Image Denoising: Learning Noise Distribution via PDE-Constrained Optimization

Juan Carlos De Los Reyes^* and Carola-Bibiane Schönlieb

We propose a PDE-constrained optimization approach for the determination of noise distribution in total variation (TV) image denoising. A bilevel optimization problem for the determination of the weights correspondent to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems.

Mathematics Subject Classification: 35Q93 94A08

Keywords: Image denoising; noise distribution; PDE-constrained optimization; Huber regularization

Minisymposion: Noise Estimation, Model Selection and Bilevel Optimization