In this paper, we propose a new and generic methodology for the analysis of the texture anisotropy. The methodology is fundamentally based on the stochastic modeling of textures by Anisotropic Fractional Brownian Fields (AFBF). It includes some original statistical tests for deciding if a texture modeled by an AFBF is clearly anisotropic. These tests are based on the estimation of directional parameters of an AFBF by generalized quadratic variations (GQV). The construction of these tests is inspired from a new theoretical result on the convergence of the GQV estimators, which is proved in the paper. The methodology is validated and discussed on simulated data. The methodology is also applied to Full-Field Digital Mammograms (FFDM). On a database composed of 118 FFDM, we show that about 60 percent of textures can be considered as anisotropic with a high level of confidence. These empirical results strongly suggest that AFBF are better-suited than the commonly used isotropic Brownian fields for the modeling of FFDM textures.