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

Published online by Cambridge University Press:  28 May 2015

Marco Donatelli
Marco Donatelli*
Affiliation:
Dipartimento di Fisica e Matematica, Università dell’Insubria, Via Valleggio, 11, Como 22100, Italia
*
*Corresponding author.Email address:[email protected]
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Abstract

Iterative regularization multigrid methods have been successful applied to signal/image deblurring problems. When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full. A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations. The smoother has to be an iterative regularization method. The grid transfer operator should preserve the regularization property of the smoother. This paper improves the iterative multigrid method proposed in [11] introducing a wavelet soft-thresholding denoising post-smoother. Such post-smoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects. The resulting iterative multigrid regularization method stabilizes the iterations so that and imprecise (over) estimate of the stopping iteration does not have a deleterious effect on the computed solution. Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.

Keywords

65F2265N5515B05Multigrid methodsToeplitz matricesdiscrete ill-posed problems
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 1 , February 2012 , pp. 43 - 61
Copyright
Copyright © Global Science Press Limited 2012

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