The transmission fluctuations of a grainy film can be measured in several ways, for instance:
(a) the standard deviation of the transmission of a single aperture as a function of its size,
(b) the standard deviation of the difference in transmission between two adjacent apertures as a function of their size,
(c) the correlation curve: the variation of the mean of the product of the transmissions of two apertures as a function of their separation.
When the apertures are slits, it is shown that if any one of these is known, the others can be calculated from it. These conclusions are valid also if a reasonable approximation is made to the effect of finite resolving power on the edge of the slits.
For circular apertures, (a) can be calculated from (c) and vice versa. It is suggested that the correlation curve can always in principle be found from (a), when the aperture edge is sharp. An analogous result may also be possible for the case of unsharp edges.
The analogous density measurements can be similarly interrelated if the form of the frequency distribution of individual readings is known.
It is concluded that the various methods of computing granularity as a function of the linear dimensions of the optical system all give the same information in different forms.
A. MARRIAGE and E. PITTS, "Relation between Granularity and Autocorrelation," J. Opt. Soc. Am. 46, 1019-1026 (1956)