosc. 66(11), 1334–1346 (2012).
[Crossref] [PubMed]

H. Liu, T. Zhang, L. Yan, H. Fang, and Y. Chang, “A MAP-based algorithm for spectroscopic semi-blind deconvolution,” Analyst (Lond.) 137(16), 3862–3873 (2012).
[Crossref] [PubMed]

2010 (2)

A. Mukherjee and A. Sengupta, “Estimating the Probability Density Function of a Nonstationary Non-Gaussian Noise,” IEEE Trans. Ind. Electron. 57(4), 1429–1435 (2010).
[Crossref]

J. Katrašnik, F. Pernuš, and B. Likar, “Deconvolution in Acousto-Optical Tunable Filter Spectrometry,” Appl. Spectrosc. 64(11), 1265–1273 (2010).
[Crossref] [PubMed]

2009 (2)

2006 (2)

2005 (1)

2002 (2)

L. Shao, X. Lin, and X. Shao, “A wavelet transform and its application to spectroscopic analysis,” Appl. Spectrosc. Rev. 37(4), 429–450 (2002).
[Crossref]

J. L. Starck, E. J. Candès, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11(6), 670–684 (2002).
[Crossref] [PubMed]

1998 (2)

T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7(3), 370–375 (1998).
[Crossref] [PubMed]

S. Sarkar, P. K. Dutta, and N. C. Roy, “A blind-deconvolution approach for chromatographic and spectroscopic peak restoration,” IEE Trans. Instrumentation and Measurement 47(4), 941–947 (1998).
[Crossref]

1997 (2)

M. B. Slima, R. Z. Morawski, and A. Barwicz, “Kalman-filter-based algorithms of spectrophotometric data correction III. Use of splines for approximation of spectra,” IEEE Trans. Instrum. Meas. 46(3), 685–689 (1997).
[Crossref]

C. J. Manning and P. R. Griffiths, “Noise Sources in Step-Scan FT-IR Spectrometry,” Appl. Spectrosc. 51(8), 1092–1101 (1997).
[Crossref]

1996 (1)

A. Economou, P. R. Fielden, and A. J. Packham, “Deconvolution of analytical peaks by means of the fast Hartley transform,” Analyst (Lond.) 121(8), 1015–1018 (1996).
[Crossref]

1993 (1)

W. E. Snyder, M. L. Hsiao, and J. N. Campbell, “Restoration of ultrasonic NDE images,” IEEE Trans. Ind. Electron. 40(2), 250–258 (1993).
[Crossref]

1992 (1)

P. B. Crilly, “Increased throughput for process chromatography using constrained deconvolution,” IEEE Trans. Ind. Electron. 39(1), 20–24 (1992).
[Crossref]

1984 (1)

1981 (1)

1977 (1)

1967 (1)

Addison, C. J.

Auger, F.

F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, and S. Katsura, “Industrial Applications of the Kalman Filter: A Review,” IEEE Trans. Ind. Electron. 60(12), 5458–5471 (2013).
[Crossref]

Ayerden, N. P.

N. P. Ayerden and R. F. Wolffenbuttel, “The Miniaturization of an Optical Absorption Spectrometer for Smart Sensing of Natural Gas,” IEEE Trans. Ind. Electron. 64(12), 9666–9674 (2017), doi:.
[Crossref]

Baker, M. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Barwicz, A.

M. B. Slima, R. Z. Morawski, and A. Barwicz, “Kalman-filter-based algorithms of spectrophotometric data correction III. Use of splines for approximation of spectra,” IEEE Trans. Instrum. Meas. 46(3), 685–689 (1997).
[Crossref]

Bassan, P.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Baumann, E.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Bhargava, R.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Blades, M. W.

Butler, H. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Cameron, D. G.

Campbell, J. N.

W. E. Snyder, M. L. Hsiao, and J. N. Campbell, “Restoration of ultrasonic NDE images,” IEEE Trans. Ind. Electron. 40(2), 250–258 (1993).
[Crossref]

Candès, E.

E. Candès, L. Demanet, D. Donoho, and L. Ying, “Fast Discrete Curvelet Transforms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

Candès, E. J.

J. L. Starck, E. J. Candès, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11(6), 670–684 (2002).
[Crossref] [PubMed]

Chan, T. F.

T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7(3), 370–375 (1998).
[Crossref] [PubMed]

Chang, Y.

H. Liu, T. Zhang, L. Yan, H. Fang, and Y. Chang, “A MAP-based algorithm for spectroscopic semi-blind deconvolution,” Analyst (Lond.) 137(16), 3862–3873 (2012).
[Crossref] [PubMed]

Chen, K.

K. Chen, T. Wu, H. Wei, X. Wu, and Y. Li, “High spectral specificity of local chemical components characterization with multichannel shift-excitation Raman spectroscopy,” Sci. Rep. 5(1), 13952 (2015).
[Crossref] [PubMed]

Chen, S.

Chen, Y.

H. Zhu, L. Deng, G. Xu, Y. Chen, and Y. Li, “Spectral semi-blind deconvolution methods based on modified φHS regularizations,” Opt. Laser Technol. 1, 46 (2018).
[Crossref]

Chen, Z.

T. Liu, H. Liu, Z. Chen, and A. M. Lesgold, "Fast Blind Instrument Function Estimation Method for Industrial Infrared Spectrometers," IEEE Trans. Ind. Inf. 279, 4449 (2018).
[Crossref]

Coddington, I.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Crilly, P. B.

P. B. Crilly, “Increased throughput for process chromatography using constrained deconvolution,” IEEE Trans. Ind. Electron. 39(1), 20–24 (1992).
[Crossref]

Cui, X.

Demanet, L.

E. Candès, L. Demanet, D. Donoho, and L. Ying, “Fast Discrete Curvelet Transforms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

Deng, L.

H. Zhu, L. Deng, G. Xu, Y. Chen, and Y. Li, “Spectral semi-blind deconvolution methods based on modified φHS regularizations,” Opt. Laser Technol. 1, 46 (2018).
[Crossref]

Devos, O.

S. Hugelier, P. H. C. Eilers, O. Devos, and C. Ruckebusch, “Improved superresolution microscopy imaging by sparse deconvolution with an interframe penalty,” J. Chemom.  31, e2847 (2017).

Diddams, S. A.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Donoho, D.

E. Candès, L. Demanet, D. Donoho, and L. Ying, “Fast Discrete Curvelet Transforms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

Donoho, D. L.

J. L. Starck, E. J. Candès, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11(6), 670–684 (2002).
[Crossref] [PubMed]

Dorling, K. M.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Dutta, P. K.

S. Sarkar, P. K. Dutta, and N. C. Roy, “A blind-deconvolution approach for chromatographic and spectroscopic peak restoration,” IEE Trans. Instrumentation and Measurement 47(4), 941–947 (1998).
[Crossref]

Economou, A.

A. Economou, P. R. Fielden, and A. J. Packham, “Deconvolution of analytical peaks by means of the fast Hartley transform,” Analyst (Lond.) 121(8), 1015–1018 (1996).
[Crossref]

Eilers, P. H. C.

S. Hugelier, P. H. C. Eilers, O. Devos, and C. Ruckebusch, “Improved superresolution microscopy imaging by sparse deconvolution with an interframe penalty,” J. Chemom.  31, e2847 (2017).

Fang, H.

H. Liu, T. Zhang, L. Yan, H. Fang, and Y. Chang, “A MAP-based algorithm for spectroscopic semi-blind deconvolution,” Analyst (Lond.) 137(16), 3862–3873 (2012).
[Crossref] [PubMed]

L. Yan, H. Liu, S. Zhong, and H. Fang, “Semi-blind spectral deconvolution with adaptive Tikhonov regularization,” Appl. Spectrosc. 66(11), 1334–1346 (2012).
[Crossref] [PubMed]

Fielden, P. R.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

A. Economou, P. R. Fielden, and A. J. Packham, “Deconvolution of analytical peaks by means of the fast Hartley transform,” Analyst (Lond.) 121(8), 1015–1018 (1996).
[Crossref]

Fogarty, S. W.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Fullwood, N. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Gardner, P.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Ghosh, A.

Giorgetta, F. R.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Goldstein, T.

T. Goldstein and S. Osher, “The Split Bregman Method for L1-Regularized Problems,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
[Crossref]

Griffiths, P. R.

Guerrero, J. M.

F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, and S. Katsura, “Industrial Applications of the Kalman Filter: A Review,” IEEE Trans. Ind. Electron. 60(12), 5458–5471 (2013).
[Crossref]

Guo, B.

Q. Han, Q. Xie, S. Peng, and B. Guo, “Simultaneous spectrum fitting and baseline correction using sparse representation,” Analyst (Lond.) 142(13), 2460–2468 (2017).
[Crossref] [PubMed]

Han, Q.

Q. Han, Q. Xie, S. Peng, and B. Guo, “Simultaneous spectrum fitting and baseline correction using sparse representation,” Analyst (Lond.) 142(13), 2460–2468 (2017).
[Crossref] [PubMed]

Helstrom, C. W.

Herman, D.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Heys, K. A.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Hilairet, M.

F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, and S. Katsura, “Industrial Applications of the Kalman Filter: A Review,” IEEE Trans. Ind. Electron. 60(12), 5458–5471 (2013).
[Crossref]

Hsiao, M. L.

W. E. Snyder, M. L. Hsiao, and J. N. Campbell, “Restoration of ultrasonic NDE images,” IEEE Trans. Ind. Electron. 40(2), 250–258 (1993).
[Crossref]

Hu, Z.

Hugelier, S.

S. Hugelier, P. H. C. Eilers, O. Devos, and C. Ruckebusch, “Improved superresolution microscopy imaging by sparse deconvolution with an interframe penalty,” J. Chemom.  31, e2847 (2017).

Hughes, C.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Ichioka, Y.

Katrašnik, J.

Katsura, S.

F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, and S. Katsura, “Industrial Applications of the Kalman Filter: A Review,” IEEE Trans. Ind. Electron. 60(12), 5458–5471 (2013).
[Crossref]

Kauppinen, J. K.

Kawata, S.

Kondo, K.

Lasch, P.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Lesgold, A. M.

T. Liu, H. Liu, Z. Chen, and A. M. Lesgold, "Fast Blind Instrument Function Estimation Method for Industrial Infrared Spectrometers," IEEE Trans. Ind. Inf. 279, 4449 (2018).
[Crossref]

Li, Y.

H. Zhu, L. Deng, G. Xu, Y. Chen, and Y. Li, “Spectral semi-blind deconvolution methods based on modified φHS regularizations,” Opt. Laser Technol. 1, 46 (2018).
[Crossref]

K. Chen, T. Wu, H. Wei, X. Wu, and Y. Li, “High spectral specificity of local chemical components characterization with multichannel shift-excitation Raman spectroscopy,” Sci. Rep. 5(1), 13952 (2015).
[Crossref] [PubMed]

Likar, B.

Lin, X.

L. Shao, X. Lin, and X. Shao, “A wavelet transform and its application to spectroscopic analysis,” Appl. Spectrosc. Rev. 37(4), 429–450 (2002).
[Crossref]

Liu, H.

T. Liu, H. Liu, Z. Chen, and A. M. Lesgold, "Fast Blind Instrument Function Estimation Method for Industrial Infrared Spectrometers," IEEE Trans. Ind. Inf. 279, 4449 (2018).
[Crossref]

H. Liu, Z. Zhang, S. Liu, T. Liu, L. Yan, and T. Zhang, “Richardson-Lucy blind deconvolution of spectroscopic data with wavelet regularization,” Appl. Opt. 54(7), 1770–1775 (2015).
[Crossref]

L. Yan, H. Liu, S. Zhong, and H. Fang, “Semi-blind spectral deconvolution with adaptive Tikhonov regularization,” Appl. Spectrosc. 66(11), 1334–1346 (2012).
[Crossref] [PubMed]

H. Liu, T. Zhang, L. Yan, H. Fang, and Y. Chang, “A MAP-based algorithm for spectroscopic semi-blind deconvolution,” Analyst (Lond.) 137(16), 3862–3873 (2012).
[Crossref] [PubMed]

Liu, Q.

Liu, S.

Liu, T.

T. Liu, H. Liu, Z. Chen, and A. M. Lesgold, "Fast Blind Instrument Function Estimation Method for Industrial Infrared Spectrometers," IEEE Trans. Ind. Inf. 279, 4449 (2018).
[Crossref]

H. Liu, Z. Zhang, S. Liu, T. Liu, L. Yan, and T. Zhang, “Richardson-Lucy blind deconvolution of spectroscopic data with wavelet regularization,” Appl. Opt. 54(7), 1770–1775 (2015).
[Crossref]

Lórenz-Fonfría, V. A.

Manning, C. J.

Mantsch, H. H.

Martin, F. L.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Martin-Hirsch, P. L.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Minami, K.

Minami, S.

Moffatt, D. J.

Monmasson, E.

F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, and S. Katsura, “Industrial Applications of the Kalman Filter: A Review,” IEEE Trans. Ind. Electron. 60(12), 5458–5471 (2013).
[Crossref]

Morawski, R. Z.

M. B. Slima, R. Z. Morawski, and A. Barwicz, “Kalman-filter-based algorithms of spectrophotometric data correction III. Use of splines for approximation of spectra,” IEEE Trans. Instrum. Meas. 46(3), 685–689 (1997).
[Crossref]

Mukherjee, A.

A. Mukherjee and A. Sengupta, “Estimating the Probability Density Function of a Nonstationary Non-Gaussian Noise,” IEEE Trans. Ind. Electron. 57(4), 1429–1435 (2010).
[Crossref]

Newbury, N. R.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Obinaju, B.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Orlowska-Kowalska, T.

F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, and S. Katsura, “Industrial Applications of the Kalman Filter: A Review,” IEEE Trans. Ind. Electron. 60(12), 5458–5471 (2013).
[Crossref]

Osher, S.

T. Goldstein and S. Osher, “The Split Bregman Method for L1-Regularized Problems,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
[Crossref]

Ostrander, J. S.

Packham, A. J.

A. Economou, P. R. Fielden, and A. J. Packham, “Deconvolution of analytical peaks by means of the fast Hartley transform,” Analyst (Lond.) 121(8), 1015–1018 (1996).
[Crossref]

Padrós, E.

Peng, S.

Q. Han, Q. Xie, S. Peng, and B. Guo, “Simultaneous spectrum fitting and baseline correction using sparse representation,” Analyst (Lond.) 142(13), 2460–2468 (2017).
[Crossref] [PubMed]

Pernuš, F.

Roy, N. C.

S. Sarkar, P. K. Dutta, and N. C. Roy, “A blind-deconvolution approach for chromatographic and spectroscopic peak restoration,” IEE Trans. Instrumentation and Measurement 47(4), 941–947 (1998).
[Crossref]

Ruckebusch, C.

S. Hugelier, P. H. C. Eilers, O. Devos, and C. Ruckebusch, “Improved superresolution microscopy imaging by sparse deconvolution with an interframe penalty,” J. Chemom.  31, e2847 (2017).

Sarkar, S.

S. Sarkar, P. K. Dutta, and N. C. Roy, “A blind-deconvolution approach for chromatographic and spectroscopic peak restoration,” IEE Trans. Instrumentation and Measurement 47(4), 941–947 (1998).
[Crossref]

Schulze, H. G.

Senga, Y.

Sengupta, A.

A. Mukherjee and A. Sengupta, “Estimating the Probability Density Function of a Nonstationary Non-Gaussian Noise,” IEEE Trans. Ind. Electron. 57(4), 1429–1435 (2010).
[Crossref]

Serrano, A. L.

Shao, L.

L. Shao, X. Lin, and X. Shao, “A wavelet transform and its application to spectroscopic analysis,” Appl. Spectrosc. Rev. 37(4), 429–450 (2002).
[Crossref]

Shao, X.

L. Shao, X. Lin, and X. Shao, “A wavelet transform and its application to spectroscopic analysis,” Appl. Spectrosc. Rev. 37(4), 429–450 (2002).
[Crossref]

Slima, M. B.

M. B. Slima, R. Z. Morawski, and A. Barwicz, “Kalman-filter-based algorithms of spectrophotometric data correction III. Use of splines for approximation of spectra,” IEEE Trans. Instrum. Meas. 46(3), 685–689 (1997).
[Crossref]

Snyder, W. E.

W. E. Snyder, M. L. Hsiao, and J. N. Campbell, “Restoration of ultrasonic NDE images,” IEEE Trans. Ind. Electron. 40(2), 250–258 (1993).
[Crossref]

Sockalingum, G. D.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Starck, J. L.

J. L. Starck, E. J. Candès, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11(6), 670–684 (2002).
[Crossref] [PubMed]

Strong, R. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Sulé-Suso, J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Sun, J.

Suzuki, T.

Trevisan, J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Turner, R. F. B.

Walsh, M. J.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Wang, G.

Wei, H.

K. Chen, T. Wu, H. Wei, X. Wu, and Y. Li, “High spectral specificity of local chemical components characterization with multichannel shift-excitation Raman spectroscopy,” Sci. Rep. 5(1), 13952 (2015).
[Crossref] [PubMed]

Wolffenbuttel, R. F.

N. P. Ayerden and R. F. Wolffenbuttel, “The Miniaturization of an Optical Absorption Spectrometer for Smart Sensing of Natural Gas,” IEEE Trans. Ind. Electron. 64(12), 9666–9674 (2017), doi:.
[Crossref]

Wong, C. K.

T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7(3), 370–375 (1998).
[Crossref] [PubMed]

Wood, B. R.

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Wu, T.

K. Chen, T. Wu, H. Wei, X. Wu, and Y. Li, “High spectral specificity of local chemical components characterization with multichannel shift-excitation Raman spectroscopy,” Sci. Rep. 5(1), 13952 (2015).
[Crossref] [PubMed]

Wu, X.

K. Chen, T. Wu, H. Wei, X. Wu, and Y. Li, “High spectral specificity of local chemical components characterization with multichannel shift-excitation Raman spectroscopy,” Sci. Rep. 5(1), 13952 (2015).
[Crossref] [PubMed]

Xie, Q.

Q. Han, Q. Xie, S. Peng, and B. Guo, “Simultaneous spectrum fitting and baseline correction using sparse representation,” Analyst (Lond.) 142(13), 2460–2468 (2017).
[Crossref] [PubMed]

Xu, G.

H. Zhu, L. Deng, G. Xu, Y. Chen, and Y. Li, “Spectral semi-blind deconvolution methods based on modified φHS regularizations,” Opt. Laser Technol. 1, 46 (2018).
[Crossref]

Yan, L.

Ycas, G.

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Ying, L.

E. Candès, L. Demanet, D. Donoho, and L. Ying, “Fast Discrete Curvelet Transforms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

Yu, M. M. L.

Yuan, J.

Zanni, M. T.

Zhang, T.

H. Liu, Z. Zhang, S. Liu, T. Liu, L. Yan, and T. Zhang, “Richardson-Lucy blind deconvolution of spectroscopic data with wavelet regularization,” Appl. Opt. 54(7), 1770–1775 (2015).
[Crossref]

H. Liu, T. Zhang, L. Yan, H. Fang, and Y. Chang, “A MAP-based algorithm for spectroscopic semi-blind deconvolution,” Analyst (Lond.) 137(16), 3862–3873 (2012).
[Crossref] [PubMed]

Zhang, Z.

Zhong, S.

Zhu, H.

H. Zhu, L. Deng, G. Xu, Y. Chen, and Y. Li, “Spectral semi-blind deconvolution methods based on modified φHS regularizations,” Opt. Laser Technol. 1, 46 (2018).
[Crossref]

Analyst (Lond.) (3)

H. Liu, T. Zhang, L. Yan, H. Fang, and Y. Chang, “A MAP-based algorithm for spectroscopic semi-blind deconvolution,” Analyst (Lond.) 137(16), 3862–3873 (2012).
[Crossref] [PubMed]

A. Economou, P. R. Fielden, and A. J. Packham, “Deconvolution of analytical peaks by means of the fast Hartley transform,” Analyst (Lond.) 121(8), 1015–1018 (1996).
[Crossref]

Q. Han, Q. Xie, S. Peng, and B. Guo, “Simultaneous spectrum fitting and baseline correction using sparse representation,” Analyst (Lond.) 142(13), 2460–2468 (2017).
[Crossref] [PubMed]

Appl. Opt. (4)

Appl. Spectrosc. (6)

Appl. Spectrosc. Rev. (1)

L. Shao, X. Lin, and X. Shao, “A wavelet transform and its application to spectroscopic analysis,” Appl. Spectrosc. Rev. 37(4), 429–450 (2002).
[Crossref]

IEE Trans. Instrumentation and Measurement (1)

S. Sarkar, P. K. Dutta, and N. C. Roy, “A blind-deconvolution approach for chromatographic and spectroscopic peak restoration,” IEE Trans. Instrumentation and Measurement 47(4), 941–947 (1998).
[Crossref]

IEEE Trans. Image Process. (2)

J. L. Starck, E. J. Candès, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11(6), 670–684 (2002).
[Crossref] [PubMed]

T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7(3), 370–375 (1998).
[Crossref] [PubMed]

IEEE Trans. Ind. Electron. (5)

P. B. Crilly, “Increased throughput for process chromatography using constrained deconvolution,” IEEE Trans. Ind. Electron. 39(1), 20–24 (1992).
[Crossref]

F. Auger, M. Hilairet, J. M. Guerrero, E. Monmasson, T. Orlowska-Kowalska, and S. Katsura, “Industrial Applications of the Kalman Filter: A Review,” IEEE Trans. Ind. Electron. 60(12), 5458–5471 (2013).
[Crossref]

N. P. Ayerden and R. F. Wolffenbuttel, “The Miniaturization of an Optical Absorption Spectrometer for Smart Sensing of Natural Gas,” IEEE Trans. Ind. Electron. 64(12), 9666–9674 (2017), doi:.
[Crossref]

W. E. Snyder, M. L. Hsiao, and J. N. Campbell, “Restoration of ultrasonic NDE images,” IEEE Trans. Ind. Electron. 40(2), 250–258 (1993).
[Crossref]

A. Mukherjee and A. Sengupta, “Estimating the Probability Density Function of a Nonstationary Non-Gaussian Noise,” IEEE Trans. Ind. Electron. 57(4), 1429–1435 (2010).
[Crossref]

IEEE Trans. Ind. Inf. (1)

T. Liu, H. Liu, Z. Chen, and A. M. Lesgold, "Fast Blind Instrument Function Estimation Method for Industrial Infrared Spectrometers," IEEE Trans. Ind. Inf. 279, 4449 (2018).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

M. B. Slima, R. Z. Morawski, and A. Barwicz, “Kalman-filter-based algorithms of spectrophotometric data correction III. Use of splines for approximation of spectra,” IEEE Trans. Instrum. Meas. 46(3), 685–689 (1997).
[Crossref]

J. Chemom (1)

S. Hugelier, P. H. C. Eilers, O. Devos, and C. Ruckebusch, “Improved superresolution microscopy imaging by sparse deconvolution with an interframe penalty,” J. Chemom.  31, e2847 (2017).

J. Opt. Soc. Am. (1)

Multiscale Model. Simul. (1)

E. Candès, L. Demanet, D. Donoho, and L. Ying, “Fast Discrete Curvelet Transforms,” Multiscale Model. Simul. 5(3), 861–899 (2006).
[Crossref]

Nat. Photonics (1)

G. Ycas, F. R. Giorgetta, E. Baumann, I. Coddington, D. Herman, S. A. Diddams, and N. R. Newbury, “High-coherence mid-infrared dual-comb spectroscopy spanning 2.6 to 5.2 μm,” Nat. Photonics 12(4), 202–208 (2018).
[Crossref]

Nat. Protoc. (1)

M. J. Baker, J. Trevisan, P. Bassan, R. Bhargava, H. J. Butler, K. M. Dorling, P. R. Fielden, S. W. Fogarty, N. J. Fullwood, K. A. Heys, C. Hughes, P. Lasch, P. L. Martin-Hirsch, B. Obinaju, G. D. Sockalingum, J. Sulé-Suso, R. J. Strong, M. J. Walsh, B. R. Wood, P. Gardner, and F. L. Martin, “Using Fourier transform IR spectroscopy to analyze biological materials,” Nat. Protoc. 9(8), 1771–1791 (2014).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Laser Technol. (1)

H. Zhu, L. Deng, G. Xu, Y. Chen, and Y. Li, “Spectral semi-blind deconvolution methods based on modified φHS regularizations,” Opt. Laser Technol. 1, 46 (2018).
[Crossref]

Sci. Rep. (1)

K. Chen, T. Wu, H. Wei, X. Wu, and Y. Li, “High spectral specificity of local chemical components characterization with multichannel shift-excitation Raman spectroscopy,” Sci. Rep. 5(1), 13952 (2015).
[Crossref] [PubMed]

SIAM J. Imaging Sci. (1)

T. Goldstein and S. Osher, “The Split Bregman Method for L1-Regularized Problems,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
[Crossref]

Other (3)

S. B. Engelson, Infrared Spectral of Butyl propionate http://www.models.life.ku.dk/specarb (2018).

S. B. Engelson, Infrared Spectral of D (+) -Glucopyranose http://www.models.life.ku.dk/specarb (2018).

P. A. Jansson, Deconvolution: with applications in spectroscopy (Academic Press, New York, USA, 1984).

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Figures (12)

Fig. 1
Fig. 1 Spectral reconstruction for an FTIR spectrometer. Original spectrum is contaminated by the band overlap and Poisson noises, which are caused by blurring effect and resolution decimation. The degraded spectrum can be recovered by the spectral reconstruction algorithm.
Fig. 2
Fig. 2 Curvelet transform coefficient comparison between the high-resolution IR spectrum and the degraded one. (a) Observed IR spectrum of methyl (C15H29NO7), which suffers from the band overlap and random noises. This degradation often appears in applications of IR spectroscopy. (b) Measured by a high-resolution spectrometer. IR spectrum can be classified into three different regions, such as flat region, shoulder region and noises region. (c)-(d) Histogram statistics for curvelet transform coefficient in Figs. 2(a) and 2(b).
Fig. 3
Fig. 3 Probability density function of the curvelet coefficients of the IR spectra (in Fig. 2). (a) Natural logarithm of the probability density function in Fig. 2(c) and 2(d), as shown by the black and blue curves. (b) Two curves are fitted by the constructed functions in Fig. 3(a). The red, yellow, and blue regions denote the low, middle, and high frequency, respectively.
Fig. 4
Fig. 4 Simulated experiments for IR spectra. (a) methyl propionate (C4H8O2) from 3300 to 2100 cm−1. (b) IRF Gaussian function with σ = 8 cm−1. (c) Overlap spectrum, convolute with the IRF. (d) Corrupted by Poisson noises (SNR = 200).
Fig. 5
Fig. 5 Simulated experiments for the noise-free case (Fig. 4(c)). (a) RL method [15]. (b) SE-BSR method [21]. (c) WE-SR method [16]. (d) FBPSR method.
Fig. 6
Fig. 6 NMSE versus the iteration number of the three methods for the IR spectrum of methyl propionate (C4H8O2).
Fig. 7
Fig. 7 Experimental results for the noisy degraded spectrum (Fig. 4(d), SNR = 200). (a) RL method [15]. (b) SE-BSR method [21]. (c) WE-SR method [16]. (d) FBPSR method.
Fig. 8
Fig. 8 Comparison of the NMSE values of RL, WE-SR, SE-BSR and the proposed method in all SNR conditions. The lower NMSE values imply improved performance.
Fig. 9
Fig. 9 Sensitivity analysis of regularization parameters λ1, and λ2. Change of the NMSE value versus the parameters (a) λ1 value, and (b) λ2 value.
Fig. 10
Fig. 10 Real IR spectrum experiment. (a) Butyl propionate (C7H14O2) [36] from 3100 to 2150 cm−1, reconstructed by (b) SE-BSR method [21]. (c) WE-SR method [16]. (d) FBPSR. (The estimated IRF is plotted at up-top corner).
Fig. 11
Fig. 11 Reconstruction experiment for real Raman spectra. (a) IR spectrum of D( + )-Cellobiose [37] from 900 to 350 cm−1, (b) WE-SR [16] method. (c) FBPSR result. (d) IRF estimated by FBPSR method.
Fig. 12
Fig. 12 Clustering comparison between experimental before-after results based on PCA. (a) Original spectra. Experimental results by (b) WE-SR [16]. (c) FBPSR method.

Tables (5)

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Table 1 Algorithm 1. Numerical algorithm for fast blind Poissonian spectral reconstruction via curvelet transform regularization

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Table 1 Peak Distortions in Reconstructed Spectra by RL, SE-BSR, WE-SR and FBPSR in Fig. 5 (methyl propionate (C4H8O2)).

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Table 2 NMSE, NSR and RFWHM of Degraded Spectrum and the Best Reconstructed Spectrum. The larger the NSR and RFWHM values, the higher the spectral quality.

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Table 3 Comparison of the speed between the proposed method and RL, SE-BSR, and WE-SR (Unit: Second).

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Table 4 RFWHM and NSR (in Bracket) values of different reconstruction methods on the real IR spectra.

Equations (25)

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o ( v ) = P { F { g ( v ) } } = P { G { f ( v ) } }
p ( | C g | ) e x p ( λ 1 C g 1 1 ) .
g ^ M A P = arg max p ( g , f | o )
< g ^ , f ^ > = arg max { p ( o | g , f ) p ( g ) , p ( f ) }
< g ^ , f ^ > = arg max { log p ( o | g , f ^ ) log p ( g ) log p ( f ^ ) } .
p ( o | g , f ) = v N ( F g ) v o v e x p ( F g ) v o v ! .
log p ( o | g , f ) = v N log ( F g ) v o v exp ( F g ) v o v ! = v N [ o v log ( F g ) v ( F g ) v log o v ! ] = v N [ ( F g ) v o v log ( F g ) v ] + v N log o v ! o v ! .
log p ( o | g , f ) v N [ ( F g ) v - o v log ( F g ) v ]
p ( g ) e x p ( λ 1 C g 1 )
p ( f ) e x p ( λ 2 D f 2 ) ,
< g ^ , f ^ > = arg min g , h v = 1 N [ ( F g ) o log ( F g ) + λ 1 C g 1 + λ 2 D f 2 + Θ g 0 ( g ) ] ,
min f v = 1 N [ ( F g ) o log ( F g ) ] + λ 2 D f 2
f ^ k + 1 = f ^ 1 λ 2 D T D f ^ k { ( G k ) * [ o G k f ^ k ] }
f ^ k + 1 = f ^ k + 1 v = 1 N ( f ^ k + 1 ) v
arg min g R N v = 1 N [ ( F g ) o log ( F g ) ] + λ 1 C g 1 + Θ g 0 ( g ) ,
arg min g R N , d 1 , d 2 , d 3 v = 1 N [ d 1 o log d 1 ] + λ 1 d 2 1 + Θ d 3 0 ( d 3 ) s u c h t h a t d = F g , d 2 = C g , d 3 = g .
min g R N , d 1 , d 2 , d 3 v = 1 N [ d 1 o log d 1 ] + λ 1 | | d 2 | | 1 + Θ d 3 0 ( d 3 ) + 1 2 α { | | b 1 + F g d 1 | | 2 2 + | | b 2 + C g d 2 | | 2 2 + | | b 3 + g d 3 | | 2 2 }
( F T F + 2 I ) g k + 1 = F T ( d 1 k b 1 k ) + C T ( d 2 k b 2 k ) + ( d 3 k b 3 k ) ,
g k + 1 = F 1 ( F ( F T ( d 1 k b 1 k ) + C T ( d 2 k b 2 k ) + ( d 3 k b 3 k ) F ( F T F ) + 2 I )
d k + 1 = 1 2 ( η k + ( η k ) 2 + 4 α o )
{ d 2 k + 1 = max { C g k + 1 + b 2 k + 1 λ 1 α , 0 } C g k + 1 + b 2 k C g k + 1 + b 2 k 2 d 3 k + 1 = max { g k + 1 + b 3 k + 1 , 0 } .
δ = 1.4826 2 M e d i a n { | o v o v 1 | , v = 2.... , N } .
N M S E = g g ^ F 2 g F 2
N S R = | D o | / | D g ^ | ,
R F W H M = 1 N v N F W H M o ( v ) / F W H M g ^ ( v )

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