Stochastic Framework

We consider a stochastic framework that is based on an optical microscope setup, which consists of the object of interest, the microscope optics, and a detector that captures the image of the object. Here, the object of interest is a pair of point sources (e.g. single molecules) and we consider imaging experiments that capture photons during a fixed acquisition time interval. Our approach to calculate the resolution measure is based on the theory concerning the Cramer-Rao lower bound (see e.g., refs. 1 and 2). The task of determining the distance of separation d between two point sources is a parameter estimation problem. The distance of separation is obtained by using an unbiased estimation procedure, and the performance of this estimator is given by the standard deviation of the distance estimates assuming repeated experiments. According to the Cramer-Rao lower bound (1, 2), the (co)variance of any unbiased estimator θ̂ of an unknown parameter θ is always greater than or equal to the inverse Fisher information matrix, i.e.,