The ML-EM algorithm in continuum: sparse measure solutions
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Linear inverse problems A μ = δ with Poisson noise and non-negative unknown μ≥ 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≥ 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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