Exact reconstruction and reconstruction from noisy data with anisotropic total variation

TL;DR

This paper investigates the extension of superresolution techniques to anisotropic total variation regularization, demonstrating exact reconstruction and error bounds for piecewise constant images with horizontal and vertical edges under noisy Fourier measurements.

Contribution

It introduces the use of anisotropic total variation regularization in superresolution, providing theoretical guarantees for exact reconstruction and error bounds in noisy settings.

Findings

Exact reconstruction results under mutual edge distance

L^1 error bounds proportional to noise level

Applicability to piecewise constant images with edges

Abstract

It is well-known that point sources with sufficient mutual distance can be reconstructed exactly from finitely many Fourier measurements by solving a convex optimization problem with Tikhonov-regularization (this property is sometimes termed superresolution). In case of noisy measurements one can bound the reconstruction error in unbalanced Wasserstein distances or weak Sobolev-type norms. A natural question is to what other settings the phenomenon of superresolution extends. We here keep the same measurement operator, but replace the regularizer to anisotropic total variation, which is particularly suitable for regularizing piecewise constant images with horizontal and vertical edges. Under sufficient mutual distance between the horizontal and vertical edges we prove exact reconstruction results and $L^1$ error bounds in terms of the measurement noise.