An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems

Iterative regularization multigrid methods have been successfully applied to signal/image deblurring problems. When zero-Dirichletboundary conditions are imposed the deblurring matrix has a Toeplitzstructure and it is potentially full. A crucial task of a multilevelstrategy is to preserve the Toeplitz structure at the coarse levelswhich can be exploited to obtain fast computations. The smoother hasto be an iterative regularization method. The grid transfer operatorshould preserve the regularization property of the smoother.This paper improves the iterative multigrid method proposed in[11] introducing a wavelet soft-thresholding denoisingpost-smoother. Such post-smoother avoids the noise amplificationthat is the cause of the semi-convergence of iterativeregularization methods and reduces ringing effects. The resultingiterative multigrid regularization method stabilizes the iterationsso that the imprecise (over) estimate of the stopping iteration doesnot have a deleterious effect on the computed solution. Numericalexamples of signal and image deblurring problems confirm theeffectiveness of the proposed method.