New Risk Bounds for 2D Total Variation Denoising

Abstract

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.

Figures (4)

• Figure 1: The MSE of the ideally tuned TVD estimator $\hat{\theta}_{\mb V^*}$ is estimated with $50$ Monte carlo repetitions for a grid of $n = \sqrt{N}$ ranging from $500$ to $700$ in increments of $20.$ The true matrices were taken to be $\theta^{\mathrm{two}}$ ( blue), $\theta^{\mathrm{four}}$ ( red) and $\theta^{\mathrm{worst}}$ ( black). In each case, we have chosen the ideal tuning parameter to allow fair comparison. We plot log of estimated MSE versus log $N$ where log is taken in base $\e.$ The points are the estimated log MSE and the dotted lines are the least squares line fitted to the points. The least squares slope for $\theta^{\mathrm{two}}$ is $-0.73$ and for $\theta^{\mathrm{four}}$ is $- 0.68$ which is considerably lower than the slope for the matrix $\theta^{\mathrm{worst}}$ which is $-0.52.$
• Figure 2: The MSE of the ideally tuned TVD estimator is estimated with $50$ Monte carlo repetitions when $n = 800.$ The true matrices were taken to be such that the number of rectangular level sets is $2,4,8,16.$ In each case, we have chosen the ideal tuning parameter to allow fair comparison. We have also normalized the matrices so that $\mb V^* = 1$. We plot log of estimated MSE versus $\log_2 k$ where $k = 2,4,8,16.$ The points are the estimated log MSE and the dotted lines are the least squares line fitted to the points. The least squares slope is $0.81.$
• Figure 3: The MSE of our tuning free estimator is estimated with $50$ Monte carlo repetitions for a grid of $n = \sqrt{N}$ ranging from $160$ to $250$ in increments of $10.$ The true matrices were taken to be $\theta^{\mathrm{two}}$ ( blue), $\theta^{\mathrm{four}}$ ( red), and $\theta^{\mathrm{worst}}$ ( black). We plot log of estimated MSE versus log $N$ where log is taken in base $e.$ The circular points are the estimated log MSE and the dotted lines are the least squares line fitted to the points. The slopes of the least squares lines are $-0.47$,$-0.51$,$-0.40$ for $\theta^{\mathrm{two}}$, $\theta^{\mathrm{four}}$, $\theta^{\mathrm{worst}}$ respectively.
• Figure 4: The MSE of the tuning free TVD estimator is estimated with $50$ Monte carlo repetitions for a grid of $\mb V^* \in [10]$ and $n = 200.$ The true matrices were taken to be $\theta^{\mathrm{two}}$ ( blue), $\theta^{\mathrm{four}}$ ( red) and $\theta^{\mathrm{worst}}$ ( black) properly normalized. We plot log of estimated MSE versus log $N$ where log is taken in base $\e.$ The points are the estimated log MSE and the dotted lines are the least squares line fitted to the points. The least squares slope for $\theta^{\mathrm{two}}$ is $1.16$, for $\theta^{\mathrm{four}}$ is $1.07$ and for the matrix $\theta^{\mathrm{worst}}$ it is $0.94.$

Theorems & Definitions (92)

• theorem \oldthetheorem: Hütter and Rigollet
• theorem \oldthetheorem
• remark 1
• remark 2
• theorem \oldthetheorem
• remark 3
• remark 4
• remark 5
• theorem \oldthetheorem
• proposition 1