An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite-dimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even non-convexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow. 1

Image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times. In this paper we present a method for computing elastic registration and warping maps based on the Monge–Kantorovich theory of optimal mass transport. This mass transport method has a number of important characteristics. First, it is parameter free. Moreover, it utilizes all of the grayscale data in both images, places the two images on equal footing and is symmetrical: the optimal mapping from image to ¡ image being the inverse of the optimal mapping ¡ from to  £ ¢ The method does not require that landmarks be specified, and the minimizer of the distance functional involved is unique; there are no other local minimizers. Finally, optimal transport naturally takes into account changes in density that result from changes in area or volume. Although the optimal transport method is certainly not appropriate for all registration and warping problems, this mass preservation property makes the Monge-Kantorovich approach quite useful for an interesting class of warping problems, as we show in this paper. Our method for finding the registration mapping is based on a partial differential equation approach to the minimization of ¤¦ ¥ the Kantorovich–Wasserstein or “Earth Mover’s Distance ” under a mass preservation constraint. We show how this approach leads to practical algorithms, and demonstrate our method with a number of examples, including those from the medical field. We also extended this method to take into account changes in intensity, and show that it is well suited for applications such as image morphing. A. Image Registration I.

Abstract. Let M and ¯ M be n-dimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M × ¯ M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry. Using the pseudo-Riemannian structure, we extend Ma, Trudinger and Wang’s conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principal characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and ¯ M of Rn. This maximum principle plays a key role in Loeper’s Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered. 1.

In this paper, we consider probability measures µ and ν on a d–dimensional sphere in R d+1,d ≥ 1, and cost functions of the form c(x, y) = l( | x−y|2 2) that generalize those arising in geometric optics where l(t) = −log t. We prove that if µ and ν vanish on (d − 1)–rectifiable sets, if |l ′ (t) |> 0and g(t):=t(2 − t)(l ′ (t)) 2 is monotone then there exists a unique optimal map To that transports µ onto ν, where optimality is measured against c. Furthermore, infx |Tox − x |> 0. In the special case when l(t) =−log t, existence of optimal maps on the sphere was obtained earlier in [10] and [27] under more restrictive assumptions that µ, ν are absolutely continuous with respect to the d–dimensional Haussdorff measure and have disjoint supports. Without the restriction on the supports of µ and ν the existence of optimal maps in this case was also shown in [10], but the proof is indirect. Another aspect of interest in this work is that it is in contrast with the work in [9] where it is proven that when l(t) =tthen existence of an optimal map fails when µ and ν are supported by Jordan curves in the plane.

Abstract—In this paper, we present a novel method for texture mapping of closed surfaces. Our method is based on the technique of optimal mass transport (also known as the “earth-mover’s metric”). This is a classical problem that concerns determining the optimal way, in the sense of minimal transportation cost, of moving a pile of soil from one site to another. In our context, the resulting mapping is area preserving and minimizes angle distortion in the optimal mass sense. Indeed, we first begin with an angle-preserving mapping (which may greatly distort area) and then correct it using the mass transport procedure derived via a certain gradient flow. In order to obtain fast convergence to the optimal mapping, we incorporate a multiresolution scheme into our flow. We also use ideas from discrete exterior calculus in our computations. Index Terms—Texture mapping, optimal mass transport, parametrization, spherical wavelets. Ç 1

Abstract. In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Ampère type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.

continuity for optimal multivalued mappings ∗

Abstract of “The geometry of shape recognition via the Monge-Kantorovich optimal

Abstract not found

Non-existence of polar factorisations and polar inclusion of a vector-valued mapping