We consider the framework of functions of Bounded Variation for
some Imaging problems (mainly denoising and inpainting), highlighting the
main properties of Total Variation (TV) regularisation. The main challenges
when considering these models are on one hand the tradeoff between the
quality of the reconstruction and the computational costs that, for higher
regularisation models, are generally quite high and, on the other hand, the
optimal setup of the experiments in order to get accurate results.

We tackle the former issue by considering a numerical strategy that breaks
down the numerical methods solving the higher-order PDEs involved into
easier parts acting typically in one direction only (ADI splitting schemes).

The central part of the seminar will address the latter issue, i.e. the
selection of optimal parameters for a general TV-denoising model. We
formulate the problem by means of a bilevel optimisation approach where the
noise of the image is learned by means of a dictionary of images. This
approach is normally used in medical imaging applications (for example, MRI
and PET) and allows a robust and accurate estimation. Due both to the large
size of the dictionary considered and to the nonsmooth nature of the
constraints, the brute-force numerical solution of the model is
computationally demanding. We consider dynamic sampling schemes combined
with a BFGS optimisation method to solve the problem in an efficient and
accurate way. The final part of my talk will deal with the the case of
spatial dependent parameters, i.e. when the TV regularisation adapts to the
underlying image structure for the case of localised noise by the
introduction of a nonsmooth L^1-term enforcing sparsity in the cost
functional.

This is a joint work with Dr. Carola-Bibiane Schšnlieb, University of
Cambridge (UK) and Dr. Juan Carlos De Los Reyes, ModeMat, Quito (Ecuador).