A weightedH¹seminorm regularization method for Fredholm integral equations of the first kind

Abstract

Many problems in mathematics and engineering lead to Fredholm integral equations of the first kind, e.g. signal and image processing. These kinds of equations are difficult to solve numerically since they are illposed. Therefore, regularization is required to obtain a reasonable approximate solution. This paper presents a new regularization method based on a weighted H 1 seminorm. Details of numerical implementation are given. Numerical examples, including one-dimensional and two-dimensional integral equations, are presented to illustrate the efficiency of the proposed approach. Numerical results show that the proposed regularization method can restore edges as well as details.

An overview of numerical methods for the first kind Fredholm integral equation

“…Lin and Yang [52] proposed a new regularization method based on weighted H 1 seminorm. The efficiency of the method is verified by the examples of a onedimensional and two-dimensional integral equation.…”

A hybrid regularization model for linear inverse problems

“…In this paper, we consider the numerical method for solving the linear inverse problem b = Hu + n, (1) where H ∈ R n×n is a nonsingular matrix, b is the observed result, u is the true solution to be recovered and n is a noise vector. The linear inverse problem arises in a wide range of applications, such as astrophysics, signal and image processing, see [2,10,15,23] and the references therein.…”

“…For example, Tikhonov regularization is not good at judging the discontinuous points and solving the constant functions, and the numerical solutions obtained by the TV regularization often suffer the staircase effects and loss fine details. For the two regularization methods, readers can refer to [3,5,6,9,14,15,18,19,[22][23][24][25] and the references therein.…”

On some variational models and their algorithms for axially symmetric objects tomography from a single X-ray source

“…TV 解由分块常数组成, 对光滑函数, 显出 明显的不理想的阶梯效果. 为了克服解的光滑性, 也有许多利用高阶导数等作正则项替代 TV 的工作, 如文献 [9,[18][19][20][21][22][23][24]. 文献 [25] 提出了一种用 TV 来修正 H1 的方法, 以提高密度逼近精度.…”