Gaussian random fields and monogenic images

Abstract

In this paper, we focus on lighthouse anisotropic fractional Brownian fields (AFBFs), whose self-similarity depends solely on the so-called Hurst parameter, while anisotropy is revealed through the opening angle of an oriented spectral cone. This fractional field generalizes fractional Brownian motion and models rough natural phenomena. Consequently, estimating the model parameters is a crucial issue for modeling and analyzing real data. This work introduces the representation of AFBFs using the monogenic transform. Combined with a multiscale analysis, the monogenic signal is built from the Riesz transform to extract local orientation and structural information from an image at different scales. We then exploit the monogenic signal to define new estimators of AFBF parameters in the particular case of lighthouse fields. We prove that the estimators of anisotropy and self-similarity index (called the Hurst index) are strongly consistent. We demonstrate that these estimators verify asymptotic normality with explicit variance. We also introduce an estimator of the texture orientation. We propose a numerical scheme for calculating the monogenic representation and strategies for computing the estimators. Numerical results illustrate the performance of these estimators. Regarding Hurst index estimation, estimators based on the monogenic representation of random fields appear to be more robust than those using only the Riesz transform. We show that both estimation methods outperform standard estimation procedures in the isotropic case and provide excellent results for all degrees of anisotropy.