Regularization of ill-posed problems by envelope guided conjugate gradients

Journal of Computational and Graphical Statistics

Volumn 6, Issue 4, 1997, Pages 451-463

  Kaufman, Linda ^a   Neumaier, Arnold ^b  

^a LUCENT TECHNOLOGIES   (United States)

^b UNIVERSITY OF VIENNA   (Austria)

Author keywords

Envelope; Ill posed; L curve; Multiobjective optimization; Preconditioned conjugate gradients; Tikhonov regularization

Indexed keywords

EID: 0031327507     PISSN: 10618600     EISSN: 15372715     Source Type: Journal    
DOI: 10.1080/10618600.1997.10474753     Document Type: Article

References (20)

1. Engl, H.W. (1993), “Regularization Methods for the Stable Solution of Inverse Problems,” Surveys of Mathematics in Industry, 3, 71-143.
2. Geman, S., and McClure, D. (1985), “Bayesian Image Analysis: An Application to Single Photon Emission Tomography,” in Proceedings of the Statistical Computing Section, Washington, DC: American Statistical Association, pp. 12-18.
3. Green, P. (1990), “Bayesian Reconstructions from Emission Tomography Using a Modified EM Algorithm,” IEEE Transactions on Medical Imaging, 9, 84-93.
4. Hanke, M., and Hansen, P.C. (1993), “Regularization Methods for Large-Scale Problems,” Surveys of Mathematics in Industry, 3, 253-315.
5. Hansen, P.C. (1992), “Analysis of Discrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review, 34, 561-580.
6. Hansen, P.C., and O’Leary, D.P. (1993), “The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems,” SIAM Journal of Scientific Computing, 14, 1487-1506.
7. Hebert, T., and Leahy, R. (1990), “A Generalized EM Algorithm for the 3-D Bayesian Reconstruction from Poisson Data Using Gibbs Priors,” IEEE Transactions on Medical Imaging, 8, 194-202.
8. Honerkamp, J., and Weese, J. (1990), “Tikhonov Regularization Methods for Ill-Posed Problems,” Continuum Mechanics Thermodynamics, 2, 17-30.
9. Huber, P.J. (1981), Robust Statistics, New York: John Wiley.
10. Kaufman, L. (1993), “Maximum Likelihood, Least Squares, and Penalized Least Squares for PET,” IEEE Transactions on Medical Imaging, 12, 200-214.
11. Kaufman, L., and Neumaier, A. (1994), “Image Reconstruction Through Regularization by Envelope Guided Conjugate Gradients,” Technical Report CS94/4-14, AT&T Bell Laboratories, Murray Hill, NJ (available through the WWW, http://cm.bell-labs.com/cm/cs/doc/94/4-14.ps.gz, 2MB).
12. Lalush, D.S., and Tsui, B.M.W. (1992), “Simulation Evaluation of Gibbs Prior Distributions for Use in Maximum A Posteriori SPECT Reconstructions,” IEEE Transactions Medical Imaging, 11, 267-275.
13. Lawson, C.L., and Hanson, R.J. (1974), Solving Least Squares Problems, Englewood Cliffs, NJ: Prentice-Hall.
14. Morozov, V.A. (1984), Methods for Solving Incorrectly Posed Problems, Berlin: Springer-Verlag.
15. Sawaragi, Y., Nakayama, H., and Tanino, T. (1985), Theory of Multiobjective Optimization, Orlando, FL: Academic Press.
16. Symonds, M., Han, Y.-S., Santago, P., and Snyder, W. (1994), “Reconstruction of Positron Emission Tomography Images by Using Maximum a Posteriori and Mean Field Annealing Techniques,” Physics of Medical Imaging, Proceedings ofSPIE, Vol. 2613, pp. 212-222.
17. Shepp, L., and Vardi, Y. (1982), “Maximum Likelihood Reconstruction in Positron Emission Tomography,” IEEE Transactions on Medical Imaging, 1, 113-122.
18. Tikhonov, A.N. (1963a), “Solution of Incorrectly Formulated Problems and the Regularization Method,” Soviet Mathematics Doklady, 4, 1035-1038.
19. Tikhonov, A.N. (1963b), “Regularization of Incorrectly Posed Problems,” Soviet Mathematics Doklady. 4, 1624-1627.
20. Vogel, C.R., and Oman, M.E. (1996), “Iterative Methods for Total Variation Denoising,” SIAM Journal on Statistical Computing 17, 227-238.