Inverse Scale Space Theory for Inverse Problems

  • Otmar Scherzer
  • Chuck Groetsch
Conference paper
Part of the Lecture Notes in Computer Science 2106 book series (LNCS, volume 2106)


In this paper we derive scale space methods for inverse problems which satisfy the fundamental axioms of fidelity and causality and we provide numerical illustrations of the use of such methods in deblurring. These scale space methods are asymptotic formulations of the Tikhonov-Morozov regularization method. The analysis and illustrations relate difusion filtering methods in image processing to Tikhonov regularization methods in inverse theory.


Inverse Problem Regularization Method Scale Space Tikhonov Regularization Fundamental Axiom 
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  1. 1.
    A.B. Bakushinskii and A.V. Goncharskii. Ill-Posed Problems: Theory and Applications. Kluwer Academic Publishers, Dordrecht, Boston, London, 1994.CrossRefGoogle Scholar
  2. 2.
    M. Bertero and P. Boccacci. Introduction to Inverse Problems in Imaging. IOP Publishing, London, 1998.zbMATHCrossRefGoogle Scholar
  3. 3.
    H.W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht, 1996.zbMATHCrossRefGoogle Scholar
  4. 4.
    L.C. Evans and R.F. Gariepy. Measure Theory and Fine Properties of Functions. CRC-Press, Boca Raton, 1992.zbMATHGoogle Scholar
  5. 5.
    C.W. Groetsch and O. Scherzer. Nonstationary iterated tikhonov-morozov method and third order differential equations for the evaluation of unbounded operators. Math. Meth. Appl. Sci., 23:1287–1300, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    C.W. Groetsch. The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston, 1984.zbMATHGoogle Scholar
  7. 7.
    C.W. Groetsch and O. Scherzer. Optimal order of convergence for stable evaluation of differential operators. Electronic Journal of Differential Equations, 4:1–10, 1993. MathSciNetGoogle Scholar
  8. 8.
    M. Hanke and C.W. Groetsch. Nonstationary iterated Tikhonov regularization. J. Optim. Theory and Applications, 98:37–53, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    B. Hofmann. Mathematik inverser Probleme. (Mathematics of inverse problems). Teubner, Stuttgart, 1999.zbMATHGoogle Scholar
  10. 10.
    A. Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. Springer-Verlag, New York, 1996.zbMATHCrossRefGoogle Scholar
  11. 11.
    A.K. Louis. Inverse und Schlecht Gestellte Probleme. Teubner, Stuttgart, 1989.zbMATHCrossRefGoogle Scholar
  12. 12.
    V.A. Morozov. Methods for Solving Incorrectly Posed Problems. Springer Verlag, New York, Berlin, Heidelberg, 1984.CrossRefGoogle Scholar
  13. 13.
    M.Z. Nashed, editor. Generalized Inverses and Applications. Academic Press, New York, 1976.zbMATHGoogle Scholar
  14. 14.
    O. Scherzer. Stable evaluation of differential operators and linear and nonlinear milti-scale filtering. Electronic Journal of Differential Equations, 15:1–12, 1997. MathSciNetGoogle Scholar
  15. 15.
    A.N. Tikhonov and V.Y. Arsenin. Solutions of Ill-Posed Problems. John Wiley & Sons, Washington, D.C., 1977. Translation editor: Fritz John.Google Scholar
  16. 16.
    J. Weickert. Anisotropic Difusion in Image Processing. Teubner, Stuttgart, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Otmar Scherzer
    • 1
  • Chuck Groetsch
    • 2
  1. 1.Angewandte MathematikUniversität BayreuthBayreuthGermany
  2. 2.Department of MathematicsUniversity of CincinnatiOhioUSA

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