New Risk Bounds for 2D Total Variation Denoising

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