MODERN STABLE MATHEMATICAL AND SOFTWARE-BASED METHODS FOR DISTORTED SPECTRA RESTORATION

The paper presents analysis and comparison of various methods and algorithms for restoration of the spectra fine structure smoothed by the instrumental spectrometer function and/or having the overlapping of close spectral lines. Continuous and discrete spectra are considered. Successful spectra restoration enhances mathematically the resolution of spectrometers. In the case of a continuous spectrum smoothing by the instrumental function, the problem of restoration is reduced to solving integral equations of the first kind. This problem is ill-posed (essentially unstable). Therefore, to obtain a stable solution of integral equations, the Tikhonov regularization, Wiener filtering, Kalman–Bucy and other methods are used. However, in the case of close lines overlapping in the spectrum, these methods make it possible to restore only the total spectrum, but not the profiles of each line. To separate line profiles, the desired lines are modeled by the Gaussians or Lorentzians; the total spectrum is differentiated using smoothing splines; the number and parameters of the lines are estimated from the results of differentiation. To refine the line parameters, minimization of the discrepancy functional by the coordinate descent method and for comparison by the Nelder–Mead method is performed. A comparison is also made with the Fourier-self-deconvolution method, in which the line widths are artificially reduced due to apodization (the interferogram truncation), and, as a result, the true line profiles are distorted for their resolution. In the original convolution method, the parameters of lines (peaks) are determined from convolutions of experimental spectrum with model spectrum derivatives. If a discrete spectrum is smoothed by the instrumental function, then the problem of spectrum restoration is described by a system of linear-non-linear equations (SLNE) and solved by the integral approximation algorithm that is more efficient than the Prony method, the Golub–Mullen–Hegland variable projection method, and other methods. Based on the results of the review of various mathematical methods, it is proposed to create a new complex algorithm for distorted spectra restoration, which makes it possible to remove the effect of instrumental function, noise, lines overlapping and other effects. The software in MATLAB is developed and the processing of a number of spectra is performed. The stated technique can be used to enhance the spectrometer resolution via mathematical and computer processing of spectra.

1. Encyclopedic Dictionary of Physics.Ed. A.M. Prokhorov. Moscow,Sovetskaya Entsiklopediya Publ., 1984, 944 p. (in Russian)
2. Rautian S.G. Real spectral apparatus. Soviet Physics
   Uspekhi, 1958, vol. 66, no. 2, pp. 245–273. doi: 10.1070/PU1958v001n02ABEH003099
3. Stewart J.E. Resolution enhancement of X-ray fluorescence spectra with a computerized multichannel analyzer. Applied Spectroscopy, 1975, vol. 29, no. 2, pp. 171–174.
4. Starkov V.N. Constructive Methods of Computational Physics in Problems of Interpretation. Kiev, Naukova Dumka, 2002, 264 p. (in Russian)
5. Fleckl T., Jäger H., Obernberger I. Experimental verification of gas spectra calculated for high temperatures using the HITRAN/HITEMP database. Journal of Physics D: Applied Physics, 2002, vol. 35, no. 23, pp. 3138–3144. doi: 10.1088/0022-3727/35/23/315
6. Sizikov V.S. Direct and Inverse Problems of Image Restoration, Spectroscopy and Tomography with MatLab. St. Petersburg, Lan’ Publ., 2017, 412 p. (in Russian)
7. Giese A.T., French C.S. The analysis of overlapping spectral absorption bands by derivative spectrophotometry. Applied Spectroscopy, 1955, vol. 9, no. 2, pp. 78–96. doi: 10.1366/000370255774634089
8. Kauppinen J.K., Moffatt D.J., Mantsch H.H., Cameron D.G. Fourier self-deconvolution: a method for resolving intrinsically overlapped bands. Applied Spectroscopy, 1981, vol. 35, no. 3, pp. 271–276. doi: 10.1366/0003702814732634
9. Sirvidas S.I., Zarutsky I.V., Larionov A.M., Manoilov V.V. The convolution of a signal with derivatives of gaussians as an
   approach suitable for detection, separation and estimation of mass spectrometer peaks. Scientific Instrumentation, 1999, vol. 9, no. 2, pp. 71–75. (in Russian)
10. Sizikov V.S., Lavrov A.V. A comparison of different methods of separation of continuous overlapping spectral lines. Optics
    and Spectroscopy, 2018, vol. 124, no. 6, pp. 753–762. doi: 10.1134/S0030400X1806022X
11. Mikhailenko V.I., Mikhal’chuk V.V. Method of expanding spectra with unresolved structure. Journal of Applied Spectroscopy,1987, vol. 46, no. 4, pp. 327–335. doi: 10.1007/BF00660037
12. Kochikov I.V., Kuramshina G.M., Pentin Yu.A., Yagola A.G. Inverse Problems of Vibrational Spectroscopy. Moscow, MGU Publ., 1993, 204 p. (in Russian)
13. Sizikov V.S. Mathematical Methods for Processing the Results of Measurements. St. Petersburg, Polytechnika Publ., 2001, 240 p. (in Russian)
14. Sizikov V.S. Inverse Applied Problems and MatLab. St. Petersburg,Lan’ Publ., 2011, 256 p. (in Russian)
15. Sizikov V.S., Krivykh A.V. Reconstruction of continuous
    spectra by the regularization method using model spectra. Optics and Spectroscopy,2014,vol. 117,no. 6, pp.1010–1017. doi: 10.1134/S0030400X14110162
16. Sizikov V.S., Stepanov A.V. Method of training examples in solving inverse ill-posed problems of spectroscopy. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 6, pp. 1147–1154. doi: 10.17586/2226-1494-2015-15-6-1147-1154
17. Sizikov V.S., Lavrov A.V. Stable Methods for Mathematical and Computerized Processing the Images and Spectra. Tutorial. St. Petersburg, ITMO Publ., 2018, 70 p. (in Russian)
18. Sizikov V., Sidorov D. Discrete spectrum reconstruction using integral approximation algorithm. Applied Spectroscopy, 2017, vol. 71, no. 7, pp. 1640–1651. doi: 10.1177/0003702817694181
19. Bousquet P. Spectroscopy and Its Instrumentation. London, Adam Hilger, 1971, 239 p.
20. Landsberg G.S. Optics. Tutorial. 6^th ed. Moscow, Fizmatlit Publ., 2006, 848 p. (in Russian)
21. Tourin R.H., Krakow B. Applicability of infrared emission and absorption spectra to determination of hot gas temperature
    profiles. Applied Optics, 1965, vol. 4, no. 2, pp. 237–242. doi: 10.1364/ao.4.000237
22. Workman J., Springsteen A. Applied Spectroscopy: A Compact Reference for Practitioners. San Diego, Academic Press, 1998, 539 p.
23. Chalmers J.M., Griffiths P.R., eds. Handbook of Vibrational Spectroscopy. New York, Wiley, 2002, 4000 p. doi: 10.1002/0470027320
24. Glazov M.V., Bolokhova T.A. Solution of the Rayleigh reduction problem using different modifications of the regularization method.Optics and Spectroscopy, 1989, vol. 67, no. 3, pp. 312–314.
25. Vasilenko G.I. The Theory of Signal Reconstruction: On
    Reduction to Ideal Device in Physics and Technics. Moscow, Soviet Radio Publ., 1979, 272  p. (in Russian)
26. Tikhonov A.N., Arsenin V.Ya. Solutions of Ill-Posed Problems. New York, Wiley, 1977, 258 p.
27. Verlan’ A.F., Sizikov V.S. Integral Equations: Methods, Algorithms,Programs. Kiev, Naukova Dumka, 1986, 544 p. (in Russian)
28. Tikhonov A.N., Goncharsky A.N., Stepanov V.V., Yagola A.G. Numerical Methods for the Solution of Ill-Posed Problems. Dordrecht, Kluwer, 1995, 254 p. doi: 10.1007/978-94-015-8480-7
29. Engl H., Hanke M., Neubauer A. Regularization of Inverse Problems. Dordrecht, Kluwer, 1996, 328 p.
30. Manoylov V.V., Zarutsky I.V. Capability of the algorithm on the base convolution processing signals form for the estimation of mass-spectra peak parameters in multiplets. Scientific Instrumentation, 2009, vol. 19, no. 4, pp. 103–108. (in Russian)
31. Sizikov V.S., Lavrov A.V. Study of errors of some methods for separating overlapped spectral lines under noise effect. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2017, vol. 17, no. 5, pp. 879–889 (in Russian).
    doi: 10.17586/2226-1494-2017-17-5-879-889
32. Sizikov V.S., Lavrov A.V. Separation of continuous lines mutually overlapping and smoothed by the instrumental function. Optics and Spectroscopy, 2017, vol. 123, no. 5, pp. 682–691. doi: 10.1134/S0030400X17110200
33. Voskoboinikov Yu.E., Preobrazhensky N.G., Sedel’nikov A.I. Mathematical Processing of Experiment in Molecular Gas
    Dynamics. Novosibirsk, Nauka Publ., 1984, 240 p. (in Russian)
34. Petrov Yu.P., Sizikov V.S. Well-Posed, Ill-Posed, and
    Intermediate Problems with Applications. Leiden-Boston, VSP, 2005, 234 p.
35. Goldberg D.E. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, 1989, 412 p.
36. Kincaid D., Cheney W. Numerical Analysis: Mathematics of Scientific Computing. 3^rd ed. Providence, AMS, 2009, 788 p.
37. Holland J.H. Adaptation in Natural and Artificial Systems. Cambridge, MIT Press, 1992, 232 p.
38. Yan L., Liu H., Zhong S., Fang H. Semi-blind spectral deconvolution with adaptive Tikhonov regularization. Applied Spectroscopy, 2012, vol. 66, no. 11, pp. 1334–1346. doi: 10.1366/11-06256
39. D’yakonov V., Abramenkova I. MATLAB. Processing of Signals and Images. St. Petersburg, Piter Publ., 2002, 608 p. (in Russian)
40. Sizikov V.S. Infrared tomography of hot gas: mathematical model of active-passive diagnosis. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2013, no. 6, pp. 1–17. (in Russian)
41. Sizikov V.S., Evseev V., Fateev A., Clausen S. Direct and
    inverse problems of infrared tomography. Applied Optics, 2016, vol. 55, no. 1, pp. 208–220. doi: 10.1364/AO.55.000208
42. Nelder J.A., Mead R. A simplex method for function minimization.The Computer Journal, 1965, vol. 7, no. 4, pp. 308–313. doi: 10.1093/comjnl/7.4.308
43. D’yakonov V. MATLAB 6: Training Course. St. Petersburg, Piter Publ., 2001, 592 p. (in Russian)
44. Manoilov V.V., Zarutsky I.V. Outlier rejection and parameter estimation of mass spectrometric signals for high precision
    isotopic analysis. Scientific Instrumentation, 2002, vol. 12, no. 3, pp. 38–46. (in Russian)
45. Sizikov V. Use of an integral equation for solving special
    systems of linear-non-linear equations. In Integral Methods in Science and Engineering. Vol. 2. Approximation Methods. Eds.
    C. Constanda, J. Saranen, S. Seikkala. Harlow, Longman, 1997, pp. 200–205.
46. Krivykh A.V., Sizikov V.S. Discrete spectra processing by an integral approximation algorithm. Scientific and Technical
    Journal of Information Technologies, Mechanics and Optics, 2011, no. 5, pp. 14–18. (in Russian)
47. Himmelblau D.M. Applied Nonlinear Programming. New York, McGray-Hill, 1972, 416 p.
48. Rheinboldt W.C. Methods for Solving Systems of Nonlinear Equations. 2^nd ed. Rhiladelphia, SIAM, 1998, 148 p.
49. Kay S.M., Marple S.L. Spectrum analysis – a modern perspective. Proceedings of the IEEE, 1981, vol. 69, no. 11, pp. 1380–1420.doi:10.1109/PROC.1981.12184
50. Peebles P.Z., Berkowitz R.S. Multiple-target monopulse radar processing techniques. IEEE Transactions on Aerospace and Electronic Systems, 1968, vol. AES-4, no. 6, pp. 845–854.
    doi: 10.1109/TAES.1968.5409051
51. Falkovich S.E., Konovalov L.N. Estimation for unknown numbers of signals. Journal of Communications Technology and Electronics,1982, vol. 27, no. 1, pp. 92–97.
52. Golub G.H., Pereyra V. The differentiation of pseudo-inverses and nonlinear least squares whose variables separate. SIAM Journal on Numerical Analysis,1973, vol. 10, no. 2, pp. 413–432. doi: 10.1137/0710036
53. Mullen K.M., van Stokkum I.H.M. The variable projection algorithm in time-resolved spectroscopy, microscopy and mass spectrometry applications. Numerical Algorithms, 2009, vol. 51, no. 3, pp. 319–340. doi: 10.1007/s11075-008-9235-2
54. Hegland M. Error bounds for spectral enhancement which are based on variable Hilbert scale inequalities. Journal of Integral Equations and Applications. 2010, vol. 22, no. 2. pp. 285–312. doi: 10.1216/JIE-2010-22-2-285

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License