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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 5 — May. 1, 2013
  • pp: 948–955

Signal restoration combining Tikhonov regularization and multilevel method with thresholding strategy

Liang-Jian Deng, Ting-Zhu Huang, Xi-Le Zhao, Liang Zhao, and Si Wang  »View Author Affiliations

JOSA A, Vol. 30, Issue 5, pp. 948-955 (2013)

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Singular value decomposition (SVD)-based approaches, e.g., truncated SVD and Tikhonov regularization methods, are effective ways to solve problems of small or moderate size. However, SVD, in the sense of computation, is expensive when it is applied in large-sized cases. A multilevel method (MLM) combining SVD-based methods with the thresholding technique for signal restoration is proposed in this paper. Our MLM will transfer large-sized problems to small- or moderate-sized problems in order to make the SVD-based methods available. The linear systems on the coarsest level in the multilevel process will be solved by the Tikhonov regularization method. No presmoothers are implemented in the multilevel process to avoid damaging the parameter choice on the coarsest level. Furthermore, the soft-thresholding denoising technique is employed for the postsmoothers aiming to eliminate the leaving high-frequency information due to the lack of presmoothers. Finally, computational experiments show that our method outperforms other SVD-based methods in signal restoration ability at a shorter CPU-time consumption.

© 2013 Optical Society of America

OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(100.0100) Image processing : Image processing

ToC Category:
Image Processing

Original Manuscript: November 20, 2012
Revised Manuscript: March 24, 2013
Manuscript Accepted: March 24, 2013
Published: April 22, 2013

Liang-Jian Deng, Ting-Zhu Huang, Xi-Le Zhao, Liang Zhao, and Si Wang, "Signal restoration combining Tikhonov regularization and multilevel method with thresholding strategy," J. Opt. Soc. Am. A 30, 948-955 (2013)

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  1. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problem: Numerical Aspects of Linear Inversion (SIAM, 1998).
  2. P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006).
  3. M. Fuhry and L. Reichel, “A new Tikhonov regularization method,” Numer. Algorithms 59, 433–445 (2012). [CrossRef]
  4. W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed. (SIAM, 2000).
  5. M. Donatelli and S. Serra Capizzano, “On the regularizing power of multigrid-type algorithms,” SIAM J. Sci. Comput. 27, 2053–2076 (2006). [CrossRef]
  6. M. Donatelli, “A multigrid for image deblurring with Tikhonov regularization,” Numer. Linear Algebra Appl. 12, 715–729 (2005). [CrossRef]
  7. W. Zhu, Y. Wang, Y. Deng, Y. Yao, and R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imag. 16, 210–217 (1997). [CrossRef]
  8. M. Hanke and C. R. Vogel, “Two-level preconditioners for regularized inverse problems I: theory,” Numer. Math. 83, 385–402 (1999). [CrossRef]
  9. B. Kaltenbacher, “On the regularizing properties of a full multigrid method for ill-posed problems,” Inverse Probl. 17, 767–788 (2001). [CrossRef]
  10. S. Morigi, L. Reichel, F. Sgallari, and A. Shyshkov, “Cascadic multiresolution methods for image deblurring,” SIAM J. Imaging Sci. 1, 51–74 (2008). [CrossRef]
  11. L. Reichel and A. Shyshkov, “Cascadic multilevel methods for ill-posed problems,” J. Comput. Appl. Math. 233, 1314–1325 (2010). [CrossRef]
  12. M. I. Español, “Multilevel methods for discrete ill-posed problems: application to deblurring,” Ph.D. thesis (Tufts University, 2009).
  13. M. I. Español and M. E. Kilmer, “Multilevel approach for signal restoration problems with Toeplitz matrices,” SIAM J. Sci. Comput. 32, 299–319 (2010). [CrossRef]
  14. M. Donatelli, “An iterative multigrid regularization method for Toeplitz discrete ill-posed problems,” Numer. Math. Theory Methods Appl. 5, 43–61 (2012).
  15. D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995). [CrossRef]
  16. S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd. ed. (Academic, 1998).
  17. P. C. Hansen, “Regularization tools version 4.0 for Matlab 7.3,” Numer. Algorithms 46, 189–194 (2007). [CrossRef]

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