Authors’ abstract: Linear inverse problems \(A\mu=y\) with Poisson noise and non-negative unknown \(\mu\geq 0\) are ubiquitous in applications, for instance in positron emission tomography (PET) in medical imaging. The associated maximum likelihood problem is routinely solved using an expectation-maximisation algorithm (ML-EM). This typically results in images which look spiky, even with early stopping. We give an explanation for this phenomenon. We first regard the image \(\mu\) as a measure. We prove that if the measurements \(y\) are not in the cone \(\{A\mu,\mu\geqslant 0\}\), which is typical of low injected dose, likelihood maximisers must be sparse, i.e., typically a sum of point masses. We also show a weak sparsity result for cluster points of ML-EM. On the other hand, in the low noise regime, we prove that cluster points of ML-EM are optimal measures with full support. Finally, we provide concentration bounds for the probability to be in the sparse case, and a set of numerical experiments supporting our claims.