Parameter space study of optimal scale-dependent weights in TV image denoising
Juan Carlos De los Reyes, K. Herrera · Applicable Analysis · 2022
We investigate the choice of finite-dimensional parameter spaces within a bilevel optimization framework for selecting scale-dependent weights in total variation image denoising with non-uniform noise. Due to the pointwise box constraints on the parameter function, we prove existence of Lagrange multipliers in low regularity spaces, and derive a first-order optimality system. To cope with the difficulties related to the lack of regularity, a Moreau-Yosida regularization is introduced and convergence of the regularized optimal parameters towards the optimal weight for the original problem is verified. For each regularized bilevel problem a second-order quasi-Newton algorithm is proposed, together with a semismooth Newton scheme for solving the lower-level problem. Finally, several numerical tests are carried out to compare the different parameter space choices and draw some conclusions.