Asymptotics of Bayesian Error Probability and Rotating-PSF-Based Source Super-Localization in Three Dimensions

We present an asymptotic analysis of the minimum probability of error (MPE) in inferring the correct hypothesis in a Bayesian multi-hypothesis testing (MHT) formalism using many pixels of data that are corrupted by signal dependent shot noise, sensor read noise, and background illumination. We perform this error analysis for a variety of combined noise and background statistics, including a pseudo-Gaussian distribution that can be employed to treat approximately the photon-counting statistics of signal and background as well as purely Gaussian sensor read-out noise and more general, exponentially peaked distributions. We subsequently apply the MPE asymptotics to characterize the minimum conditions needed to localize a point source in three dimensions by means of a rotating-PSF imager and compare its performance with that of a conventional imager in the presence of background and sensor-noise fluctuations. In a separate paper, we apply the formalism to the related but qualitatively different problem of 2D super-resolution imaging of a closely spaced pair of point sources in the plane of best focus.