PDE constrained optimisation for learning the optimal image de-noising model

We will discuss the concept of an optimal image de-noising model made up of a data discrepancy term and an appropriate regularisation. Starting with a generic, parametrised variational approach consisting of different regularisation functionals and discrepancy terms, we phrase the quest for an optimal model as seeking those parameters that return an optimal subset of the generic model, optimal for a training set of exemplar images. Formally, this approach constitutes a bilevel optimisation with a nonlinear PDE as constraint. The analysis and numerical realisation of this approach will be presented, furnished with various examples.