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

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Abstract

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.

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Authors and Affiliations

  1. Weierstrass Institute, Mohrenstrasse 39, 10117, Berlin, Germany

    Michael Hintermüller

  2. Department of Mathematics, Humboldt-University of Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Michael Hintermüller & Carlos N. Rautenberg

Authors
  1. Michael Hintermüller
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  2. Carlos N. Rautenberg
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Correspondence to Michael Hintermüller.

Additional information

This research was carried out in the framework of MATHEON supported by the Einstein Foundation Berlin within the ECMath Projects OT1, SE5 and SE15 as well as by the DFG under Grant No. HI 1466/7-1 “Free Boundary Problems and Level Set Methods”.

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Hintermüller, M., Rautenberg, C.N. Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part I: Modelling and Theory. J Math Imaging Vis 59, 498–514 (2017). https://doi.org/10.1007/s10851-017-0744-2

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  • DOI: https://doi.org/10.1007/s10851-017-0744-2

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