The Cross Ratio: A Revisit to its Probability Density Function

The cross ratio has wide applications in computer vision because of its invariance under projective transformation. In active vision where the projections of quadruples of collinear landmark points in the scene are tracked in the image sequence for robot localisation or online camera calibration, one often needs to compute cross ratios from noisy image data for some subsequent operations. Being able to assess the reliability of each computed cross ratio value against a known level of image noise is therefore of importance. This aim motivates our research to derive the probability density function (p.d.f.) of the cross ratio based on the normality assumption of the associated random variables and to investigate into empirical cases where this assumption fails to hold. Although an analytical formula for the general p.d.f. of the cross ratio has not been achieved, our research results show that (i) the distance between the closest pair of collinear points is a significant factor that determines the shape of the p.d.f. of the cross ratio and (ii) a good estimate of the cross ratio can be obtained if the points of the quadruple are sufficiently far apart.