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Iterative Regularization Techniques in Image Reconstruction

  • Martin Hanke

Abstract

In this survey we review recent developments concerning the efficient iterative regularization of image reconstruction problems in atmospheric imaging. We present a number of preconditioners for the minimization of the corresponding Tikhonov functional, and discuss the alternative of terminating the iteration early, rather than adding a stabilizing term in the Tikhonov functional. The methods are examplified for a (synthetic) model problem.

Keywords

Point Spread Function Coarse Grid Conjugate Gradient Method Tikhonov Regularization Signal Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Å. Björck, A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations, BIT, 28 (1988), pp. 659–670.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    R. Chan AND M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), pp. 427–482.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    R. H. Chan, J. G. Nagy and R. J. Plemmons, FFT-based preconditioners for Toeplitz-block least squares problems, SIAM J. Numer. Anal., 30 (1993), pp. 1740–1768.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    M. Defrise AND C. De Mol, A note on stopping rules for iterative regularization methods and filtered SVD, in: P. C. Sabatier, ed., Inverse Problems: An Interdisciplinary Study, Academic Press, London, Orlando, San Diego, New York, 1987, 261–268.Google Scholar
  5. 5.
    H. W. Engl, M. Hanke, AND A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.Google Scholar
  6. 6.
    A. Frommer AND P. Maass. Fast CG-based methods for Tikhonov regularization, SIAM J. Sci. Comput. (1999), to appear.Google Scholar
  7. 7.
    G. H. Golub AND C. F. Van Loan, Matrix Computations, 3rd. ed., John Hopkins Univ. Press, Baltimore, 1996.MATHGoogle Scholar
  8. 8.
    G. H. Golub AND U. Von Matt, Generalized Cross-Validation for large-scale problems, J. Comput. Graph. Statist., 6 (1997), pp. 1–34.MathSciNetCrossRefGoogle Scholar
  9. 9.
    C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.MATHGoogle Scholar
  10. 10.
    W. Hackbusch, Integral Equations. Theory and Numerical Treatment, Birkhauser Verlag, Basel, 1995.Google Scholar
  11. 11.
    M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems, Longman Scientific & Technical, Harlow, Essex, 1995.MATHGoogle Scholar
  12. 12.
    M. Hanke AND J. G. Nagy, Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques, Inverse Problems, 12 (1996), pp. 157–173.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    —, Inverse Toeplitz preconditioners for ill-posed problems, Linear Algebra Appl., 284 (1998), pp. 137–156.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    M. Hanke, J. G. Nagy AND R. J. Plemmons, Preconditioned iterative regularization, in Numerical Linear Algebra, L. Reichel, A. Ruttan, and R. S. Varga, eds., de Gruyter, Berlin, 1993, pp. 141–163.Google Scholar
  15. 15.
    L. Kaufman AND A. Neumaier, Regularization of ill-posed problems by envelope guided conjugate gradients, J. Comput. Graph. Statist., 6 (1997), pp. 451–463.MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. E. Kilmer, Cauchy-like preconditioners for 2-dimensional ill-posed problems, manuscript, 1997.Google Scholar
  17. 17.
    R. L. Lagenduk AND J. Biemond, Iterative Identification and Restoration of Images, Kluwer, Boston, 1991.CrossRefGoogle Scholar
  18. 18.
    D. P. O’leary AND J. A. Simmons, A bidiagonalization-regularization procedure for large scale discretizations of ill-posed problems, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 474–489.MathSciNetCrossRefGoogle Scholar
  19. 19.
    R. Plato, Die Effizienz des Diskrepanzverfahrens für Verfahren vom Typ der konjugierten Gradienten, in: E. Schock, ed., Beiträge zur Angewandten Analysis und Informatik, Shaker Verlag, Aachen, 1994, pp. 288–297. In German.Google Scholar
  20. 20.
    K. Riley AND C. R. Vogel, Preconditioners for linear systems arising in image reconstruction, in: F. T. Luk, ed., Advanced Signal Processing Algorithms, Architectures, and Implementations VIII, Proceedings of SPIE Vol. 3461 (1998), pp. 372–380.Google Scholar
  21. 21.
    M. C. Roggemann AND B. Welsh, Imaging Through Turbulence, CRC Press, Boca Raton, Florida, 1996.Google Scholar
  22. 22.
    D. Slepian, Some comments on Fourier analysis, uncertainty and modelling, SIAM Rev., 25 (1985), pp. 379–393.MathSciNetCrossRefGoogle Scholar
  23. 23.
    C. R. Vogel AND M. Hanke, Two-level preconditioners for regularized inverse problems II: Implementation and numerical results, manuscript, 1998.Google Scholar
  24. 24.
    C. R. Vogel AND M. E. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Proc., 7 (1998), pp. 813–824.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag/Wien 2000

Authors and Affiliations

  • Martin Hanke
    • 1
  1. 1.Fachbereich MathematikJohannes-Gutenberg-Universität MainzMainzGermany

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