The Fisher-Rao Metric for Projective Transformations of the Line

A conditional probability density function is defined for measurements arising from a projective transformation of the line. The conditional density is a member of a parameterised family of densities in which the parameter takes values in the three dimensional manifold of projective transformations of the line. The Fisher information of the family defines on the manifold a Riemannian metric known as the Fisher-Rao metric. The Fisher-Rao metric has an approximation which is accurate if the variance of the measurement errors is small. It is shown that the manifold of parameter values has a finite volume under the approximating metric. These results are the basis of a simple algorithm for detecting those projective transformations of the line which are compatible with a given set of measurements. The algorithm searches a finite list of representative parameter values for those values compatible with the measurements. Experiments with the algorithm suggest that it can detect a projective transformation of the line even when the correspondences between the components of the measurements in the domain and the range of the projective transformation are unknown.