A Comparison between Wasserstein Distance and a Distance Induced by Fisher-Rao Metric in Complex Shapes Clustering

Shape Analysis studies geometrical objects, as for example a flat fish in the plane or a human head in the space. The applications range from structural biology, computer vision, medical imaging to archaeology. We focus on the selection of an appropriate measurement of distance among observations with the aim of obtaining an unsupervised classification of shapes. Data from a shape are often realized as a set of representative points, called landmarks. For planar shapes, we assume that each landmark is modeled via a bivariate Gaussian, where the means capture uncertainties that arise in landmarks placement and the variances the natural variability across the population of shapes. At first we consider the Fisher-Rao metric as a Riemannian metric on the Statistical Manifold of the Gaussian distributions. The induced geodesic-distance is related with the minimization of information in the Fisher sense and we can use it to discriminate shapes. Another suitable distance is the Wasserstein distance, which is induced by a Riemannian metric and is related with the minimal transportation cost. In this work, a simulation study is conducted in order to make a comparison between Wasserstein and Fisher-Rao metrics when used in shapes clustering.