Exact recovery of the support of piecewise constant images via total variation regularization

TL;DR

This paper proves that total variation regularization can exactly recover the support of piecewise constant images from noisy measurements, under certain conditions, with reconstructions converging to the true image as noise diminishes.

Contribution

It establishes conditions under which total variation regularization guarantees exact support recovery of piecewise constant images in noisy settings.

Findings

Reconstructed images preserve the number and shape of original components in low noise.

Reconstructed shapes and intensities converge to true values as noise approaches zero.

Support recovery is robust under a non-degenerate source condition.

Abstract

This work is concerned with the recovery of piecewise constant images from noisy linear measurements. We study the noise robustness of a variational reconstruction method, which is based on total (gradient) variation regularization. We show that, if the unknown image is the superposition of a few simple shapes, and if a non-degenerate source condition holds, then, in the low noise regime, the reconstructed images have the same structure: they are the superposition of the same number of shapes, each a smooth deformation of one of the unknown shapes. Moreover, the reconstructed shapes and the associated intensities converge to the unknown ones as the noise goes to zero.