Learning variational regularization models for image denoising by PDE constrained optimization

We review some recent learning approaches in variational image regularisation — in particular for variational image denoising — based on PDE constrained (also called bilevel) optimisation. Optimal solutions are typically parameters determining the type of regularisation and data discrepancy. A general type of regulariser is considered, which encompasses total variation (TV), total generalized variation (TGV) and infimal-convolution total variation (ICTV), as well as general data discrepancies encoding different noise distributions. We analyse those models for existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are employed to solve the models numerically. The computational verification of the developed techniques will be discussed, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.
This is joint work with Luca Calatroni, Chung Cao, Juan Carlos De los Reyes, and Tuomo Valkonen.