Linear inverse problems with nonnegativity constraints: Singularity of optimisers

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 cover 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\{A x \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.