The structure of optimal parameters for image restoration problems
de, los Reyes JC
Journal of Mathematical Analysis and Applications
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de, l. R. J., Schönlieb, C., & Valkonen, T. (2015). The structure of optimal parameters for image restoration problems. Journal of Mathematical Analysis and Applications, 434 464-500. https://doi.org/10.1016/j.jmaa.2015.09.023
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 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.
total variation, total generalised variation, bi-level optimisation, optimality, parameter choice
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).
Alan Turing Institute (unknown)
European Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (691070)
External DOI: https://doi.org/10.1016/j.jmaa.2015.09.023
This record's URL: https://www.repository.cam.ac.uk/handle/1810/250589
Creative Commons Attribution 4.0 International License
Licence URL: http://creativecommons.org/licenses/by/4.0/
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