Adaptive Rates for Total Variation Image Denoising

TL;DR

This paper analyzes the theoretical performance of total variation regularization in image denoising, demonstrating its adaptivity and convergence properties for piecewise constant images.

Contribution

It provides a theoretical framework showing that TV denoising adapts to the true image structure and achieves near-optimal convergence rates.

Findings

Denoised images are piecewise constant on rectangular sets.

Convergence to the true image at a near-parametric rate for piecewise constant images.

Denoised images have oracle properties, performing nearly as well as if the true structure was known.

Abstract

We study the theoretical properties of image denoising via total variation penalized least-squares. We define the total vatiation in terms of the two-dimensional total discrete derivative of the image and show that it gives rise to denoised images that are piecewise constant on rectangular sets. We prove that, if the true image is piecewise constant on just a few rectangular sets, the denoised image converges to the true image at a parametric rate, up to a log factor. More generally, we show that the denoised image enjoys oracle properties, that is, it is almost as good as if some aspects of the true image were known. In other words, image denoising with total variation regularization leads to an adaptive reconstruction of the true image.