Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T19:14:47.080Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aricò, A.; and Donatelli, M., A V-cycle Multigrid for multilevel matrix algebras: proof of optimality, Numer. Math., vol. 105, (2007), pp. 511–547.CrossRefGoogle Scholar
[2] Aricò, A., Donatelli, M., and Serra Capizzano, S., V-cycle optimal convergence for certain (multilevel) structured linear systems, SIAM J. Matrix Anal. Appl., vol. 26, (2004) pp. 186–214.CrossRefGoogle Scholar
[3] Chan, R. H., Chang, Q., and Sun, H., Multigrid method for ill-conditioned symmetric Toeplitz systems, SIAM J. Sci. Comput., vol. 19 (1998), pp. 516–529.CrossRefGoogle Scholar
[4] Chan, R. H., Chan, T.F., and Wan, W., Multigrid for differential-convolution problems arising from image processing, in Scientific Computing, Golub, G., Lui, S.H., Luk, F., and Plemmons, R., eds., Springer-Verlag, Singapore, 1999, pp. 58–72.Google Scholar
[5] Chan, R. H. and Chen, K., A multilevel algorithm for simultaneously denoising and deblurring images SIAM J. Sci. Comput., vol. 32, (2010), pp. 1043Ű1063,CrossRefGoogle Scholar
[6] Cheng, L., Wang, H., and Zhang, Z., The solution of ill-conditioned symmetric Toeplitz systems via two-grid and wavelet methods, Comput. Math. Appl., vol. 46 (2003), 793–804.Google Scholar
[7] Daubechies, I., Ten lectures on wavelets, Society for Industrial and Applied Mathematics, Philadelphia, USA, 1992.Google Scholar
[8] Daubechies, I., Han, B., Ron, A., and Shen, Z., Framelets: MRA-based constructions of wavelet frames, Applied and Computational Harmonic Analysis, vol. 14 (2003), pp. 1–46.CrossRefGoogle Scholar
[9] Donatelli, M., An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices, Numer. Linear Algebra Appl., vol. 17 (2010), pp. 179–197.CrossRefGoogle Scholar
[10] Donatelli, M., A Multigrid for image deblurring with Tikhonov regularization, Numer. Linear Algebra Appl., vol. 12, (2005), pp. 715–729.CrossRefGoogle Scholar
[11] Donatelli, M. and Capizzano, S. Serra, On the regularizing power of multigrid-type algorithms, SIAM J. Sci. Comput., vol. 27, (2006), pp. 2053–2076.CrossRefGoogle Scholar
[12] Donatelli, M. and Serra-CAPIZZANO, S., Filter factor analysis of an iterative multilevel regularizing method, Electron. Trans. Numer. Anal., vol. 29 (2007/2008), pp. 163–177.Google Scholar
[13] Donoho, D. L., De-Noising by Soft-Thresholding, IEEE Trans. Inform. Theory, vol. 41, (1995), pp. 613–627.CrossRefGoogle Scholar
[14] Engl, H. W., Hanke, M., and Neubauer, A., Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.CrossRefGoogle Scholar
[15] Español, M. I. and Kilmer, M. E., Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices, SIAM J. Sci. Comput., vol. 32, (2010), pp. 299–319.CrossRefGoogle Scholar
[16] Fiorentino, G. and Serra-capizzano, S., Multigrid methods for Toeplitz matrices, Calcolo, vol. 28 (1991), pp. 283–305.CrossRefGoogle Scholar
[17] Goldstein, T. and Osher, S., The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci. vol. 2 (2009), pp. 323–343.CrossRefGoogle Scholar
[18] Gonzalez, R. and Woods, R., Digital Image Processing, Addison-Wesley, Reading, MA, 1992.Google Scholar
[19] Grenander, U. and Szegö, G., Toeplitz Forms and Their Applications, Second Edition, Chelsea, New York, 1984.Google Scholar
[20] Hanke, M., Conjugate Gradient Type Methods for Ill-Posed Problems, Longman Scientific and Technical, Harlow, UK, 1995.Google Scholar
[21] Hanke, M. and Vogel, C. R., Two-Level Preconditioners for Regularized Inverse Problems I: Theory, Numer. Math, vol. 83 (1998), pp. 385–402.CrossRefGoogle Scholar
[22] Hansen, P. C., Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998.CrossRefGoogle Scholar
[23] Hansen, P. C., Regularization Tools Version 4.0 for Matlab 7.3, Numer. Algo., vol. 46 (2007), pp. 189Ű194.CrossRefGoogle Scholar
[24] Huckle, T. and Staudacher, J., Multigrid preconditioning and Toeplitz matrices, Electron. Trans. Numer. Anal., vol. 13 (2002), pp. 81–105.Google Scholar
[25] Kaltenbacher, B., On the regularizing properties of a full multigrid method for ill-posed problems, Inverse Problems, vol. 17 (2001), pp. 767–788.CrossRefGoogle Scholar
[26] King, J.T., Multilevel algorithms for ill-posed problems, Numer. Math., vol. 61 (1992), pp. 311–334.CrossRefGoogle Scholar
[27] Morigi, S., Reichel, L., Sgallari, F., and Shyshkov, A., Cascadic multiresolution methods for image deblurring, SIAM J. Imaging Sci., vol. 1 (2008), pp. 51–74.CrossRefGoogle Scholar
[28] Neelamani, R., Choi, H., and Baraniuk, R., ForWaRD: Fourier-Wavelet Regularized Deconvolu-tionforIll-Conditioned Systems IEEE Trans. Signal Process., vol. 52, (2004), pp. 418–433.CrossRefGoogle Scholar
[29] Ng, M., Chan, R. H., and Tang, W. C., A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., vol. 21 (1999), pp. 851–866.CrossRefGoogle Scholar
[30] Reichel, L. and Shyshkov, A., Cascadic multilevel methods for ill-posed problems, J. Comput. Appl. Math., vol. 233 (2010), pp. 1314–1325.CrossRefGoogle Scholar
[31] Rieder, A., A wavelet multilevel methodfor ill-posed problems stabilized by Tikhonov regularization, Numerische Mathematik, vol. 75 (1997), pp. 501–522.CrossRefGoogle Scholar
[32] Serra Capizzano, S., A note on anti-reflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput. vol. 25, (2003) pp. 1307–1325.Google Scholar
[33] Trottenberg, U., Oosterlee, C. W. and Schüller, A., Multigrid, Academic Press, 2001.Google Scholar
[34] Wang, Y. L., Yang, J. F., Yin, W. T., and Zhang, Y., A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imag. Sci., vol. 1 (2008), pp. 248–272.CrossRefGoogle Scholar