An EM Approach for Time-Variant Poisson-Gaussian Model Parameter Estimation

The problem of estimating the parameters of a Poisson-Gaussian model from experimental data has recently raised much interest in various applications, for instance in confocal fluorescence microscopy. In this context, a field of independent random variables is observed, which is varying both in time and space. Each variable is a sum of two components, one following a Poisson and the other a Gaussian distribution. In this paper, a general formulation is considered where the associated Poisson process is nonstationary in space and also exhibits an exponential decay in time, whereas the Gaussian component corresponds to a stationary white noise with arbitrary mean. To solve the considered parametric estimation problem, we follow an iterative Expectation-Maximization (EM) approach. The parameter update equations involve deriving finite approximation of infinite sums. Expressions for the maximum error incurred in the process are also given. Since the problem is non-convex, we pay attention to the EM initialization, using a moment-based method where recent optimization tools come into play. We carry out a performance analysis by computing the Cramer-Rao bounds on the estimated variables. The practical performance of the proposed estimation procedure is illustrated on both synthetic data and real fluorescence macroscopy image sequences. The algorithm is shown to provide reliable estimates of the mean/variance of the Gaussian noise and of the scale parameter of the Poisson component, as well as of its exponential decay rate. In particular, the mean estimate of the Poisson component can be interpreted as a good-quality denoised version of the data.