  Describing shapes by suitable measures in object segmentation, as proposed in
[24], allows to combine the advantages of the representations as parametrized
contours and indicator functions. The pseudo-Riemannian structure of optimal
transport can be used to model shapes in ways similar as with contours, while
the Kantorovich functional enables the application of convex optimization
methods for global optimality of the segmentation functional.
  In this paper we provide a mathematical study of the shape measure
representation and its relation to the contour description. In particular we
show that the pseudo-Riemannian structure of optimal transport, when restricted
to the set of shape measures, yields a manifold which is diffeomorphic to the
manifold of closed contours. A discussion of the metric induced by optimal
transport and the corresponding geodesic equation is given.
