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

37 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

Bibliographical note

KAUST Repository Item: Exported on 2021-04-02
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: In Cambridge, this project has been supported by King Abdullah University of Science and Technology (KAUST) Award No. KUK-I1-007-43, EPSRC grants Nr. EP/J009539/1 “Sparse & Higher-order Image Restoration”, and Nr. EP/M00483X/1 “Efficient computational tools for inverse imaging problems”. In Quito, the project has been supported by the Escuela Politécnica Nacional de Quito under award PIS 12-14 and the MATHAmSud project SOCDE “Sparse Optimal Control of Differential Equations”. When in Quito, T. Valkonen was moreover supported by a Prometeo scholarship of the Senescyt (Ecuadorian Ministry of Science, Technology, Education, and Innovation).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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