  In this paper, we leverage the properties of non-Euclidean Geometry to define
the Geodesic distance (GD) on the space of statistical manifolds. The Geodesic
distance is a real and intuitive similarity measure that is a good alternative
to the purely statistical and extensively used Kullback-Leibler divergence
(KLD). Despite the effectiveness of the GD, a closed-form does not exist for
many manifolds, since the geodesic equations are hard to solve. This explains
that the major studies have been content to use numerical approximations.
Nevertheless, most of those do not take account of the manifold properties,
which leads to a loss of information and thus to low performances. We propose
an approximation of the Geodesic distance through a graph-based method. This
latter permits to well represent the structure of the statistical manifold, and
respects its geometrical properties. Our main aim is to compare the graph-based
approximation to the state of the art approximations. Thus, the proposed
approach is evaluated for two statistical manifolds, namely the Weibull
manifold and the Gamma manifold, considering the Content-Based Texture
Retrieval application on different databases.
