Digital image deblurring with SOR

We address the numerical performance of the successive overrelaxation technique (SOR) in the restoration of astronomical images. Local analysis of the convergence rates reveals resonant properties, with convergence enhancement at a spatial frequency which is determined by the SOR relaxation parameter τ. The analysis can serve as a guide for the practical choice of the relaxation parameter(s), which governs the regularization properties of the SOR algorithm. One particular prediction is that in typical implementations, fast image deblurring requires deep underrelaxation (τ << 1). This theoretical result allows us to better understand the productive properties of underrelaxation, which have been discovered in earlier work. Comparison of SOR in artificial inversions with conjugate gradients and related methods (GMRES) indicates that a solution of similar or better quality may be obtained in a comparable or smaller number of iterations. Restricting the solution with non-negativity constraint (+SOR) enhances both the quality of the solutions and the convergence rate. Theoretical analysis of the convergence properties of +SOR, however, remains a challenge which cannot be addressed by the simple analysis implemented in this paper