, Volume 81, Issue 2–3, pp 109–135 | Cite as

Error estimation for Bregman iterations and inverse scale space methods in image restoration



In this paper, we consider error estimation for image restoration problems based on generalized Bregman distances. This error estimation technique has been used to derive convergence rates of variational regularization schemes for linear and nonlinear inverse problems by the authors before (cf. Burger in Inverse Prob 20: 1411–1421, 2004; Resmerita in Inverse Prob 21: 1303–1314, 2005; Inverse Prob 22: 801–814, 2006), but so far it was not applied to image restoration in a systematic way. Due to the flexibility of the Bregman distances, this approach is particularly attractive for imaging tasks, where often singular energies (non-differentiable, not strictly convex) are used to achieve certain tasks such as preservation of edges. Besides the discussion of the variational image restoration schemes, our main goal in this paper is to extend the error estimation approach to iterative regularization schemes (and time-continuous flows) that have emerged recently as multiscale restoration techniques and could improve some shortcomings of the variational schemes. We derive error estimates between the iterates and the exact image both in the case of clean and noisy data, the latter also giving indications on the choice of termination criteria. The error estimates are applied to various image restoration approaches such as denoising and decomposition by total variation and wavelet methods. We shall see that interesting results for various restoration approaches can be deduced from our general results by just exploring the structure of subgradients.


image restoration error estimation iterative regularization Bregman distance total variation wavelets 

AMS Subject Classifications

Primary 47A52 Secondary 49M30 94A08 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Acar R. and Vogel C.R. (1994). Analysis of bounded variation penalty method for ill-posed problems. Inverse Prob 10: 1217–1229 MATHCrossRefMathSciNetGoogle Scholar
  2. Ambrosio L., Fusco N. and Pallara D. (2000). Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford MATHGoogle Scholar
  3. Andreu F., Ballester C., Caselles V. and Mazon J.M. (2001). Minimizing total variation flow. Diff Int Equ 14: 321–360 MATHMathSciNetGoogle Scholar
  4. Aubert G. and Aujol J. F. (2005). Modeling very oscillating signals, application to image processing. Appl Math Optim 51: 163–182 MATHCrossRefMathSciNetGoogle Scholar
  5. Bachmayr M. (2007). Iterative total variation methods for nonlinear inverse problems. Master Thesis. Johannes Kepler University, Linz Google Scholar
  6. Bregman L.M. (1967). 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 Math Phys 7: 200–217 CrossRefGoogle Scholar
  7. Burger M. and Osher S. (2004). Convergence rates for convex variational regularization. Inverse Prob 20: 1411–1421 MATHCrossRefMathSciNetGoogle Scholar
  8. Burger M., Frick K., Osher S. and Scherzer O. (2007). Inverse total variation flow. SIAM Multiscale Mod Simul 6(2): 366–395 CrossRefGoogle Scholar
  9. Burger M., Gilboa G., Osher S. and Xu J. (2006). Nonlinear inverse scale space methods. Commun Math Sci 4: 179–212 MATHMathSciNetGoogle Scholar
  10. Chambolle A. (2004). An algorithm for total variation regularization and denoising. J Math Imaging Vis 20: 89–97 CrossRefMathSciNetGoogle Scholar
  11. Chambolle A., DeVore R., Lee N.Y. and Lucier B. (1998). Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans Image Proc. 7: 319–335 MATHCrossRefMathSciNetGoogle Scholar
  12. Chan T. and Shen J. (2005). Image processing and analysis. SIAM, Philadelphia MATHGoogle Scholar
  13. Daubechies, I., Teschke, G.: Wavelet-based image decompositions by variational functionals. In: (Truchetet, F., ed) Wavelet applications in industrial processing. Proc SPIE 5266, pp. 94–105 (2004).Google Scholar
  14. Donoho D. and Johnstone I. (1994). Ideal spatial adaptation via wavelet shrinkage. Biometrika 81: 425–455 MATHCrossRefMathSciNetGoogle Scholar
  15. Engl, H. W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Kluwer, Dordrecht (1996) (Paperback edition 2000)Google Scholar
  16. Ekeland I. and Temam R. (1999). Convex analysis and variational problems. Corrected reprint edition. SIAM, Philadelphia Google Scholar
  17. Feng X. and Prohl A. (2003). Analysis of total variation flow and its finite element approximations. Math Meth M2AN 37: 533–556 MATHMathSciNetGoogle Scholar
  18. Gao H.Y. and Bruce A.G. (1997). WaveShrink with firm shrinkage. Stat Sin 7: 855–874 MATHMathSciNetGoogle Scholar
  19. Gilboa G., Sochen N. and Zeevi Y.Y. (2006). Estimation of optimal PDE-based denoising in the SNR sense. IEEE TIP 15(8): 2269–2280 Google Scholar
  20. He, L., Chung, T. C., Osher, S., Fang, T., Speier, P.: MR image reconstruction by using the iterative refinement method and nonlinear inverse scale space methods. UCLA, CAM 06-35Google Scholar
  21. Holmes R.B. (1975). Geometric functional analysis and its applications. Springer, New York MATHGoogle Scholar
  22. Kindermann S., Osher S. and Xu J. (2006). Denoising by BV-duality. J Sci Comput 28: 411–444 MATHCrossRefMathSciNetGoogle Scholar
  23. Koenderink J.J. (1988). Scale-time. Biol Cybern 58: 159–162 MATHCrossRefMathSciNetGoogle Scholar
  24. Lie J. and Nordbotten J.M. (2007). Inverse scale spaces for nonlinear regularization. J Math Imaging Vis 27(1): 41–50 CrossRefMathSciNetGoogle Scholar
  25. Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. AMS, Providence (2001)Google Scholar
  26. Osher S., Burger M., Goldfarb D., Xu J. and Yin W. (2005). An iterative regularization method for total variation-based image restoration. SIAM Multiscale Model Simul 4: 460–489 MATHCrossRefMathSciNetGoogle Scholar
  27. Osher S.J., Sole A. and Vese L. (2003). Image decomposition and restoration using total variation minimization and the H −1 norm. SIAM Multiscale Model Simul 1: 349–370 MATHCrossRefMathSciNetGoogle Scholar
  28. Perona P. and Malik J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Mach Intell 12: 629–639 CrossRefGoogle Scholar
  29. Resmerita E. (2005). Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Prob 21: 1303–1314 MATHCrossRefMathSciNetGoogle Scholar
  30. Resmerita E. and Scherzer O. (2006). Error estimates for non-quadratic regularization and the relation to enhancing. Inverse Prob 22: 801–814 MATHCrossRefMathSciNetGoogle Scholar
  31. Rudin L.I., Osher S.J. and Fatemi E. (1992). Nonlinear total variation based noise removal algorithms. Physica D 60: 259–268 MATHCrossRefGoogle Scholar
  32. Scherzer O. and Groetsch C. (2001). Inverse scale space theory for inverse problems. In: Kerckhove, M. (eds) Scale-space and morphology in computer vision. Proc. 3rd Int. Conf. Scale-space, pp 317–325. Springer, Berlin Google Scholar
  33. Schoepfer F., Louis A.K. and Schuster T. (2006). Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Prob 22: 311–329 MATHCrossRefGoogle Scholar
  34. Tadmor E., Nezzar S. and Vese L. (2004). A multiscale image representation using hierarchical (BV;L2) decompositions. Multiscale Model Simul 2: 554–579 MATHCrossRefMathSciNetGoogle Scholar
  35. Witkin, A. P.: Scale-space filtering. In: Proc. Int. Joint Conf. on Artificial Intelligence, Karlsruhe 1983, pp. 1019–1023Google Scholar
  36. Xu J. and Osher S. (2007). Iterative regularization and nonlinear inverse scale space applied to wavelet based denoising. IEEE Trans Image Proc 16(2): 534–544 CrossRefGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Institut für Numerische und Angewandte MathematikWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria

Personalised recommendations