  The use of the fractional Laplacian in image denoising and regularization of
inverse problems has enjoyed a recent surge in popularity, since for
discontinuous functions it can behave less aggressively than methods based on
$H^1$ norms, while being linear and computable with fast spectral numerical
methods. In this work, we examine denoising and linear inverse problems
regularized with fractional Laplacian in the vanishing noise and regularization
parameter regime. The clean data is assumed piecewise constant in the first
case, and continuous and satisfying a source condition in the second. In these
settings, we prove results of convergence of level set boundaries with respect
to Hausdorff distance, and additionally convergence rates in the case of
denoising and indicatrix clean data. The main technical tool for this purpose
is a family of barriers constructed by Savin and Valdinoci for studying the
fractional Allen-Cahn equation. To help put these fractional methods in
context, comparisons with the total variation and classical Laplacian are
provided throughout.
