  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.
