We review some recent learning approaches in variational imaging, based on bilevel optimisation, and emphasize the importance of their treatment in function space. The paper covers both analytical and numerical techniques. Analytically, we include results on the existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are studied for the solution of the problems at hand, combining them with sampling techniques in case of large databases. The computational verification of the developed techniques is extensively documented, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.

We consider a bilevel optimization approach in function space for the choice of spatially dependent regularization parameters in TV image denoising models. First-and second-order optimality conditions for the bilevel problem are studied when the spatiallydependent parameter belongs to the Sobolev space H 1 (Ω). A combined Schwarz domain decomposition-semismooth Newton method is proposed for the solution of the full optimality system and local superlinear convergence of the semismooth Newton method is verified. Exhaustive numerical computations are finally carried out to show the suitability of the approach.2010 Mathematics Subject Classification. 47N40, 65D18, 65N06, 68W10, 65M55.

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