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dc.contributor.authorde, los Reyes JCen
dc.contributor.authorSchönlieb, CBen
dc.contributor.authorValkonen, Ten
dc.date.accessioned2015-09-16T10:41:44Z
dc.date.available2015-09-16T10:41:44Z
dc.date.issued2015-09-16en
dc.identifier.citationJournal of Mathematical Analysis and Applications 2015, 434(1): 464-500. doi:10.1016/j.jmaa.2015.09.023en
dc.identifier.issn0022-247X
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/250589
dc.description.abstractWe 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 generalized 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.
dc.description.sponsorshipIn 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).
dc.languageEnglishen
dc.language.isoenen
dc.publisherElsevier
dc.rightsCreative Commons Attribution 4.0 International License
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/
dc.subjecttotal variationen
dc.subjecttotal generalised variationen
dc.subjectbi-level optimisationen
dc.subjectoptimalityen
dc.subjectparameter choiceen
dc.titleThe structure of optimal parameters for image restoration problemsen
dc.typeArticle
dc.description.versionThis is the final version of the article. It first appeared from Elsevier via http://dx.doi.org/10.1016/j.jmaa.2015.09.023en
prism.endingPage500
prism.publicationDate2015en
prism.publicationNameJournal of Mathematical Analysis and Applicationsen
prism.startingPage464
prism.volume434en
dc.rioxxterms.funderEPSRC
dc.rioxxterms.funderEP/J009539/1
dc.rioxxterms.funderEP/M00483X/1
rioxxterms.versionofrecord10.1016/j.jmaa.2015.09.023en
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserveden
rioxxterms.licenseref.startdate2015-09-16en
dc.identifier.eissn1096-0813
rioxxterms.typeJournal Article/Reviewen
pubs.funder-project-idEPSRC (EP/J009539/1)
pubs.funder-project-idEPSRC (EP/M00483X/1)
pubs.funder-project-idEPSRC (EP/N014588/1)
pubs.funder-project-idAlan Turing Institute (unknown)
pubs.funder-project-idEuropean Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (691070)


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