Relaxation of Randomness in Two-Dimensional Statistical Model of Overlap: Theory and Verification

Abstract

An equation for the expected number of spots in a two-dimensional (2-D) separation containing randomly distributed single-component spots (SCSs) was modified to predict the expected number of spots when the local density of SCSs is random but varies continuously throughout the separation. The modified equation was expressed by a double integral, whose value depends on the mean number of SCSs, the average saturation of the separation, and a dimensionless frequency proportional to SCS density. This equation is much more useful for interpretation of separations than its predecessor, because SCSs in 2-D separations rarely are distributed with constant density but rather with variable density. The modified equation was verified by two types of computer simulations, in which SCSs were represented alternatively by constant-diameter circles and by bi-Gaussians having circular contours and exponentially distributed amplitudes. An excellent agreement between simulation and theory was obtained over a wide range of saturations when SCSs were represented by circles ; a good agreement was obtained for saturations less than a critical threshold when SCSs were represented by bi-Gaussians. The equation also was used to predict the number of SCSs in separations of low saturation, in which SCSs were represented by either 30 or 250 bi-Gaussians. This prediction required estimating frequencies from the coordinates of maxima, and two procedures for this estimation were proposed and tested. The predictions on average were very good, as long as the saturation was below a critical threshold. The modified theory was shown to be insensitive to arbitrary deflation of the separation's borders, which has practical importance.