Random Cross Ratios

The main result of this paper is the derivation of the probability density function of the cross ratio k(a;b;c;d) = (c \Gamma a) (d \Gamma b) (d \Gamma a) (c \Gamma b) when the numbers a, b, c and d are independent stochastic variables with identical rectangular distribution. The probability density function of the cross ratio is needed to study the effectiveness of invariant based recognition systems. A short background into invariant based recognition is given. The use of the result is illustrated by examples and a short comment is given on generalizing the result to other distributions on a, b, c and d. Although theoretical, this result is useful in understanding algorithms based on the cross ratio, e.g. in autonomous vehicle navigation, object recognition and reconstruction from images