Calculus of Variations and Geometric Measure Theory

K. Bessas - G. Stefani

Non-local $BV$ functions and a denoising model with $L^1$ fidelity

created by stefani on 21 Oct 2022
modified on 20 Nov 2022



Inserted: 21 oct 2022
Last Updated: 20 nov 2022

Year: 2022

ArXiv: 2210.11958 PDF


We study a general total variation denoising model with weighted $L^1$ fidelity, where the regularizing term is a non-local variation induced by a suitable (non-integrable) kernel $K$, and the approximation term is given by the $L^1$ norm with respect to a non-singular measure with positively lower-bounded $L^\infty$ density. We provide a detailed analysis of the space of non-local $BV$ functions with finite total $K$-variation, with special emphasis on compactness, Lusin-type estimates, Sobolev embeddings and isoperimetric and monotonicity properties of the $K$-variation and the associated $K$-perimeter. Finally, we deal with the theory of Cheeger sets in this non-local setting and we apply it to the study of the fidelity in our model.

Keywords: Image denoising, total variation denoising models, non-local variation, non-local perimeter, non-local Cheeger problem, non-local Laplacian operator