Contour Manifolds and Optimal Transport
This work provides a mathematical foundation for shape representation in segmentation, which is incremental as it builds on existing methods.
The paper tackles the problem of representing shapes in object segmentation by studying the relationship between shape measures and contour descriptions, showing that the pseudo-Riemannian structure of optimal transport yields a manifold diffeomorphic to closed contours.
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.