The Wasserstein-Fisher-Rao metric

This note gives a summary of the presentation that I gave at the workshop on shape analysis. Based on [CSPV15, CPSV15], we present a generalization of optimal transport to measures that have different total masses. This generalization enjoys most of the properties of standard optimal transport but we will focus on the geometric formulation of the model. We expect this new metric to have interesting applications in imaging. 1 Motivation and a Dynamical Model In several contexts of applications including imaging, it is natural to consider data that can be represented by densities and these densities might have different masses. Often, optimal transport has been used in these applications (for instance, [HZTA04, AKS15]) since it provides an "easily computable" (at least, an efficient approximation [Cut13]) distance between probability measures that reflects a geometric displacement between them. Therefore, the mass constraint on the densities has to be taken into account and this problem seems to bring renewed interest in the applied mathematics literature [PR13, PR14, FG10, LM13, MRSS15] although this issue has been addressed since Kantorovich [Gui02]. In the following, we describe a dynamical approach to define optimal transport between general non-negative Radon measures. We will present the model only in a smooth setting although it is well defined on the space of Radon measures. The Benamou-Brenier formulation: In [BB00], the authors formulated the Wasserstein L distance as a convex variational problem, inspired by a fluid dynamic approach. In what follows, M will be a compact manifold without boundary. Let ρ ∈ C(M,R+) be a positive function, note that all the quantities will be implicitly time dependent. The dynamic formulation of the Wasserstein distance consists in minimizing