  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 $\Gamma$ converges to the original problem as the smoothing
vanishes. All analysis is done in function spaces rather than on the
discretised learning problem.
