  This paper defines a new transport metric over the space of non-negative
measures. This metric interpolates between the quadratic Wasserstein and the
Fisher-Rao metrics and generalizes optimal transport to measures with different
masses. It is defined as a generalization of the dynamical formulation of
optimal transport of Benamou and Brenier, by introducing a source term in the
continuity equation. The influence of this source term is measured using the
Fisher-Rao metric, and is averaged with the transportation term. This gives
rise to a convex variational problem defining our metric. Our first
contribution is a proof of the existence of geodesics (i.e. solutions to this
variational problem). We then show that (generalized) optimal transport and
Fisher-Rao metrics are obtained as limiting cases of our metric. Our last
theoretical contribution is a proof that geodesics between mixtures of
sufficiently close Diracs are made of translating mixtures of Diracs. Lastly,
we propose a numerical scheme making use of first order proximal splitting
methods and we show an application of this new distance to image interpolation.
