  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.
