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 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.