Anisotropic Analysis of Some Gaussian Models

Abstract

Although the classical Fractional Brownian Motion is often used to describe porosity, it is not adapted to anisotropic situations. In the present work, we study a class of Gaussian fields with stationary increments and “spectral density.” They present asymptotic self-similarity properties and are good candidates to model a homogeneous anisotropic material, or its radiographic images. Unfortunately, the paths of all Gaussian fields with stationary increments have the same apparent regularity in all directions (except at most one). Hence we propose here a procedure to recover anisotropy from one realization: computing averages over all the hyperplanes which are orthogonal to a fixed direction, we get a process whose Hölder regularity depends explicitly on the asymptotic behavior of the spectral density in this direction.