Estimation of anisotropic Gaussian fields through Radon transform

Abstract

Abstract.We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. Finally we use simulated fields to test the proposed estimator in different anisotropic and isotropic cases. Results show that the estimator behaves similarly in all cases and is able to detect anisotropy quite accurately.Mathematics Subject Classification. 60G60,62M40,60G15,60G10,60G17,60G35,44A12.

Statistical Tests of Anisotropy for Fractional Brownian Textures. Application to Full-field Digital Mammography

“…We denote V N,u (θ) the variations of the projection Y = R ρ X(θ, ·) defined by Equations (8) and (12). In Theorem 2.3 of [11], we established that (14)…”

“…In [35] and Proposition 1.3 of [11], the convergence of this estimator to H with asymptotic normality was shown under some appropriate assumptions on the variogram of Y or on its spectral density. In the context of the EFBF, we use the generalized quadratic variations of the projected fields.…”

“…Such estimators are particularly interesting: (i) they can deal with a large class of Gaussian fields which includes all fBm without any restriction on the range of H, (ii) they are also consistent and have an asymptotic normality. In [11], we constructed a technique for the estimation of the directional Hurst index based on generalized quadratic variations [11].…”

“…It covers numerous open issues related to the definition and the analysis of anisotropy, the estimation of anisotropic model parameters, and the simulation of anisotropic fields [4,8,12,25,11,34,37,42,10,50]. The work presented here concerns the anisotropic fields defined in [12] by Bonami and Estrade, by considering a large class of Gaussian fields (with stationary increments), for which the variogram v is characterized by a positive even measurable function f satisfying the relation…”

“…Some simulations of EFBF are shown in Figure 1. In our previous works [11], we addressed the problem of estimating the directional Hurst index of an EFBF. We constructed and studied some orientation-dependant estimators based on generalized quadratic variations.…”