In this paper we explicitly derive a level set formulation for mean curvature flow in a Riemannian metric space. This extends the traditional geodesic active contour framework which is based on conformal flows. Curve evolution for image segmentation can be posed as a Riemannian evolution process where the induced metric is related to the local structure tensor. Examples on both synthetic and re...

The fast marching algorithm, and its variants, solves numerically the generalized eikonal equation associated to an underlying riemannian metric M. A major challenge for these algorithms is the non-isotropy of the riemannian metric, which magnitude is characterized by the anisotropy ratio κ(M) ∈ [1,∞]. Applications of the eikonal equation to image processing [1, 3] often involve large anisotrop...

Abstract. The space which consists of measures having finite second moment is an infinite dimensional metric space endowed with Wasserstein distance, while the space of Gaussian measures on Euclidean space is parameterized by mean and covariance matrices, hence a finite dimensional manifold. By restricting to the space of Gaussian measures inside the space of probability measures, we manage to ...

Ray Singer torsion is a numerical invariant associated with a compact manifold equipped with a flat bundle, a Riemannian metric on the manifold and a Hermitian structure on the bundle. In this note we show how one can remove the dependence on the Riemannian metric and the Hermitian structure with the help of a base point and of an Euler structure, and obtain a topological invariant. A numerical...

The geodesics for a sub-Riemannian metric on a threedimensional contact manifold M form a 1-parameter family of curves along each contact direction. However, a collection of such contact curves on M , locally equivalent to the solutions of a fourth-order ODE, are the geodesics of a sub-Riemannian metric only if a sequence of invariants vanish. The first of these, which was first identified by F...

We show that in the analytic category, given a Riemannian metric g on a hypersurface M ⊂ Z and a symmetric tensor W on M , the metric g can be locally extended to a Riemannian Einstein metric on Z with second fundamental form W , provided that g and W satisfy the constraints on M imposed by the contracted Gauss and CodazziMainardi equations. We use this fact to study the Cauchy problem for metr...

A Riemann-Lie algebra is a Lie algebra G such that its dual G∗ carries a Riemannian metric compatible (in the sense introduced by the author in C. R. Acad. Sci. Paris, t. 333, Série I, (2001) 763–768) with the canonical linear Poisson structure of G∗ . The notion of Riemann-Lie algebra has its origins in the study, by the author, of Riemann-Poisson manifolds (see Differential Geometry and its A...

Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzianmanifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically asso...

A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fiel...

This paper develops a theory of clustering and coding which combines a geometric model with a probabilistic model in a principled way. The geometric model is a Riemannian manifold with a Riemannian metric, gij(x), which we interpret as a measure of dissimilarity. The probabilistic model consists of a stochastic process with an invariant probability measure which matches the density of the sampl...