Regularization parameter estimation for large-scale Tikhonov regularization using a priori information

Computational Statistics and Data Analysis

Volumn 54, Issue 12, 2010, Pages 3430-3445

  Renaut, Rosemary A ^a   Hnětynková, Iveta ^a   Mead, Jodi ^a  

^a ARIZONA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COVARIANCE MATRIX; DEGREES OF FREEDOM (MECHANICS); IMAGE ENHANCEMENT; ITERATIVE METHODS; MATRIX ALGEBRA; NONLINEAR EQUATIONS; NORMAL DISTRIBUTION; PARAMETERIZATION; SINGULAR VALUE DECOMPOSITION;

COVARIANCE STRUCTURES; DISTRIBUTED MEASUREMENTS; GENERALIZED SINGULAR VALUE DECOMPOSITION; LEAST SQUARES PROBLEMS; ORIGINAL ALGORITHMS; REGULARIZATION PARAMETERS; REGULARIZED LEAST SQUARES; TIKHONOV REGULARIZATION;

INVERSE PROBLEMS;

EID: 78049324264     PISSN: 01679473     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.csda.2009.05.026     Document Type: Article

References (30)

1. Benito, M., Pea, D., 2007. Detecting defects with image data. Comput. Statist. Data Anal. 51 (12), 6395-6403. doi:10.1016/j.csda.2007.02.016.
2. Chung, J., Nagy, J., O'Leary, D.P., 2008. A weighted GCV method for Lanczos hybrid regularization. ETNA 28, 149-167.
3. Eidén, L., 1982. A weighted pseudoinverse, generalized singular values, and constrained least squares problems. BIT 22,487-502.
4. Fierro, R.D., Golub, G.H., Hansen, P.C., O'Leary, D.P., 1997. Regularization by truncated total least squares. SIAM J. Sci. Stat. Comput. 18, 1225-1241.
5. Golub, G.H., van Loan, C., 1996. Matrix Computations, 3rd ed. John Hopkins Press, Baltimore.
6. Hanke, M, Hansen, P.C., 1993. Regularization methods for large-scale problems. Surv. Math. Indust. 3,253-315.
7. Hansen, P.C., 1989. Regularization, GSVD and truncated GSVD. BIT 6, 491-504.
8. Hansen, P.C., 1994. Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6, 1-35.
9. Hansen, P.C., 1998. Rank-Deficient and Discrete III-Posed Problems: Numerical Aspects of Linear Inversion. In: SIAM Monographs on Mathematical Modeling and Computation, vol. 4.
10. Hansen, P.C., Nagy, J.G., O'Leary, D.P., 2006a. Deblurring Images, Matrices, Spectra and Filtering. In: SIAM Fundamentals of Algorithms.
11. Hansen, P.C., Kilmer, M.E., Kjeldsen, R.H., 2006b. Exploiting residual information in the parameter choice for discrete ill-posed problems. BIT 46, 41-59.
12. Hnětynková, I., Plešinger, M., Strakoš, Z., 2009. Golub-Kahan iterative bidiagonalization and determining the size of the noise in the data. http://www.cs.cas.cz/strakos/download/2009-HnPlSt.pdf.
13. Kilmer, M.E., O'Leary, D.P., 2001. Choosing regularization parameters in iterative methods for ill-posed problems. SIAMJ. Numer. Anal. Appl. 22, 1204-1221.
14. Kilmer, M.E., Hansen, P.C., Español, M.I., 2007. A projection-based approach to general-form Tikhonov regularization. SIAM J. Sci. Comput. 29 ( 1), 315-330.
15. Marquardt, D.W., 1970. Generalized inverses, ridge regression, biased linear estimation, and nonlinear estimation. Technometrics 12 (3), 591-612.
16. MATLAB is a registered mark of MathWorks, Inc., MathWorks Web Site. http://.mathworks.com.
17. Mead, J., 2008. Parameter estimation: A new approach to weighting a priori information. J. Inv. III-posed Problems 16(2), 175-194.
18. Mead, J., Renaut, R.A., 2009. A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems. Inverse Problems 25, 025002. doi:10.1088/0266-5611/25/2/ 025002.
19. Nagy, J.G., Palmer, K, Perrone, L., 2004. Iterative methods for image deblurring: A Matlab object-oriented approach. Numer. Algorithms 36, 73-93. (Pubitemid 39072750)
20. Obereder, A., Scherzer, O., Kovac, A., 2007. Bivariate density estimation using BV regularisation. Comput. Statist. Data Anal. 51 (12), 5622-5634. doi:10.1016/j.csda.2007.04.019.
21. O'Leary, D.P., 2001. Near-optimal parameters for Tikhonov and other regularization methods. SIAM J. on Sci. Comput. 23, 1161-1171.
22. O'Leary, D.P., Simmons, J.A., 1989. A bidiagonalization-regularization procedure for large scale discretizations of ill-posed problems. SIAM J. Sci. Stat. Comput. 2 (4), 474-489.
23. Paige, C.C., Saunders, M.A., 1982a. LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software 8, 43-71.
24. Paige, C.C., Saunders, M.A., 1982b. ALGORITHM 583 LSQR: Sparse linear equations and least squares problems. ACM Trans. Math. Software 8, 195-209.
25. Qiu, P., 2008. A nonparametric procedure for blind image deblurring. Comput. Statist. Data Anal. 52 (10), 4828-4841. doi:10.1016/j.csda.2008.03.027.
26. Rao, C.R., 1973. Linear Statistical Inference and its Applications. Wiley, New York.
27. Rust, B.W., O'Leary, D.P., 2008. Residual periodograms for choosing regularization parameters for ill-posed problems. Inverse Problems 24, 034005.
28. Stefan, W., 2008. Total variation regularization for linear ill-posed inverse problems: Extensions and applications, Ph.D. Dissertation, Department of Mathematics and Statistics, Arizona State University, http://mathpost.la.asu. edu/~stefan.
29. Thompson, A.M., Kay, J.W., Titterington, D.M., 1991. Noise estimation in signal restoration using regularization. Biometrika 78 (3), 475-488.
30. Vogel, C.R., 2002. Computational Methods for Inverse Problems. In: SIAM Frontiers in Applied Mathematics.