Optimizability of loglikelihoods for the estimation of detector efficiencies and singles rates in PET

In various prior work, Poisson models have been proposed for normalization scan and delayed coincidence measurements in PET systems. In these models, the mean measured counts are given by pair-wise products of a set of unknown detector efficiency or singles rate parameters. Additionally, several kinds of iterative algorithms have been proposed for solving the associated maximum likelihood estimation problem, and have exhibited strong empirical performance. To our knowledge, however, no analysis has been given proving that these algorithms actually converge, but only that convergence, should it hypothetically occur, would have to be to a stationary point of the loglikelihood. One of the main obstacles to a complete convergence analysis is showing that the loglikelihood actually possesses a finite maximizer. As we will show, there are cases where the loglikelihood function for these applications can approach its maximum value only by letting the unknown parameters tend asymptotically to infinity. In such cases, convergent behavior is not possible with any algorithm because there is nothing to converge to. In this article, we derive sufficient conditions - for cylindrical PET system geometries - guaranteeing not only that a unique and finite maximizer exists, but also that the loglikelihood maximization is equivalent to minimizing a strictly convex function. The latter obviates many questions of convergent algorithm design, since virtually any minimization algorithm with standard monotonicity and stationary properties will convergently minimize a strictly convex function when the minimizer exists. The conditions that we derive are, moreover, readily verifiable by pre-analysis of the projection measurements and show that failure of a solution to exist is a very rare event, except for unrealistically low count scans or for systems with unrealistically dense patterns of malfunctioning detectors.