  As a general rule of thumb the resolution of a light microscope (i.e. the
ability to discern objects) is predominantly described by the full width at
half maximum (FWHM) of its point spread function (psf)---the diameter of the
blurring density at half of its maximum. Classical wave optics suggests a
linear relationship between FWHM and resolution also manifested in the well
known Abbe and Rayleigh criteria, dating back to the end of 19th century.
However, during the last two decades conventional light microscopy has
undergone a shift from microscopic scales to nanoscales. This increase in
resolution comes with the need to incorporate the random nature of observations
(light photons) and challenges the classical view of discernability, as we
argue in this paper. Instead, we suggest a statistical description of
resolution obtained from such random data. Our notion of discernability is
based on statistical testing whether one or two objects with the same total
intensity are present. For Poisson measurements we get linear dependence of the
(minimax) detection boundary on the FWHM, whereas for a homogeneous Gaussian
model the dependence of resolution is nonlinear. Hence, at small physical
scales modeling by homogeneous gaussians is inadequate, although often
implicitly assumed in many reconstruction algorithms. In contrast, the Poisson
model and its variance stabilized Gaussian approximation seem to provide a
statistically sound description of resolution at the nanoscale. Our theory is
also applicable to other imaging setups, such as telescopes.
