Shape recognition via Wasserstein distance

The Kantorovich-Rubinstein-Wasserstein metric defines the distance between two probability measures $\mu$ and $\nu$ on ${R^{d + 1}}$ by computing the cheapest way to transport the mass of $\mu$ onto $\nu$, where the cost per unit mass transported is a given function $c\left ( x, y \right )$ on ${R^{2d + 2}}$. Motivated by applications to shape recognition, we analyze this transportation problem with the cost $c\left ( x, y \right ) = {\left | {x - y} \right |^2}$ and measures supported on two curves in the plane, or more generally on the boundaries of two domains $\Omega , \Lambda \subset {R^{d + 1}}$. Unlike the theory for measures that are absolutely continuous with respect to Lebesgue, it turns out not to be the case that $\mu - a.e.x \in \partial \Omega$ is transported to a single image $y \in \partial \Lambda$; however, we show that the images of $x$ are almost surely collinear and parallel the normal to $\partial \Omega$ at $x$. If either domain is strictly convex, we deduce that the solution to the optimization problem is unique. When both domains are uniformly convex, we prove a regularity result showing that the images of $x \in \partial \Omega$ are always collinear, and both images depend on $x$ in a continuous and (continuously) invertible way. This produces some unusual extremal doubly stochastic measures.