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Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models.

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De Los Reyes, JC 
Schönlieb, C-B 
Valkonen, T 


We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasi-Newton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV 2 and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.



Bilevel optimisation, Image quality measures, Total variation regularisers

Journal Title

J Math Imaging Vis

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Springer Science and Business Media LLC
Engineering and Physical Sciences Research Council (EP/M00483X/1)
Engineering and Physical Sciences Research Council (EP/N014588/1)
Engineering and Physical Sciences Research Council (EP/J009539/1)
Alan Turing Institute (unknown)
European Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (691070)
King Abdullah University of Science and Technology (KAUST) (Grant ID: KUKI1-007-43), Engineering and Physical Sciences Research Council (Grant IDs: Nr. EP/J009539/1 “Sparse & Higher-order Image Restoration” and Nr. EP/M00483X/1 “Efficient computational tools for inverse imaging problems”), Escuela Politécnica Nacional de Quito (Grant ID: PIS 12-14, MATHAmSud project SOCDE “Sparse Optimal Control of Differential Equations”), Leverhulme Trust (project on “Breaking the non-convexity barrier”), SENESCYT (Ecuadorian Ministry of Higher Education, Science, Technology and Innovation) (Prometeo Fellowship)