Classification based on the cross ratio

Abstract

The performance of the cross ratio in model based vision is quantified for a decision rule which is computationally expensive, but which makes good use of the available information. The probabilities of rejection, false alarm and misclassification are calculated, correct to leading order in the noise level. The probability of false alarm is closely related to the complete elliptic integral of the third kind. An expression is obtained for the probability density function of the cross ratio of four points with independent identical Gaussian distributions. A general framework is sketched to show how the methods of calculation employed here for the cross ratio can in principle be extended to more complicated scalar invariants or to vectors of invariants.