Semi-Blind Spectral Deconvolution with Adaptive Tikhonov Regularization

Abstract

Deconvolution has become one of the most used methods for improving spectral resolution. Deconvolution is an ill-posed problem, especially when the point spread function (PSF) is unknown. Non-blind deconvolution methods use a predefined PSF, but in practice the PSF is not known exactly. Blind deconvolution methods estimate the PSF and spectrum simultaneously from the observed spectra, which become even more difficult in the presence of strong noise. In this paper, we present a semi-blind deconvolution method to improve the spectral resolution that does not assume a known PSF but models it as a parametric function in combination with the a priori knowledge about the characteristics of the instrumental response. First, we construct the energy functional, including Tikhonov regularization terms for both the spectrum and the parametric PSF. Moreover, an adaptive weighting term is devised in terms of the magnitude of the first derivative of spectral data to adjust the Tikhonov regularization for the spectrum. Then we minimize the energy functional to obtain the spectrum and the parameters of the PSF. We also discuss how to select the regularization parameters. Comparative results with other deconvolution methods on simulated degraded spectra, as well as on experimental infrared spectra, are presented.