Resolving Power and Decision Theory*†

The fundamental resolving power of optical systems is defined by use of decision theory. For binary decisions in the presence of Gaussian noise, the probability of a correct decision is determined by the quadratic content of the difference image and the noise variance per unit of area. In the spatial frequency domain the probability of correct decision involves the integral of the power spectrum of the difference image and the noise variance per unit of area. The merit of a particular optical system for the performance of a specific binary decision task may be evaluated by use of a formula involving its modulation transfer function. The resolution of two monochromatic, incoherent point sources under conditions of high background radiation is discussed, as an illustrative example. The resolution of two point sources is limited only by the precision with which the flux density at all points in the image plane can be determined. The manner in which this increased precision can be obtained by increasing the period of observation is discussed, for the case of a photon-counting detector.