Optimal rates for total variation denoising

TL;DR

This paper proves that the two-dimensional total variation denoiser achieves near-optimal estimation rates for various image models, supported by spectral analysis of grid Laplacians and extended to higher dimensions and different graphs.

Contribution

It establishes a sharp oracle inequality for the 2D TV denoiser, demonstrating near-optimal rates and extending the analysis to higher dimensions and other graph structures.

Findings

Near-optimal estimation rates for 2D TV denoising.

Spectral properties of grid Laplacians underpin the analysis.

Extensions to higher dimensions and diverse graph structures.

Abstract

Motivated by its practical success, we show that the two-dimensional total variation denoiser satisfies a sharp oracle inequality that leads to near optimal rates of estimation for a large class of image models such as bi-isotonic, H\"older smooth and cartoons. Our analysis hinges on properties of the unnormalized Laplacian of the two-dimensional grid such as eigenvector delocalization and spectral decay. We also present extensions to more than two dimensions as well as several other graphs.