https://hal.inria.fr/hal-02192480v2Pouchol, CamilleCamillePoucholKTH Royal Institute of Technology - Department of Mathematics - KTH - Royal Institute of Technology [Stockholm]Verdier, OlivierOlivierVerdierKTH Royal Institute of Technology - Department of Mathematics - KTH - Royal Institute of Technology [Stockholm]Western Norway University of Applied Sciences, NorwayThe ML-EM algorithm in continuum: sparse measure solutionsHAL CCSD2019[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC][MATH.MATH-ST] Mathematics [math]/Statistics [math.ST]Pouchol, Camille2019-08-12 15:55:052020-03-05 15:31:332019-08-13 11:04:55enPreprints, Working Papers, ...https://hal.inria.fr/hal-02192480v1application/pdf2Linear inverse problems $A \mu = y$ with Poisson noise and non-negative unknown $\mu \geq 0$ are ubiquitous in applications where light is involved, such as Positron Emission Tomography (PET) in medical imaging. The associated maximum likelihood problem is most often solved by the celebrated iterative ML-EM algorithm, which is known to yield results which look spiky even when stopped early. This work provides an explanation for this phenomenon by going into continuum in image space when the operator $A$ is integral, such as in PET. This is done by considering the image $\mu$ as a measure rather than an element of a Lebesgue space. We prove that if the measurements $y$ are not in the cone $\{A \mu, \mu \geq 0\}$, which is typical of high noise or short exposure times, likelihood maximisers as well as ML-EM cluster points must be sparse, i.e., typically a sum of Dirac masses. In the better case where $y$ is in the cone, we prove that cluster points of ML-EM will be measures without singular part, as long as the initial guess is smooth. Finally, we provide concentration bounds for the probability to be in the undesirable sparse case, proving that it is exponentially small with exposure time, with a rate controlled by the distance of the noiseless data to the boundary of the cone.