Restoration by deconvolution of three-dimensional images that have been contaminated by noise and spatially invariant blur is computationally demanding. We describe efficient parallel implementations of iterative methods for image deconvolution on a distributed memory computing cluster.
Total variation-penalized Tikhonov regularization is a popular method for the restoration of images that have been degraded by noise and blur. The method is particularly effective, when the desired noise- and blur-free image has edges between smooth surfaces. The method, however, is computationally expensive. We describe a hybrid regularization method that combines a few steps of the GMRES iterative method with total variation-penalized Tikhonov regularization on a space of small dimension. This hybrid method requires much less computational work than available methods for total variation-penalized Tikhonov regularization and can
produce restorations of similar quality.
The BiCG and QMR methods are well-known Krylov subspace iterative methods for the solution of linear systems of equations with a large nonsymmetric, nonsingular matrix. However, little is known of the performance of these methods when they are applied to the computation of approximate solutions of linear systems of equations with a matrix of ill-determined rank. Such linear systems are known as linear discrete ill-posed problems. We describe an application of the BiCG and QMR methods to the solution of linear discrete ill-posed problems that arise in image restoration, and compare these methods to the conjugate gradient method applied to the associated normal equations and to total variation-penalized Tikhonov regularization.
A variant of the MINRES method, often referred to as the MR-II method, has in the last few years become a popular iterative scheme for computing approximate solutions of large linear discrete ill- posed problems with a symmetric matrix. It is important to terminate the iterations sufficiently early in order to avoid severe amplification of measurement and round-off errors. We present a new L-curve for determining when to terminate the iterations with the MINRES and MR-II method.
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.
In this paper we compare a new regularizing scheme based on the exponential filter function with two classical regularizing methods: Tikhonov regularization and a variant of truncated singular value regularization. The filter functions for the former methods are smooth, but for the latter discontinuous. These regularization methods are applied to the restoration of images degraded by blur and noise. The norm of the noise is assumed to be known, and this allows application of the Morozov discrepancy principle to determine the amount of regularization. We compare the restored images produced by the three regularization methods with optimal values of the regularization parameter. This comparison sheds light on how these different approaches are related.