An Operator Factorization Method for Restoration of Blurred Images

Abstract

A problem of restoration of images blurred by space-invariant point-spread functions (SIPSF) is considered. The SIPSF operator is factorized as a sum of two matrices. The first term is a polynomial of a noncirculant operator P and the second term is a Hankel matrix which affects only the boundary observations. The image covariance matrix is also factorized into two terms; the covariance of the first term is a polynomial in P and the second term depends on the boundary values of the image. Thus, by modifying the image matrix by its boundary terms and the observations by the boundary observations, it is shown that the wieWir filter equation is a function of the operator P and can be solved exactly via the eigenvector expansion of P. The eigenvectors of the noncirculant matrix P are a set of orthronormal harmonic sinusoids called the sine transform, and the eigenvector expansion of the Wiener filter equation can be numerically achieved via a fast-sine-transform algorithm which is related to the fast-Fourier-transform (FFT) algorithm. The factorization therefore provides a fast Wiener restoration scheme for images and other random processes. Examples on 255 X 255 images are given.