Journal of Scientific Computing

, Volume 54, Issue 2–3, pp 269–310 | Cite as

Higher-Order TV Methods—Enhancement via Bregman Iteration

  • Martin Benning
  • Christoph Brune
  • Martin BurgerEmail author
  • Jahn Müller


In this work we analyze and compare two recent variational models for image denoising and improve their reconstructions by applying a Bregman iteration strategy. One of the standard techniques in image denoising, the ROF-model (cf. Rudin et al. in Physica D 60:259–268, 1992), is well known for recovering sharp edges of a signal or image, but also for producing staircase-like artifacts. In order to overcome these model-dependent deficiencies, total variation modifications that incorporate higher-order derivatives have been proposed (cf. Chambolle and Lions in Numer. Math. 76:167–188, 1997; Bredies et al. in SIAM J. Imaging Sci. 3(3):492–526, 2010). These models reduce staircasing for reasonable parameter choices. However, the combination of derivatives of different order leads to other undesired side effects, which we shall also highlight in several examples.

The goal of this paper is to analyze capabilities and limitations of the different models and to improve their reconstructions in quality by introducing Bregman iterations. Besides general modeling and analysis we discuss efficient numerical realizations of Bregman iterations and modified versions thereof.


Total variation regularization Higher order methods Staircasing Exact solutions Bregman iteration 



Financial support is acknowledged to the German Science Foundation (DFG) via grants SFB 656 (Subproject B2) and BU 2327/1. The third author thanks Stanley Osher for initiating his interest in variational methods for image processing.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Martin Benning
    • 1
  • Christoph Brune
    • 2
  • Martin Burger
    • 1
    Email author
  • Jahn Müller
    • 1
  1. 1.Institut für Numerische und Angewandte MathematikWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Department of MathematicsUniversity of California Los AngelesLos AngelesUSA

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