Linear inverse problems with nonnegativity constraints: singularity of optimisers

TL;DR

This paper investigates the singularity of solutions in nonnegative linear inverse problems under high noise, explaining why algorithms produce spiky images at high resolution, supported by theoretical results and medical imaging examples.

Contribution

It provides a general singularity result for optimizers in nonnegative inverse problems with various noise models, explaining the spike artifacts in high-resolution reconstructions.

Findings

Optimizers develop singular parts under high noise conditions.

Spiky artifacts are inherent in solutions as resolution increases.

Numerical examples demonstrate theoretical phenomena in medical imaging.

Abstract

We look at continuum solutions in optimisation problems associated to linear inverse problems $y = Ax$ with non-negativity constraint $x \geq 0$. We focus on the case where the noise model leads to maximum likelihood estimation through general divergences, which covers a wide range of common noise statistics such as Gaussian and Poisson. Considering $x$ as a Radon measure over the domain on which the reconstruction is taking place, we show a general singularity result. In the high noise regime corresponding to $y \notin \{{Ax}\mid{x \geq 0}\}$ and under a key assumption on the divergence as well as on the operator $A$, any optimiser has a singular part with respect to the Lebesgue measure. We hence provide an explanation as to why any possible algorithm successfully solving the optimisation problem will lead to undesirably spiky-looking images when the image resolution gets finer, a phenomenon well documented in the literature. We illustrate these results with several numerical examples inspired by medical imaging.