  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
quasi-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.
