Maximum-probable restoration of photon-limited images

We define a computer algorithm for obtaining physically maximum-probable estimates of incoherent object scenes from their photon-count imges. The aim is to combine Bayesian maximum-probable (MAP) estimation with the known physics of image formation. A previous MAP approach to the problem assumed Poisson statistics to be obeyed by the image data. This is correct in the presence of negligible quantum degeneracy. Our aim is to extend this MAP approach into the domain of finite quantum degeneracy where a negative binomial law replaces the Poisson law. The resulting algorithm is a partial solution to the problem, being optimal or suboptimal depending on the object involved. To keep the algorithm simple and hence efficient to implement on a computer, the joint likelihood law for the image data is assumed to have a simple product form. This may be realized by either (a) ignoring interpixel correlations when they are finite (resulting in a suboptimal estimate) or (b) using two sets of data in an interlaced mode. These qualifications limit the applicability of the approach. The result of(a) or (b) is ajoint likelihood law which is a simple product of negative binomial laws. Its maximum defines the MAP object estimate sought. Such object estimates are found to be regularized by prior knowledge of a variance or autocorrelation value across the object. Variance in particular is shown to directly exert a controlling influence on the fluctuation in the estimates. Test cases on computer-generated images are shown.