Truncated spectral decomposition preconditioner for image restoration

In this paper, we consider the solution of linear systems which arise from the discretization of large scale ill-posed inverse problems, such as image restoration. We propose and study a truncated spectral decomposition preconditioner for the blurring matrix enforcing anti-reflective boundary conditions (AR-BCs), which have been shown to produce superior restorations compared to other commonly used boundary conditions, such as zero, periodic and reflective. The preconditioner is based on the spectral decomposition of the blurring matrix with AR-BCs. It clusters the large eigenvalues around one, but leaves the small eigenvalues alone as well. Hence, conjugate gradient type methods, when applied to solving these preconditioned systems, converge very fast. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner.