  2D Total Variation Denoising (TVD) is a widely used technique for image
denoising. It is also an important nonparametric regression method for
estimating functions with heterogenous smoothness. Recent results have shown
the TVD estimator to be nearly minimax rate optimal for the class of functions
with bounded variation. In this paper, we complement these worst case
guarantees by investigating the adaptivity of the TVD estimator to functions
which are piecewise constant on axis aligned rectangles. We rigorously show
that, when the truth is piecewise constant, the ideally tuned TVD estimator
performs better than in the worst case. We also study the issue of choosing the
tuning parameter. In particular, we propose a fully data driven version of the
TVD estimator which enjoys similar worst case risk guarantees as the ideally
tuned TVD estimator.
