Regularizing Inverse Preconditioners for Symmetric Band Toeplitz Matrices

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(n2 logn) operations, where n2 is the pixel number of the 2D image. In this paper, we propose inverse preconditioners with a band Toeplitz structure, which lower the cost to O(n2) and in experiments showed the same speed of convergence and reconstruction efficiency as the inverse Toeplitz preconditioner.