  Due to their ability to handle discontinuous images while having a
well-understood behavior, regularizations with total variation (TV) and total
generalized variation (TGV) are some of the best-known methods in image
denoising. However, like other variational models including a fidelity term,
they crucially depend on the choice of their tuning parameters. A remedy is to
choose these automatically through multilevel approaches, for example by
optimizing performance on noisy/clean image pairs. In this work, we consider
such methods with space-dependent parameters which are piecewise constant on
dyadic grids, with the grid itself being part of the minimization. We prove
existence of minimizers for fixed discontinuous parameters under mild
assumptions on the data, which lead to existence of finite optimal partitions.
We further establish that these assumptions are equivalent to the commonly used
box constraints on the parameters. On the numerical side, we consider a simple
subdivision scheme for optimal partitions built on top of any other bilevel
optimization method for scalar parameters, and demonstrate its improved
performance on some representative test images when compared with constant
optimized parameters.
