  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.
