WIAS Preprint No. 2611, (2019)

Generating structured non-smooth priors and associated primal-dual methods


  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Papafitsoros, Kostas
    ORCID: 0000-0001-9691-4576

2010 Mathematics Subject Classification

  • 94A08 68U10 49K20 49M37 49M15 26A45


  • Non-smooth priors, image processing, total variation, total generalized variation, bilevel optimization, regularization parameter selection




The purpose of the present chapter is to bind together and extend some recent developments regarding data-driven non-smooth regularization techniques in image processing through the means of a bilevel minimization scheme. The scheme, considered in function space, takes advantage of a dualization framework and it is designed to produce spatially varying regularization parameters adapted to the data for well-known regularizers, e.g. Total Variation and Total Generalized variation, leading to automated (monolithic), image reconstruction workflows. An inclusion of the theory of bilevel optimization and the theoretical background of the dualization framework, as well as a brief review of the aforementioned regularizers and their parameterization, makes this chapter a self-contained one. Aspects of the numerical implementation of the scheme are discussed and numerical examples are provided.

Appeared in

  • M. Hintermüller, K. Papafitsoros, Chapter 11: Generating Structured Nonsmooth Priors and Associated Primal-dual Methods, in: Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2 , R. Kimmel, X.-Ch. Tai, eds., vol. 20 of Handbook of Numerical Analysis, Elsevier, 2019, pp. 437--502, DOI 10.1016/bs.hna.2019.08.001 .

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