Analysis of Piecewise Fractional Brownian Motion Signals and Textures

Abstract

Piecewise Fractional Brownian motion (p-fBm) is a continuous non-stationary Gaussian process having stationary Gaussian increments, named piecewise fractional Gaussian noise (p-fGn). Unlike fractional Brownian motion (fBm) governed by a unique parameter (Hurst exponent), p-fBm is defined by three parameters: the Hurst exponent in low frequencies, the Hurst exponent in high frequencies and the threshold frequency, which separates the two regimes. In this paper, we present a synthesis method that generates a finite approximation of p-fBm series. Moreover, we analyze the synthesized p-fBm series and test the Gaussianity of both p-fBm and p-fGn. We test the stationarity of the first order increments (p-fGn) and explore an approach to estimation of the process Hurst parameters. Our contribution is relevant to modeling and analysis of certain textures that are characteristic of certain medical and other natural images.