  We address the problem of optimal scale-dependent parameter learning in total
variation image denoising. Such problems are formulated as bilevel optimization
instances with total variation denoising problems as lower-level constraints.
For the bilevel problem, we are able to derive M-stationarity conditions, after
characterizing the corresponding Mordukhovich generalized normal cone and
verifying suitable constraint qualification conditions. We also derive
B-stationarity conditions, after investigating the Lipschitz continuity and
directional differentiability of the lower-level solution operator. A
characterization of the Bouligand subdifferential of the solution mapping, by
means of a properly defined linear system, is provided as well. Based on this
characterization, we propose a two-phase non-smooth trust-region algorithm for
the numerical solution of the bilevel problem and test it computationally for
two particular experimental settings.
