Deconvolution of astronomical images using SOR with adaptive relaxation

Abstract

We address the potential performance of the successive overrelaxation technique (SOR) in image deconvolution, focusing our attention on the restoration of astronomical images distorted by atmospheric turbulence. SOR is the classical Gauss-Seidel iteration, supplemented with relaxation. As indicated by earlier work, the convergence properties of SOR, and its ultimate performance in the deconvolution of blurred and noisy images, can be made competitive to other iterative techniques, including conjugate gradients, by a proper choice of the relaxation parameter. The question of how to choose the relaxation parameter, however, remained open, and in the practical work one had to rely on experimentation. In this paper, using constructive (rather than exact) arguments, we suggest a simple strategy for choosing the relaxation parameter and for updating its value in consecutive iterations to optimize the performance of the SOR algorithm (and its positivity-constrained version, +SOR) at finite iteration counts. We suggest an extension of the algorithm to the notoriously difficult problem of "blind" deconvolution, where both the true object and the point-spread function have to be recovered from the blurred image. We report the results of numerical inversions with artificial and real data, where the algorithm is compared with techniques based on conjugate gradients. In all of our experiments +SOR provides the highest quality results. In addition +SOR is found to be able to detect moderately small changes in the true object between separate data frames: an important quality for multi-frame blind deconvolution where stationarity of the object is a necesessity.

Iterative Methods for Image Restoration

“…They have recently been proposed for use in image restoration [93,103], so we give a brief description of these methods here.…”

“…We mention that the properties of SOR for image restoration, in the simplified case of spatially invariant blurs with periodic boundary conditions, were studied in [93,103]. They show that, contrary to well-posed problems arising in partial differential equations, the relaxation parameter should satisfy τ 1; that is, the method should use severe under-relaxation instead of over-relaxation.…”

“…The specific case of projected SOR for well posed problems was first studied by Cryer [28], while its use for ill-posed problems in image restoration has been considered more recently in [93,103]. Properties of a general projected steepest (gradient) descent method can be found in one of many references on numerical optimization methods, such as [55,74], and a discussion in the context of image restoration can be found in [102].…”

A new approach to blind deconvolution of astronomical images

“…In [30], a modification to the inexact alternating minimization was suggested which allowed a significant improvement in the quality of the deconvolution. Alternating minimization was interrupted, at regular intervals in the iteration count, by a 'refresh' procedure, which itself was performed in a few cycles.…”

Image Deconvolution by Means of Frequency Blur Invariant Concept

“…But in most cases, the PSF is unknown. Hence, in order to estimate the PSF and the original image, blind deconvolution technique is adopted [7–13]. The general model which is used for the observed blurred image, g ( x , y ) of a scene, f ( x , y ), is described by a convolution integral: $\begin{array}{c}g\left(x,y\right)=f\left(x,y\right)\otimes h\left(x,y\right)+n\left(x,y\right),\end{array}$ where h ( x , y ) and n ( x , y ) are the PSF kernel and random noise, respectively.…”