<title>Restoration of images with spatially variant blur by the GMRES method</title>

Abstract

The GMRES method is a popular iterative method for the solution of linear systems of equations with a large nonsymmetric nonsingular matrix. However, little is known about the performance of the GMRES method when the matrix of the linear system is of ill-determined rank, i.e., when the matrix has many singular values of different orders of magnitude close to the origin. Linear systems with such matrices arise, for instance, in image restoration, when the image to be restored is contaminated by noise and blur. We describe how the GMRES method can be applied to the restoration of such images. The GMRES method is compared to the conjugate gradient method applied to the normal equations associated with the given linear system of equations. The numerical examples show the GMRES method to require less computational work and to give restored images of higher quality than the conjugate gradient method.

A hybrid GMRES and TV-norm-based method for image restoration

“…The algorithm appeared in Calvetti et al 4 and is repeated here for the reader's convenience. The GMRES implementation is due to Saad and Schultz.…”

“…Regularization methods produce solutionsx α of a problem different from, but related to (4). The solutionx α is referred to as a regularized solution of (4), and it depends on A,b, the regularization method, and the value of the regularization parameter α (and sometimes on the values of additional parameters as well).…”

FGMRES for linear discrete ill-posed problems

“…Image restoration examples for which GMRES outperforms LSQR both in terms of the quality of the computed restorations and the number of matrix-vector product evaluations required are reported in [5,7].…”

“…GMRES performs better than the conjugate gradient method applied to the normal equations for some linear discrete ill-posed problems but worse for others; see [5,6,7,13] for computed examples. The range restricted GMRES (RRGMRES) method performs better than GMRES for many linear discrete ill-posed problems.…”

Efficient shift-variant image restoration using deformable filtering (Part I)

“…This requires some approximations to make the SVD tractable, like hierarchically extracting singular vectors until their associated singular values become irrelevant (see, e.g., [2,9]). Other researchers have addressed the problem of inverting the linear transformation in a stable, iterative way, mostly by using the so-called Krylov sub-spaces and related techniques [5,10]. A serious problem of all these methods, besides their high computational cost, is their lack of modeling upon which to establish a criterion for fixing the amount of required regularization.…”