Application of ADI Iterative Methods to the Restoration of Noisy Images

The restoration of two-dimensional images in the presence of noise by Wiener’s minimum mean square error filter requires the solution of large linear systems of equations. When the noise is white and Gaussian, and under suitable assumptions on the image, these equations can be written as a Sylvester’s equation \[ T_1^{ - 1} \hat F + \hat FT_2 = C \] for the matrix $\hat F$ representing the restored image. The matrices $T_1 $ and $T_2 $ are symmetric positive definite Toeplitz matrices. We show that the ADI iterative method is well suited for the solution of these Sylvester’s equations, and illustrate this with computed examples for the case when the image is described by a separable first-order Markov process. We also consider generalizations of the ADI iterative method, propose new algorithms for the generation of iteration parameters, and illustrate the competitiveness of these schemes.