DISTRIBUTIONS OF PROJECTIVE INVARIANTS AND MODEL-BASED MACHINE VISION

In machine vision, objects are observed subject to an unknown projecti ve transformation, and it is usual to use projective invariants for ei ther testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is i nvariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means . The case of identical means, which has appeared previously in the li terature, is derived as a particular case. Various alternative forms o f the cross-ratio density are obtained, e.g. under the Casey arccos tr ansformation, and under an arctan transformation from the real project ive line of cross-ratios to the unit circle. The cross-ratio distribut ions are novel to the probability literature; surprisingly various typ es of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also g ive some results for the projective invariants of five coplanar points . We discuss the general moment properties of the cross-ratio, and con sider some inference problems, including maximum likelihood estimation of the parameters.