This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.W. Hilgers and W.R. Reynolds, Instabilities in the optimization parameter relating to image recovery problems, J. Opt. Soc. Amer. A 9 (1992), 1273–1279.
C.W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Res. Notes in Math. 105, Pitman, London, 1977.
V.A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, Berlin-Heidelberg, 1984.
S.J. Reeves and R.M. Mersereau, Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration, Optical Engng. 29 (1990), 446–454.
G.H. Golub, M. Heath, and G. Wahba, Generalized cross-validation as method for choosing a good ridge parameter, Technometrics 21 (1979), 215–223.
J. Hilgers, B. Bertram, and W. Reynolds, Cross-referencing different regularized families of approximate solutions to ill-posed problems for solving the parameter choice problem in generalized CLS, in Integral Methods in Science and Engineering, P. Schiavone, C. Constanda, and A. Mioduchowski, eds., Birkhauser, Boston, 2002, 105–110.
J. Hilgers and B. Bertram, Comparing different regularizers for choosing the parameters in CLS, Computers and Math. with Appl. (to appear).
J.W. Hilgers, B.S. Bertram, and W.R. Reynolds, Extension of constrained least squares for obtaining regularized solutions to first-kind integral equations, in Integral Methods in Science and Engineering, B. Bertram, C. Constanda, and A. Struthers, eds., Chapman & Hall/CRC, Boca Raton, FL, 2000, 185–189.
E.M. Wilcheck, A comparison of cross-referencing and generalized cross-validation in the optimal regularization of first kind equations, Master’s Thesis, Michigan Tech. Univ., 1992.
J.W. Hilgers, B.S. Bertram, M.M. Alger, and W.R. Reynolds, Extensions of the cross-referencing method for choosing good regularized solutions to image recovery problems, in Proc. SPIE 3171, Barbour et al., eds., 1997, 234–237.
M.M. Alger, J.W. Hilgers, B.S. Bertram, and W.R. Reynolds, An extension of the Tikhonov regularization based on varying the singular values of the regularization operator, Proc. SPIE 3171, Barbour et al., eds., (1997), 225–231.
H.W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.
A.N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Soviet Math. Dokl. 4 (1963), 1035–1038.
C.K. Rushforth, Signal restoration, functional analysis and Fredholm integral equations of the first kind, in Image Recovery: Theory and Application, H. Stark, ed., Academic Press, New York, 1987.
A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-posed Problems, Winston and Sons, Washington, DC, 1977.
A.M. Thompson et. al., A Study of methods of choosing the smoothing parameter in image restoration by regularization, IEEE Trans. on PAMI 13 (1991), 326–339.
M. Bertero, Regularization Methods for Linear Inverse Problems, Lect. Notes in Math., Springer-Verlag, Berlin-Heidelberg, 1986.
A.K. Katsaggelos, Iterative image restoration algorithms, Optical Engng. 28 (1989), 735–738.
J.W. Hilgers, Non-iterative methods for solving operator equations of the first kind, TSR 1413 Math. Res. Center, Univ. of Wisconsin-Madison, 1974.
B. Bertram, On the use of wavelet expansions and the conjugate gradient method for solving first kind integral equations, in Integral Methods in Science and Engineering, B. Bertram, C. Constanda, and A. Struthers, eds., Chapman & Hall/CRC, Boca Raton, FL, 2000, 62–66.
B. Bertram and H. Cheng, On the use of the conjugate gradient method for the numerical solution of first kind integral equations in two variables, in Integral Methods in Science and Engineering, P. Shiavone, C. Constanda, and A. Mioduchowski, eds., Birkhauser, Boston, 2002, 51–56.
H. Cheng and B. Bertram, On the stopping criteria for conjugate gradient solutions of first kind integral equations in two variables, in Integral Methods in Science and Engineering, P. Schiavone, C. Constanda, and A. Mioduchowski, eds., Birkhauser, Boston, 2002, 57–62.
L. Desbat and D. Girard, The “minimum reconstruction error” choice of reconstruction parameters: some more efficient methods and the application to deconvolution problems, SIAM J. Sci. Comput. 16 (1995),1387–1403.
M. Hanke and T. Raus, A general heuristic for choosing the regularization parameter in ill-posed problems, SIAM J. Sci. Comput. 17 (1996),956–972.
A. Frommer and P. Maass, Fast CG-based methods for Tikhonov-Phillips regularization, SIAM. J. Sci. Comput. 20 (1999), 1831–1850.
M.E. Kilmer and D.P. O’Leary, Choosing regularization parameters in iterative methods for ill-posed problems, SIAM J. Matrix Anal. Appl. 22 (2001), 1204–1221.
D.P. O’Leary, Near-optimal parameters for Tikhonov and other regularization methods, SIAM J. Sci. Comput. 23 (2001), 1161–1171.
S. Saitoh, Integral Transformations, Reproducing Kernels, and Their Applications, Addison-Wesley Longman, London, 1997.
M. Rojas and D.C. Sorensen, A trust region approach to the regularization of large-scale discrete forms of ill-posed problems, SIAM J. Sci.Comput. 23 (2002), 1842–1860.
P.R. Johnston and R.M. Julrajani, An analysis of the zero-crossing method for choosing regularization parameters, SIAM J. Sci. Comput. 24 (2002), 428–442.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Boston
About this chapter
Cite this chapter
Hilgers, J.W., Bertram, B.S. (2006). Cross-referencing for Determining Regularization Parameters in Ill-Posed Imaging Problems. In: Constanda, C., Nashed, Z., Rollins, D. (eds) Integral Methods in Science and Engineering. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4450-4_9
Download citation
DOI: https://doi.org/10.1007/0-8176-4450-4_9
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4377-5
Online ISBN: 978-0-8176-4450-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)