Advertisement

Nonlinear Inverse Scale Space Methods for Image Restoration

  • Martin Burger
  • Stanley Osher
  • Jinjun Xu
  • Guy Gilboa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3752)

Abstract

In this paper we generalize the iterated refinement method, introduced by the authors in [8],to a time-continuous inverse scale-space formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image.

The inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noise-free image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known.

The inverse flow is computed directly for one-dimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow accurate, efficient and straightforward implementation.

Keywords

Noisy Image Small Scale Feature Iterative Regularization Bregman Distance Quadratic Regularization 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bregman, L.M.: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comp. Math. and Math. Phys. 7, 200–217 (1967)CrossRefGoogle Scholar
  2. 2.
    Burger, M., Goldfarb, D., Osher, S., Xu, J., Yin, W.: Inverse total variation flow (in preparation)Google Scholar
  3. 3.
    Chen, G., Teboulle, M.: Convergence analysis of a proximal-like minimization algorithm using bregman functions. SIAM J. Optim. 3, 538–543 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishers, Amsterdam (1976)zbMATHGoogle Scholar
  5. 5.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (1996)zbMATHGoogle Scholar
  6. 6.
    Groetsch, C., Scherzer, O.: Nonstationary iterated Tikhonov-Morozov method and third order differential equations for the evaluation of unbounded operators. Math. Methods Appl. Sci. 23, 1287–1300 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    He, L., Marquina, A., Osher, S.: Blind deconvolution using TV regularization and Bregman iteration. Int. J. of Imaging Systems and Technology 5, 74–83 (2005)CrossRefGoogle Scholar
  8. 8.
    Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation based image restoration. Multiscale Model. and Simul. 4, 460–489 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. PAMI 12(7), 629–639 (1990)Google Scholar
  10. 10.
    Plato, R.: The discrepancy principle for iterative and parametric methods to solve linear ill-posed problems. Numer. Math. 75, 99–120 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rudin, L.I., Osher, S.J., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)zbMATHCrossRefGoogle Scholar
  12. 12.
    Scherzer, O., Groetsch, C.: Inverse scale space theory for inverse problems. In: Kerckhove, M. (ed.) Scale-Space 2001. LNCS, vol. 2106, pp. 317–325. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Scherzer, O., Weickert, J.: Relations between regularization and diffusion filtering. J. Math. Imaging and Vision 12, 43–63 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Tadmor, E., Nezzar, S., Vese, L.: A multiscale image representation using hierarchical (BV,L 2) decompositions. Multiscale Model. Simul. 2, 554–579 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Martin Burger
    • 1
  • Stanley Osher
    • 2
  • Jinjun Xu
    • 2
  • Guy Gilboa
    • 2
  1. 1.Industrial Mathematics InstituteJohannes Kepler UniversityLinzAustria
  2. 2.Department of MathematicsUCLALos AngelesUSA

Personalised recommendations