Wavelet-Based Entropy Measures to Characterize Two-Dimensional Fractional Brownian Fields

The aim of this work was to extend the results of Perez et al. (Physica A (2006), 365 (2), 282–288) to the two-dimensional (2D) fractional Brownian field. In particular, we defined Shannon entropy using the wavelet spectrum from which the Hurst exponent is estimated by the regression of the logarithm of the square coefficients over the levels of resolutions. Using the same methodology. we also defined two other entropies in 2D: Tsallis and the Rényi entropies. A simulation study was performed for showing the ability of the method to characterize 2D (in this case, α = 2 ) self-similar processes.

ARFBF MORPHOLOGICAL ANALYSIS - APPLICATION TO THE DISCRIMINATION OF CATALYST ACTIVE PHASES

This paper addresses the characterization of spatial arrangements of fringes in catalysts imaged by High Resolution Transmission Electron Microscopy (HRTEM). It presents a statistical model-based approach for analyzing these fringes. The proposed approach involves Fractional Brownian Field (FBF) and 2-D AutoRegressive (AR) modeling, as well as morphological analysis. The originality of the approach consists in identifying the image background as an FBF, subtracting this background, modeling the residual by 2-D AR so as to capture fringe information and, finally, discriminating catalysts from fringe characterizations obtained by morphological analysis. The overall analysis is called ARFBF (Auto-Regressive Fractional Brownian Field) based morphology characterization.

Piecewise-Multilinear Interpolation of a Random Field

We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

Piecewise-Multilinear Interpolation of a Random Field

We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

A FRACTIONAL POISSON EQUATION: EXISTENCE, REGULARITY AND APPROXIMATIONS OF THE SOLUTION

We consider a stochastic boundary value elliptic problem on a bounded domain D ⊂ ℝk, driven by a fractional Brownian field with Hurst parameter H = (H1,…,Hk) ∈ [½, 1[k. First, we define the stochastic convolution derived from the Green kernel and prove some properties. Using monotonicity methods, we prove the existence and uniqueness of solution along with regularity of the sample paths. Finally, we propose a sequence of lattice approximations and prove its convergence to the solution of the SPDE at a given rate.