The ML-EM algorithm in continuum: sparse measure solutions

Linear inverse problems $A μ= δ$ with Poisson noise and non-negative unknown $μ\geq 0$ 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 $μ$ as a measure. We prove that if the measurements $δ$ are not in the cone $\{A μ, μ\geq 0\}$, which is typical of short exposure times, likelihood maximisers as well as ML-EM cluster points must be sparse, i.e., typically a sum of point masses. On the other hand, in the long exposure regime, we prove that cluster points of ML-EM will be measures without singular part. Finally, we provide concentration bounds for the probability to be in the sparse case.

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