  This paper is devoted to the study of the Monge-Kantorovich theory of optimal
mass transport and its applications, in the special case of one-dimensional and
circular distributions. More precisely, we study the Monge-Kantorovich
distances between discrete sets of points on the unit circle, in the case where
the ground distance between two points x and y is defined as h(d(x,y)), where d
is the geodesic distance on the circle and h a convex and increasing function.
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 obtain a metric between 1D and circular discrete histograms, which can
be computed in linear time. A particular case of this formula has already been
used in [Rabin, Delon and Gousseau SIAM 09}] for the matching of local features
between images, involving circular histograms of gradient orientations. In this
paper, other applications are investigated, in particular dealing with the hue
component of color images. In a last part, a study is conducted to compare the
advantages and drawbacks of transportation distances relying on convex or
concave cost functions, and of the classical L-1 distance.
