We propose a bilevel learning approach for the determination of optimal weights in nonlocal image denoising. We consider both spatial weights in front of the nonlocal regularizer, as well as weights within the kernel of the nonlocal operator. In both cases we investigate the differentiability of the solution operator in function spaces and derive a first order optimality system that characterizes local minima. For the numerical solution of the problems, we propose a second- order optimization algorithm in combination with a suitable finite element discretization of the nonlocal denoising models.
Opportunities and challenges of bilevel optimization for applications in image processing are addressed. On the mathematical side, analytical and algorithmic difficulties as well as solution concepts are highlighted. Concerning imaging applications, statistics based automated regularization function choice rules, and blind deconvolution problems are in the focus.
We will bind together and extend some recent developments regarding data-driven non-smooth regularization techniques in image processing through the means of bilevel minimization schemes. The schemes, considered in function space, take advantage of dualization frameworks and they are designed to produce spatially varying regularization parameters adapted to the data for well-known regularizers, e.g. Total Variation and Total Generalized Variation, leading to automated (monolithic), image reconstruction workflows.