New Risk Bounds for 2D Total Variation Denoising
This work provides incremental improvements in risk bounds for image denoising and nonparametric regression, benefiting researchers in statistics and signal processing.
The paper tackled the problem of improving risk bounds for 2D Total Variation Denoising (TVD) by showing that when the true function is piecewise constant on axis-aligned rectangles, the ideally tuned TVD estimator performs better than worst-case guarantees, and proposed a data-driven version with similar worst-case risk guarantees.
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.