, Volume 60, Issue 1, pp 1–27 | Cite as

Denoising with higher order derivatives of bounded variation and an application to parameter estimation

  • O. Scherzer


Regularization with functions of bounded variation has been proven to be effective for denoising signals and images. This nonlinear regularization technique, in contrast with linear regularization techniques like Tikhonov regularization, has the advantage that discontinuities in signals and images can be located very precisely. In this paper bounded variation regularization is generalized to functions with higher order derivatives of bounded variation. This concept is applied to locate discontinuities in derivatives, which has important applications in parameter estimation problems.

AMS Subject Classifications

65J10 65J15 65J20 65K10 26A45 

Key words

Nondifferentiable optimization problems nonreflexive spaces regularization bounded variation norm and generalizations 


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  1. [1]
    Barrett, R., Berry, M., Chan, T., Demmel, J., Donato, J., Dongarra, J., Eijkhoui, V., Pozo, R., Romine, Ch., van der Vorst, H.: TEMPLATES for the solution of linear systems: building blocks for iterative methods. Software Package for the solution of linear systems.Google Scholar
  2. [2]
    Deimling, K.: Nonlinear functional analysis. Berlin, Heidelberg, New York: Springer 1980.Google Scholar
  3. [3]
    Dobson, D., Santosa, F.: An image enhancement technique for electrical impedance tomography. Inverse Problems10, 317–334 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Dobson, D., Scherzer, O.: Analysis of regularized total variation penalty methods for denoising. Inverse Problems12, 601–617 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Dobson, D., Vogel, C. R.: Convergence of an iterative method for total variation denoising. SIAM J. Numer. Anal.34, 1779–1791 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Engl, H. W., Kunisch, K., Neubauer, A.: Convergence rates for Tikhonov regularization of non-linear ill-posed problems. Inverse Problems5, 523–540 (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Evans, L. C., Gariepy, R. F.: Measure theory and fine properties of functions. Ann Arbor: CRC Press 1995.Google Scholar
  8. [8]
    Groetsch, C. W.: Spectral methods for linear inverse problems with unbounded operators. J. Approx. Theory70, 16–28 (1992).zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Isakov, V.: Inverse source problems. Rhode Island: AMS 1990.zbMATHGoogle Scholar
  10. [10]
    Kohn, R. V., Lowe, B. D.: A variational method for parameter identification. RAIRO. Math. Modell. Numer. Anal.22, 119–158 (1988).zbMATHMathSciNetGoogle Scholar
  11. [11]
    Kunisch, K.: Inherent identifiability of parameters in elliptic differential equations. J. Math. Anal. Appl.132, 453–472 (1988).zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Morel, J.-M., Solimini, S.: Variational Methods in Image Segmentation. Basel: Birkhäuser 1995.Google Scholar
  13. [13]
    Neubauer, A.: Tikhonov regularisation for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation. Inverse Problems5, 541–557 (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Osher, S., Rudin, L.: Feature oriented image enhancement using shock filters. SIAM J. Numer. Anal.27, 919–940 (1990).zbMATHCrossRefGoogle Scholar
  15. [15]
    Scherzer, O., Engl, H. W., Kunisch, K.: Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems. SIAM J. Numer. Anal.30, 1796–1838 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Vainikko, G.: On the discretization and regularization of ill-posed problems with noncompact operators. Num. Funct. Anal. Opt.13, 381–396 (1992).zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Vogel, C., Oman, M.: Iterative methods for total variation denoising. SIAM J. Sci. Comput.17, 227–238 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Weidmann, J.: Linear operators in Hilbert spaces. Berlin, Heidelberg, New York, Tokyo: Springer 1990.Google Scholar
  19. [19]
    Ziemer, W. P.: Weakly Differentiable functions. Berlin, Heidelberg, New York, Tokyo: Springer 1980.Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • O. Scherzer
    • 1
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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