Application of numerical solution of linear Fredholm integral equation of the first kind for image restoration

In this paper, we consider integral equation model for image restoration. Out-of-focus models are formulated to Fredholm integral equations of the first kind. This inverse problem is ill-posed, and regularization methods must be applied for approximating the reconstructed image. In general, extrapolation for increasing the accuracy of regularization methods is much less considered. In this paper, we apply linear extrapolation for Lavrentiev regularization for image restoration. This linear extrapolation is presented in Hamarik et al. (J Inv Ill-Posed Probl 15:277–294, 2007) in order to increase accuracy of solution of Fredholm integral equation of the first kind. Also, we apply Graves and Prenter method as another regularization method because kernel of integral equation is self-adjoint and symmetric. In addition, since kernel of integral equation in out-of-focus models is a Gaussian function with small standard deviation, approximating these integrals requires careful numerical treatment, and to meet this challenge, we apply numerical quadrature based on graded mesh that is applied in Ma and Xu (J Sci Comput, 2018). Finally, experimental results of extrapolated Lavrentiev method and Graves and Prenter method in terms of visual quality of reconstructed images and relative L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^2}$$\end{document}error are presented.