Wavelet Packets of Fractional Brownian Motion: Asymptotic Analysis and Spectrum Estimation

Abstract

This work provides asymptotic properties of the autocorrelation functions of the wavelet packet coefficients of a fractional Brownian motion. It also discusses the convergence speed to the limit autocorrelation function, when the input random process is either a fractional Brownian motion or a wide-sense stationary second-order random process. The analysis concerns some families of wavelet paraunitary filters that converge almost everywhere to the Shannon paraunitary filters. From this analysis, we derive wavelet packet based spectrum estimation for fractional Brownian motions and wide-sense stationary random processes. Experimental tests show good results for estimating the spectrum of 1/f processes.

Structuring of large and heterogeneous texture databases

“…In order to approximately attain this condition, measurements are performed in the wavelet domain. Indeed, wavelet based transforms have appreciable statistical properties such as stationarization, decorrelation and higher order dependency reduction for many random processes (see [1], [2], among others). These properties are obtained with respect to some key parameters that are: the shape of the polyspectra of the input random process, the wavelet order and the wavelet decomposition level (see the above references for more details).…”

“…When p = ∞, the right hand side expression of the above equation is exactly the probability density function of X , under regularity assumptions on this function [11]. For modeling the SWT detail coefficients, we need to highlight the following facts: in the detail wavelet domain, stochastic processes tend to yield distributions that are regular with respect to the Gaussian distribution (see the literature on the wavelet transforms of stochastic processes, among which references [1] and [2] concern central limit theorems for wavelet decompositions). In contrast, geometrically regular functions tend to yield sparse distributions in the detail wavelet domain [12] and significant coefficients are those with large amplitudes (extremes, tails of the statistical distributions).…”

“…This is due to the suitable statistical properties characterizing the wavelet coefficients issued from the decomposition of stochastic processes: dependency reduction occurs among the wavelet coefficients (see [1], [2], [3], among others) and makes relevant, the use of statistical modeling under independence assumption on the wavelet subbands (see the overview given in [4]). …”

Multifractional Vector Brownian Motions, Their Decompositions, and Generalizations

“…It has been used to model natural and social phenomena in various areas of applications including communication network [1,50], finance [12], image processing [39,51], information theory [5,10,24], among others. There are two extensions of the fractional Brownian motion, the multifractional Brownian motion that is a scalar Gaussian process and exhibits local self-similarity, and the fractional vector (multivariate) Brownian motion that is a Gaussian vector process whose components are scalar fractional Brownian motions.…”

Stochasticity: A Feature for the Structuring of Large and Heterogeneous Image Databases

“…(1) Considering the wavelet packet transform, a transform that makes it possible to distribute many random processes as stationary, independent and identically distributed sequences (see, for instance, [18][19][20]);…”