Asymptotics of Bayesian Error Probability and 2D Pair Superresolution

This paper employs a recently developed asymptotic Bayesian multi-hypothesis testing (MHT) error analysis to treat the problem of superresolution imaging of a pair of closely spaced, equally bright point sources. The analysis exploits the notion of the minimum probability of error (MPE) in discriminating between two competing equi-probable hypotheses, a single point source of a certain brightness at the origin vs. a pair of point sources, each of half the brightness of the single source and located symmetrically about the origin, as the distance between the source pair is changed. For a Gaussian point-spread function (PSF), the analysis makes predictions on the scaling of the minimum source strength, expressed in units of photon number, required to disambiguate the pair as a function of their separation, in both the signal-dominated and background-dominated regimes. Certain logarithmic corrections to the quartic scaling of the minimum source strength with respect to the degree of superresolution characterize the signal-dominated regime, while the scaling is purely quadratic in the background-dominated regime. For the Gaussian PSF, general results for arbitrary strengths of the signal, background, and sensor noise levels are also presented.