Optical microscopy has been an important tool in neurobiology because it enables us to observe molecules in living cells. However, nano-scale imaging definitely suffers from blur due to diffraction of light passing through the lens. Conventionally, the Rayleigh's criterion has been known as such a two-point resolution of microscope that quantifies the blur caused by Fraunhofer diffraction. Although the Rayleigh's criterion gives us a practical criterion of the resolution limit of two-point sources in a deterministic way, it neglects the stochastic nature of the optical imaging processes. For instance, the diffraction itself is a stochastic diffusion process of photons, and the measurement of photons is also disturbed by stochastic fluctuation such as shot noise. In the existing studies (Ram, et al., 2006; Shahram and Milanfar, 2006), these stochastic aspects are just neglected or inappropriately treated.
In this study, we redefine the two-point resolution that considers various stochastic fluctuations from a statistical point of view. The two-point resolution is characterized by an error rate of decision about the number of point sources in a given image. The error rate is derived based on the statistical hypothesis testing, in particular, the likelihood ratio test, which is the most powerful test (aka Neyman-Pearson lemma). In our practical application, we put our focus on the case that either one or two point sources are assumed to be in a given image. Because analytical evaluation of the likelihood ratio is difficult, we utilize Monte Carlo sampling and kernel density estimation for evaluating the likelihood ratio. The estimation of the error rate leads to the optimal resolution for determining the number of point sources in the microscope image. We also show the results when applied to confocal microscope images.