Volume 5, Issue 1
An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems

Marco Donatelli

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 43-61.

Published online: 2012-05

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  • Abstract

Iterative regularization multigrid methods have been successfully 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 the 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

Multigrid methods, Toeplitz matrices, discrete ill-posed problems.

  • AMS Subject Headings

65F22, 65N55, 15B05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-43, author = {}, title = {An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {1}, pages = {43--61}, abstract = {

Iterative regularization multigrid methods have been successfully 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 the 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.m12si03}, url = {http://global-sci.org/intro/article_detail/nmtma/5927.html} }
TY - JOUR T1 - An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 43 EP - 61 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2011.m12si03 UR - https://global-sci.org/intro/article_detail/nmtma/5927.html KW - Multigrid methods, Toeplitz matrices, discrete ill-posed problems. AB -

Iterative regularization multigrid methods have been successfully 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 the 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.

Marco Donatelli. (2020). An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems. Numerical Mathematics: Theory, Methods and Applications. 5 (1). 43-61. doi:10.4208/nmtma.2011.m12si03
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