A weighted H¹ seminorm 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 ill-posed. Therefore, regularization is required to obtain a reasonable approximate solution. This paper presents a new regularization method based on a weighted H¹ 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.

Keywords:

• Fredholm integral equation of the first kind
• piecewise continuous solution
• weighted H¹ seminorm regularization
• alternative iterative algorithm
• image restoration

2010 AMS Subject Classifications:

• 65R32
• 65F22

Acknowledgements

We thank the anonymous referees for their valuable suggestions. The research was supported by NSF of China No. 11271238 and Guangdong Provincial NSF No. 10151503101000023.