Model-based two-object resolution from observations having counting statistics

This paper considers two-object resolution from the viewpoint of model fitting theory. The studied experiment consists in counting events, for example, an electron hitting a detector pixel. It is stated that the precision and the accuracy with which the locations of the objects can be estimated will determine the attainable resolution. Two different approaches are followed. For both, the special case of Gaussian peaks is further investigated. The first approach leads to the maximally attainable precision. It is shown that this precision is determined by a certain factor, which is a function of the distance of the peaks, their widths and the number of counts. This factor will be called the resolution factor. The influence of each of the quantities involved is determined by the way they enter this factor. The second approach defines a probability of resolution, i.e., the probability that the maximum likelihood estimates of the locations will be distinct. It is shown that the resolution factor, which resulted from the first approach, also determines the probability of resolution.