A statistical approach to the resolution of point sources

The Rayleigh criterion in optics states that two point sources of equal intensity are 'barely resolved' when the maximum of the diffraction pattern of one source overlaps the first minimum of the diffraction pattern of the second source. Although useful for rough comparisons of optical systems, such a criterion does not take into account the randomness in the detection process and does not tell whether sources can actually be distinguished. We present a statistical approach that addressed these issues. From quantum optics, the photon counts in the pixels are independent Poisson random variables with means that depend on the distance 2theta between the sources. Resolving the sources corresponds to testing H0: theta =0 vs Ha: theta >0, under conditions that make the information number zero at theta =0. We define resolution as the (asymptotic) power function of the likelihood ratio test rather than as a single number. The asymptotic distribution of the test statistic is derived under H0 and under contiguous alternatives. The results are illustrated by an application to a sky survey to detect binary stars using the Hubble space telescope.