Regularizing inverse preconditioners for symmetric band Toeplitz Systems

Image restoration is a widely studied discrete ill-posed problem. Among the many regularization methods used for treating the problem, iterative methods have been shown to be effective. In this paper we consider the case of a blurring function defined by space invariant and band limited PSF, modeled by a linear system that has a band block Toeplitz structure with band Toeplitz blocks. In order to reduce the number of iterations required to obtain acceptable reconstructions, in [1] an inverse Toeplitz preconditioner for problems with a Toeplitz structure was proposed. The cost per iteration is of O(n^2\log n) operations, where n^2 is the pixels number of the 2D image. In this paper, we propose inverse preconditioners with a band Toeplitz structure, which lower the cost to O(n^2) and in the experiments showed the same speed of convergence and reconstruction efficiency as the inverse Toeplitz preconditioner.