# Notes on performance of bidiagonalization-based noise level estimator in image deblurring

## Main Article Content

## Abstract

Image deblurring represents one of important areas of image processing. When information about the amount of noise in the given blurred image is available, it can significantly improve the performance of image deblurring algorithms. The paper [11] introduced an iterative method for estimating the noise level in linear algebraic ill-posed problems contaminated by white noise. Here we study applicability of this approach to image deblurring problems with various types of blurring operators. White as well as data-correlated noise of various sizes is considered.

## Article Details

How to Cite

HNĚTYNKOVÁ, Iveta; KUBÍNOVÁ, Marie; PLEŠINGER, Martin.
Notes on performance of bidiagonalization-based noise level estimator in image deblurring.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 333-342, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/422>. Date accessed: 22 sep. 2017.
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## References

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[11] 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 Numerical Math- ematics, 49 (2009), pp. 669-696.

[12] M. Michenková, Regularization techniques based on the least squares method, Diploma thesis, Charles University in Prague, 2013.

[13] V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), pp. 414–417.

[14] F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons and Teub- ner, Stuttgart, 1986.

[15] C. C. Paige, Bidiagonalization of matrices and solution of linear equations, SIAM J. Numer. Anal., 11 (1974), pp. 197–209.

[16] C. B. Shaw, Jr., Improvements of the resolution of an instrument by numerical solution of an integral equation, J. Math. Anal. Appl., 37 (1972), pp. 83–112.

[17] K. Vasilík, Linear algebraic modeling of problems with noisy data, Diploma thesis, Charles University in Prague, 2011.

[2] L. Desbat and D. Girard, The “minimum reconstruction error” choice of regularization parameters: Some more efficient methods and their application to deconvolution problems, SIAM J. Sci. Comput., 16 (1995), pp. 1387–1403.

[3] S. Gazzola, P. Novati, and M. R. Russo, On Krylov projection methods and Tikhonov regularization, ETNA, 44 (2015), pp. 83–123.

[4] G. H. Golub and W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Numer. Anal., Ser. B 2 (1965), pp. 205–224.

[5] P. C. Hansen, Regularization Tools Version 4.1 (for MATLAB Version 7.3). A MATLAB package for analysis and solution of discrete ill-posed problems (available at http://www.imm.dtu.dk/~pcha/Regutools).

[6] P. C. Hansen , Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of Linear Inver- sion, SIAM Publications, Philadelphia, PA, 1998.

[7] P. C. Hansen and T. K. Jensen, Noise propagation in regularizing iterations for image de- blurring, ETNA, 31 (2008), pp. 204–220.

[8] P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering, SIAM Publications, Philadelphia, PA, 2006.

[9] P. C. Hansen, J. G. Nagy, and K. Tigkos, Rotational image deblurring with sparse matrices. BIT Numerical Mathematics, 54 (2013), pp. 649–671.

[10] I. Hnětynková, M. Kubínová, and M. Plešinger, On noise propagation in residuals of Krylov subspace iterative regularization methods, submitted.

[11] 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 Numerical Math- ematics, 49 (2009), pp. 669-696.

[12] M. Michenková, Regularization techniques based on the least squares method, Diploma thesis, Charles University in Prague, 2013.

[13] V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), pp. 414–417.

[14] F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons and Teub- ner, Stuttgart, 1986.

[15] C. C. Paige, Bidiagonalization of matrices and solution of linear equations, SIAM J. Numer. Anal., 11 (1974), pp. 197–209.

[16] C. B. Shaw, Jr., Improvements of the resolution of an instrument by numerical solution of an integral equation, J. Math. Anal. Appl., 37 (1972), pp. 83–112.

[17] K. Vasilík, Linear algebraic modeling of problems with noisy data, Diploma thesis, Charles University in Prague, 2011.