Fast transport optimization on the circle

Julie Delon (1) , Julien Salomon (2) , Andrei Sobolevskii (3)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 ISCP - Interdisciplinary Scientific Center Poncelet

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. 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. This theory is applied to a transportation problem arising in image processing: for two sets of point masses, both of which have the same total mass, find an optimal transport plan with respect to a given cost function that satisfies the Monge condition. For the case of N real-valued point masses we present an O(N log epsilon) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.

Domaines: Optimisation et contrôle [math.OC]

Julie Delon, Julien Salomon, Andrei Sobolevskii. Fast transport optimization on the circle. 2009. ⟨hal-00362834v1⟩