A well-known, simple theory predicts that, for a given grain size, the granularity of a developed photographic image is proportional to the square root of the density. In this paper, it is shown that, in the case of a two-dimensional grain model consisting of randomly placed opaque disks (dots), this simple relation progressively underestimates granularity as density is increased. The disparity exists because, with the random-dot model, density is not solely proportional to the number of dots in the aperture, but may vary at random according to the amount of overlap among the dots. It is suggested that the effect of random overlap may be relatively small in a developed photographic image, which differs from a random-dot model in that the grains are distributed in depth.
B. E. BAYER, "Relation Between Granularity and Density for a Random-Dot Model," J. Opt. Soc. Am. 54, 1485-1487 (1964)