Adaptive image processing: first order PDE constraint regularizers and a bilevel training scheme

A bilevel training scheme is used to introduce a novel class of regularizers, providing a unified approach to standard regularizers $TV$, $TGV^2$ and $NsTGV^2$. Optimal parameters and regularizers are identified, and the existence of a solution for any given set of training imaging data is proved by $\Gamma$-convergence. Explicit examples and numerical results are given.