Nonlocal Bounded Variations with Applications

TL;DR

This paper explores fractional bounded variation spaces for modeling sharp transitions and interfaces, establishing their properties and applying them to develop new image denoising models with proven mathematical foundations.

Contribution

It introduces and compares two fractional BV spaces, proves their properties, and applies these to create novel image denoising models with rigorous theoretical support.

Findings

The space $bv^\alpha$ equals the Gagliardo-Slobodeckij space $W^{\alpha,1}$.

Density of smooth functions with compact support is established for convex domains.

New image denoising models based on fractional BV spaces are proposed and analyzed.

Abstract

Motivated by problems where jumps across lower dimensional subsets and sharp transitions across interfaces are of interest, this paper studies the properties of fractional bounded variation ($BV$)-type spaces. Two different natural fractional analogs of classical $BV$ are considered: $BV^\alpha$, a space induced from the Riesz-fractional gradient that has been recently studied by Comi-Stefani; and $bv^\alpha$, induced by the Gagliardo-type fractional gradient often used in Dirichlet forms and Peridynamics - this one is naturally related to the Caffarelli-Roquejoffre-Savin fractional perimeter. Our main theoretical result is that the latter $bv^\alpha$ actually corresponds to the Gagliardo-Slobodeckij space $W^{\alpha,1}$. As an application, using the properties of these spaces, novel image denoising models are introduced and their corresponding Fenchel pre-dual formulations are derived. The latter requires density of smooth functions with compact support. We establish this density property for convex domains.