Journal of Mathematical Imaging and Vision

, Volume 59, Issue 3, pp 498–514 | Cite as

Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part I: Modelling and Theory

  • Michael Hintermüller
  • Carlos N. Rautenberg


A weighted 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 weighted 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.


Image restoration Weighted total variation regularization Spatially distributed regularization weight Fenchel predual Bilevel optimization Variance corridor 

Mathematics Subject Classification

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


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Weierstrass InstituteBerlinGermany
  2. 2.Department of MathematicsHumboldt-University of BerlinBerlinGermany

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