Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization

Abstract

We propose a PDE-constrained optimization approach for the determination of noise distribution in total variation (TV) image denoising. An 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.

On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

“…Optimal control problems within this context have been considered in [23,24,28] with applications to viscoplastic fluids, contact mechanics and imaging. Existing results concern the characterization of Clarke stationary points and their numerical approximation.…”

Minimization Based Formulations of Inverse Problems and Their Regularization

“…Variational methods have been extensively developed and studied in the context of imaging applications, cf., e.g., [41] and the references therein, where usually illposedness plays a less pronounced role than here and the purpose of the term R is rather to enhance certain image features; note that also J can be chosen to have several components in order to take into account different types of noise, cf., e.g., [12]. We also wish to point to the recent paper [30], which relies on minimization based formulations of inverse problems as well; there, the main emphasis is put on the regularized iterative solution of the resulting minimization problems by gradient type methods.…”

Techniques for Gradient-Based Bilevel Optimization with Non-smooth Lower Level Problems

“…In [23,10,11] a bilevel approach was used to learn a model of natural image statistics, which was then applied to various image restoration tasks. A variational formulation for learning a good noise model was addressed in [35] in a PDE-constrained optimization framework, with some follow-up works [6,34,7]. In machine learning bilevel optimization was used to train a SVM [4] and other techniques [27].…”