  This article presents a multiscale, non-linear and directional statistical
characterization of images based on the estimation of the skewness, flatness,
entropy and distance from Gaussianity of the spatial increments. These
increments are characterized by their magnitude and direction; they allow us to
characterize the multiscale properties directionally and to explore anisotropy.
To describe the evolution of the probability density function of the increments
with their magnitude and direction, we use the skewness to probe the symmetry,
the entropy to measure the complexity, and both the flatness and distance from
Gaussianity to describe the shape. These four quantities allow us to explore
the anisotropy of the linear correlations and non-linear dependencies of the
field across scales. First, we validate the methodology on two-dimensional
synthetic scale-invariant fields with different multiscale properties and
anisotropic characteristics. Then, we apply it on two synthetic turbulent
velocity fields: a perfectly isotropic and homogeneous one, and a channel flow
where boundaries induce inhomogeneity and anisotropy. Our characterization
unambiguously detects the anisotropy in the second case, where our quantities
report scaling properties that depend on the direction of analysis.
Furthermore, we show in both cases that turbulent velocity fluctuations are
always isotropic, when the mean velocity profile is adequately removed.
