The Fisher-Rao Metric in Computer Vision
Stephen J. Maybank · 2019
The Fisher-Rao metric is a Riemannian metric defined on any manifold that forms the parameter space for a family of probability distributions. The metric is specified by quadratic forms defined on the tangent spaces of the manifold. If a parameterisation of the manifold is chosen then each quadratic form is given by a symmetric positive definite matrix. Lengths, areas, volumes and hyper-volumes calculated using the Fisher-Rao metric are invariant under reparameterisation. This invariance is essential in practice because the parameterisation can be changed arbitrarily while keeping the data unchanged. The Fisher-Rao metric is obtained as a limit of the expected value of the log likelihood ratio for two nearby probability distributions. The inverse of the Fisher-Rao matrix is the Cramer-Rao lower bound on the covariance of an unbiased estimate of a parameter. The Fisher-Rao metric is used to divide the parameter space for the Hough transform method for detecting structures in data. Each division or accumulator is invariant under reparametrerisation, and the number of accumulators is proportional to the volume of the parameter space. Accurate approximations to the Fisher Rao metric are obtained for lines, catadioptric images of lines, circles, ellipses and the cross ratio. It is shown that the Fisher-Rao metric can be used to compare the amount of information in point features with the amount of information in edge element features.