WIAS Preprint No. 2235, (2016)

Optimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory


  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Rautenberg, Carlos N.
    ORCID: 0000-0001-9497-9296

2010 Mathematics Subject Classification

  • 94A08 68U10 49K20 49K30 49K40 49M37 65K15


  • Image restoration, generalized total variation regularization, spatially distributed regularization weight, Fenchel predual, bilevel optimization, variance corridor




A generalized total variation model with a spatially varying regularization weight is considered. Existence of a solution is shown, and the associated Fenchel-predual problem is derived. For automatically selecting the regularization function, a bilevel optimization framework is proposed. In this context, the lower-level problem, which is parameterized by the regularization weight, is the Fenchel predual of the generalized total variation model and the upper-level objective penalizes violations of a variance corridor. The latter object relies on a localization of the image residual as well as on lower and upper bounds inspired by the statistics of the extremes.

Appeared in

  • J. Math. Imaging Vision, 59 (2017), pp. 498--514.

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