Transportation Distances on the Circle

Julien Rabin (1) , Julie Delon (1) , Yann Gousseau (1)

1 LTCI - Laboratoire Traitement et Communication de l'Information

In this contribution, we study Monge-Kantorovich distances between discrete set of points on the unit circle, when the ground distance between two points x and y on the circle is defined as the L1-shortest path. We first prove that computing a Monge-Kantorovich distance between two given sets of pairwise different points boils down to cut the circle at a well chosen point and to compute the same distance on the real line. This result is then used to prove a formula on the Earth Mover's Distance which is a particular Monge-Kantorovich distance. This formula asserts that the Earth Mover's Distance between two discrete circular normalized histograms can be computed from their cumulative histograms. This formula is used in recent papers on the matching of local features between images, where the Earth Mover's Distance is used to compare circular histograms of gradient orientations.

Mots clés: transportation problem; earth mover's distance

Domaines: Analyse fonctionnelle [math.FA]; Optimisation et contrôle [math.OC]

Julien Rabin, Julie Delon, Yann Gousseau. Transportation Distances on the Circle. 2009. ⟨hal-00399832v2⟩