## Abstract and Applied Analysis

### Two New Efficient Iterative Regularization Methods for Image Restoration Problems

#### Abstract

Iterative regularization methods are efficient regularization tools for image restoration problems. The IDR($s$) and LSMR methods are state-of-the-arts iterative methods for solving large linear systems. Recently, they have attracted considerable attention. Little is known of them as iterative regularization methods for image restoration. In this paper, we study the regularization properties of the IDR($s$) and LSMR methods for image restoration problems. Comparative numerical experiments show that IDR($s$) can give a satisfactory solution with much less computational cost in some situations than the classic method LSQR when the discrepancy principle is used as a stopping criterion. Compared to LSQR, LSMR usually produces a more accurate solution by using the $L$-curve method to choose the regularization parameter.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 129652, 9 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511953

Digital Object Identifier
doi:10.1155/2013/129652

Mathematical Reviews number (MathSciNet)
MR3073472

Zentralblatt MATH identifier
1371.68315

#### Citation

Zhao, Chao; Huang, Ting-Zhu; Zhao, Xi-Le; Deng, Liang-Jian. Two New Efficient Iterative Regularization Methods for Image Restoration Problems. Abstr. Appl. Anal. 2013 (2013), Article ID 129652, 9 pages. doi:10.1155/2013/129652. https://projecteuclid.org/euclid.aaa/1393511953

#### References

• P. C. Hansen, J. G. Nagy, and D. P. O'Leary, Deblurring Images: Matries, Spectra and Filtering, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2006.
• R. J. Hanson, “A numerical method for solving Fredholm integral equations of the first kind using singular values,” SIAM Journal on Numerical Analysis, vol. 8, pp. 616–622, 1971.
• P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1998.
• Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2nd edition, 2003.
• R. Plato, “Optimal algorithms for linear ill-posed problems yield regularization methods,” Numerical Functional Analysis and Optimization, vol. 11, no. 1-2, pp. 111–118, 1990.
• P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2010.
• D. Calvetti, B. Lewis, and L. Reichel, “On the regularizing properties of the GMRES method,” Numerische Mathematik, vol. 91, no. 4, pp. 605–625, 2002.
• D. Calvetti, B. Lewis, and L. Reichel, “Krylov subspace iterative methods for nonsymmetric discrete ill-posed problems in image restoration,” in Advanced Signal Processing: Algorithms, Architectures and Implementations XI, vol. 4474 of Proceedings of SPIE, pp. 224–233, The International Society for Optical Engineering, Bellingham, Wash, USA, August 2001.
• P. Brianzi, P. Favati, O. Menchi, and F. Romani, “A framework for studying the regularizing properties of Krylov subspace methods,” Inverse Problems, vol. 22, no. 3, pp. 1007–1021, 2006.
• P. Sonneveld and M. B. van Gijzen, “IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations,” SIAM Journal on Scientific Computing, vol. 31, no. 2, pp. 1035–1062, 2008.
• D. C.-L. Fong and M. Saunders, “LSMR: an iterative algorithm for sparse least-squares problems,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2950–2971, 2011.
• M. B. van Gijzen and P. Sonneveld, “An elegant IDR(s) variant that efficiently exploits bi-orthogonality properties,” Tech. Rep. 08-21, Department of Applied Mathematical Analysis, Delft University of Technology, Delft, The Netherlands, 2008.
• V. Simoncini and D. B. Szyld, “Interpreting IDR as a Petrov-Galerkin method,” SIAM Journal on Scientific Computing, vol. 32, no. 4, pp. 1898–1912, 2010.
• M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems, vol. 327 of Pitman Research Notes in Mathematics Series, Longman, Harlow, UK, 1995.
• T. K. Jensen and P. C. Hansen, “Iterative regularization with minimum-residual methods,” BIT, vol. 47, no. 1, pp. 103–120, 2007.
• P. Sonneveld, “On the convergence behaviour of IDR(s),” Tech. Rep. 10-08, Department of Applied Mathematical Analysis, Delft University of Technology, Delft, The Netherlands, 2010.
• M. Donatelli and S. Serra-Capizzano, “On the regularizing power of multigrid-type algorithms,” SIAM Journal on Scientific Computing, vol. 27, no. 6, pp. 2053–2076, 2006.
• M. Donatelli and S. Serra-Capizzano, “Filter factor analysis of an iterative multilevel regularizing method,” Electronic Transactions on Numerical Analysis, vol. 29, pp. 163–177, 2007/08.
• V. A. Morozov, “On the solution of functional equations by the method of regularization,” Soviet Mathematics. Doklady, vol. 7, pp. 414–417, 1966.
• P. C. Hansen, “Analysis of discrete ill-posed problems by means of the \emphL-curve,” SIAM Review, vol. 34, pp. 658–672, 1992.
• G. Wahba, “Practical approximate solutions to linear operator equations when the data are noisy,” SIAM Journal on Numerical Analysis, vol. 14, no. 4, pp. 651–667, 1977.
• I. Hnětynková, M. Plešinger, and Z. Strakoš, “The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data,” BIT, vol. 49, no. 4, pp. 669–696, 2009.
• L. Reichel and G. Rodriguez, “Old and new parameter choice rules for discrete ill-posed problems,” Numerical Algorithms, vol. 63, no. 1, pp. 65–87, 2013.
• P. C. Hansen, T. K. Jensen, and G. Rodriguez, “An adaptive pruning algorithm for the discrete \emphL-curve criterion,” Journal of Computational and Applied Mathematics, vol. 198, no. 2, pp. 483–492, 2007.
• P. C. Hansen, “The \emphL-curve and its use in the numerical treatment of inverse problems,” in Computational Inverse Problems in Electrocardiology, P. Johnston, Ed., pp. 119–142, WIT Press, Southampton, Ceremonial, 2001.
• C. R. Vogel, “Non-convergence of the \emphL-curve regularization parameter selection method,” Inverse Problems, vol. 12, no. 4, pp. 535–547, 1996.
• M. Hanke, “Limitations of the \emphL-curve method in ill-posed problems,” BIT, vol. 36, no. 2, pp. 287–301, 1996.
• D. C. L. Fong, Minimum-residual methods for sparse least-squares using golub-kahan bidiagonalization [Ph.D. thesis], Stanford University, Stanford, Calif, USA, 2011.
• P. C. Hansen, “Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems,” Numerical Algorithms, vol. 6, no. 1-2, pp. 1–35, 1994.