Fast Transport Optimization for Monge Costs on the Circle

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on R, and suppose that the cost $c(x,y)$ of matching two points x, y satisfies the Monge condition: c(x₁,y₁)+c(x₂,y₂)<c(x₁,y₂)+c(x₂,y₁) whenever x₁<x₂ and y₁<y₂. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry–Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circle, both of which have the same total mass, find an optimal transport plan with respect to a given cost function c satisfying the Monge condition. In the circular case the sorting strategy fails to provide a unique candidate solution, and a naive approach requires a quadratic number of operations. For the case of N real-valued point masses we present an O(N|logε|) algorithm that approximates the optimal cost within ε; when all masses are integer multiples of $1/M$, the algorithm gives an exact solution in O(Nlog M) operations.