Maximum likelihood and minimum mean squared error estimations for measurement of light intensity

The electrical signal at the output of an optical detector is mathematically modeled by a stochastic process consisted of a marked and filtered Poisson process and an additive white Gaussian noise. The measurement of a constant light intensity using an optical detector is formulated in terms of estimating the rate of the Poisson process involved in this model. Two cases are considered: minimum mean squared error estimation when the prior density of the rate is known, and maximum likelihood estimation when a priori information about the rate does not exist. The solutions to these estimation problems are determined in terms of an expression which can be computed only by means of a nonlinear infinite-dimensional system. Under a high intensity or low signal to noise ratio regime, this expression is approximated by a simpler expression which can be implemented using a linear filter. Based on this approximation, an explicit form for the maximum likelihood estimator is developed.