  We propose a bilevel optimization approach for the estimation of parameters
in nonlocal image denoising models. The parameters we consider are both the
fidelity weight and 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 use a second-order
trust-region algorithm in combination with a finite element discretization of
the nonlocal denoising models and we introduce a computational strategy for the
solution of the resulting dense linear systems. Several experiments illustrate
the applicability and effectiveness of our approach.
