Due to the arrival of photons according to a Poisson process, repeat acquisitions using the same exposure time will produce images consisting of different numbers of detected photons. Suppose,...

Due to the arrival of photons according to a Poisson process, repeat acquisitions using the same exposure time will produce images consisting of different numbers of detected photons. Suppose, however, that a hypothetical detector is used that guarantees the detection of exactly N0 photons per image, N0 being some positive integer. Further suppose that the imaged object is stationary. In this case, if the detector records the locations at which the photons are detected, then the loglikelihood function corresponding to an image is given by

where rk = (xk, yk) ∈ C denotes the location on the detector C at which the kth photon is detected, and fθ(x, y), (x, y) ∈ C, is the photon distribution profile. Using this log-likelihood function, show that the Cram´er-Rao lower bound is given by

which is just the expression of Eq. (17.13) with N_photons
replaced by N₀.