Shape analysis using the Fisher-Rao Riemannian metric: unifying shape representation and deformation

We show that the Fisher-Rao Riemannian metric is a natural, intrinsic tool for computing shape geodesics. When a parameterized probability density function is used to represent a landmark-based shape, the modes of deformation are automatically established through the Fisher information of the density. Consequently, given two shapes parameterized by the same density model, the geodesic distance between them under the action of the Fisher-Rao metric is a convenient shape distance measure. It has the advantage of being an intrinsic distance measure and invariant to reparameterization. We first model shape landmarks using a Gaussian mixture model and then compute geodesic distances between two shapes using the Fisher-Rao metric corresponding to the mixture model. We illustrate our approach by computing Fisher geodesics between 2D corpus callosum shapes. Shape representation via the mixture model and shape deformation via the Fisher geodesic are hereby unified in this approach.