# Robustness of Sparse Optimal Control in Total Variation Image Denoising

Luca Calatroni^{*}, Juan Carlos De Los Reyes and Carola SchĂ¶nlieb

We consider the optimal control approach proposed by De Los Reyes-SchĂ¶nlieb (2012) to estimate the distribution of noise and the optimal weights for the corresponding total variation denoising model. To check the robustness of the estimation we base the optimal control approach upon a training set of variable size of ``perfect'' images $u_k$ and their corresponding noisy versions $f_k$. This requires the solution of a bilevel optimsation problem constrained to a large system of nonlinear PDEs (the system being as large as the size of the training set). In this presentation we discuss the well-posedness of this optimal control approach, its numerical solution and its extension to sparse optimal weights. Depending on time, we shall also consider a slightly different modification of the original model where the optimal weights are scalar functions which adapt to the underlying image structure.

Mathematics Subject Classification: 49J20

Keywords: Image denoising; noise distribution; PDE-constrained optimisation; Huber regularisation; sparsity.

Minisymposion: Noise Estimation, Model Selection and Bilevel Optimization