Nonlocal total variation denoising model with weighted $L^1$ fidelity

Konstantinos Bessas (Università di Pavia)

Image denoising is a core problem in image processing: it consists in finding an approximation of a distorted image which is at the same time more regular. The competition between fidelity and regularity endows the problem with a natural variational structure, which gave rise to a broad investigation of minimization-based denoising models.  In particular, in the last years, more and more attention was given to nonlocal ones. In this talk I will present a general class of nonlocal denoising models, whose regularizing term is a nonlocal variation induced by a suitable (non-integrable) kernel K, and the fidelity term is given by the $L^1$ norm with respect to an absolutely continuous measure with positively lower-bounded $L^\infty$ density. I will discuss existence and uniqueness of solutions and regularity of their level sets, both for high and low values of the fidelity parameter. Finally, I will analyse in detail the fidelity of the model in the case of binary data given by the characteristic functions of convex sets. Part of the results that I will present were obtained in collaboration with Giorgio Stefani (SISSA, Trieste).