Summation formulas are derived for the precise calculation of certain quality parameters of the image space: (a) radius of gyration of the point spread function, (b) O’Neill’s sharpness criterion, and (c) the Linfoot quality parameters. These formulas are applicable to any optical system that exists either as a design or as a fabricated object. Also, in the case of a bandwidth-limited object spectrum it is shown that the image may be exactly represented by a summation—in place of the usual convolution integral—over the object space. All derivations employ the sampling theorem.
B. ROY FRIEDEN, "Image Evaluation by Use of the Sampling Theorem," J. Opt. Soc. Am. 56, 1355-1361 (1966)