The structure of optimal parameters for image restoration problems

J.C. De Los Reyes, C.-B. Schönlieb, Tuomo Valkonen

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


We study the qualitative properties of optimal regularisation parameters in variational models for image restoration. The parameters are solutions of bilevel optimisation problems with the image restoration problem as constraint. A general type of regulariser is considered, which encompasses total variation (TV), total generalised variation (TGV) and infimal-convolution total variation (ICTV). We prove that under certain conditions on the given data optimal parameters derived by bilevel optimisation problems exist. A crucial point in the existence proof turns out to be the boundedness of the optimal parameters away from 0 which we prove in this paper. The analysis is done on the original - in image restoration typically non-smooth variational problem - as well as on a smoothed approximation set in Hilbert space which is the one considered in numerical computations. For the smoothed bilevel problem we also prove that it Γ converges to the original problem as the smoothing vanishes. All analysis is done in function spaces rather than on the discretised learning problem.
Original languageEnglish (US)
Pages (from-to)464-500
Number of pages37
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - Feb 2016
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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