The ML-EM algorithm in continuum: sparse measure solutions
This addresses a fundamental issue in medical imaging and inverse problems, providing theoretical insights into algorithm behavior, but it is incremental as it builds on existing ML-EM methods.
The paper explains why the ML-EM algorithm for Poisson noise linear inverse problems, such as in PET imaging, often produces spiky images, showing that under short exposure times, solutions are sparse measures (sum of point masses), while under long exposure times, they are non-singular measures, with concentration bounds provided for the sparse case.
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.