A collocation method solving integral equation models for image restoration

Abstract

We propose a collocation method for solving integral equations which model image restoration from out-of-focus images. Restoration of images from out-of-focus images can be formulated as an integral equation of the first kind, which is an ill-posed problem. We employ the Tikhonov regularization to treat the ill-posedness and obtain results of a well-posed second kind integral equation whose integral operator is the square of the original operator. The present of the square of the integral operator requires high computational cost to solve the equation. To overcome this difficulty, we convert the resulting second kind integral equation into an equivalent system of integral equations which do not involve the square of the integral operator. A multiscale collocation method is then applied to solve the system. A truncation strategy for the matrices appearing in the resulting discrete linear system is proposed to design a fast numerical solver for the system of integral equations. A quadrature method is used to compute the entries of the resulting matrices. We estimate the computational cost of the numerical method and its approximate accuracy. Numerical experiments are presented to demonstrate the performance of the proposed method for image restoration.