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