A Multigrid method for restoration and regularization of images with Dirichlet boundary conditions

In many 2D image restoration problems, such as image deblurring with Dirichlet boundary conditions, we deal with two-level linear systems whose coefficient matrix is a banded block Toeplitz matrix with banded Toeplitz blocks (BTTB). Usually, these matrices are ill-conditioned since they are associated to generating functions which vanish at (π, π) or in neighborhood of it. In this work, we solve such BTTB systems by applying an Algebraic Multi-Grid method (AMG). The technique we propose has an optimality property, i.e., its cost is of O(n1 " n2) arithmetic operations, where n1 x n2 is the size of the images. Unfortunately, in the case of images affected by noise, our AMG method does not provide satisfactory regularization effects. Therefore, we propose two Tikhonov-like techniques, based on a regularization parameter, which can be applied to AMG method in order to reduce the noise effects.