  We study the regularity of solutions to a nonlocal variational problem, which
is related to the image denoising model, and we show that, in two dimensions,
minimizers have the same H\"older regularity as the original image. More
precisely, if the datum is (locally) $\beta$-H\"older continuous for some
$\beta \in (1-s,\,1]$, where $s \in (0,1)$ is a parameter related to the
nonlocal operator, we prove that the solution is also $\beta$-H\"older
continuous.
