Fractional Brownian Fields Research Articles

Full inference for the anisotropic fractional Brownian field

The anisotropic fractional Brownian field (AFBF) is a non-stationary Gaussian random field which has been used for the modeling of textured images. In this paper, we address the open issue of estimating the functional parameters of this field, namely the topothesy and Hurst functions. We propose an original method which fits the empirical semi-variogram of an image to the semi-variogram of a turning-band field that approximates the AFBF. Expressing the fitting criterion in terms of a separable non-linear least square criterion, we design a minimization algorithm inspired by the variable projection approach. This algorithm also includes a coarse-to-fine multigrid strategy based on approximations of functional parameters. Compared to existing methods, the new method enables to estimate both functional parameters on their whole definition domain. On simulated textures, we show that it has a low estimation error, even when the parameters are approximated with a high precision. We also apply the method to characterize mammograms and sample images with synthetic parenchymal patterns.

BSDEs generated by fractional space-time noise and related SPDEs

This paper is concerned with the backward stochastic differential equations whose generator is a weighted fractional Brownian field: Yt=ξ+∫tTYsW(ds,Bs)−∫tTZsdBs, 0≤t≤T, where W is a (d+1)-parameter weighted fractional Brownian field of Hurst parameter H=(H0,H1,⋯,Hd), which provide probabilistic interpretations (Feynman-Kac formulas) for certain linear stochastic partial differential equations with colored space-time noise. Conditions on the Hurst parameter H and on the decay rate of the weight are given to ensure the existence and uniqueness of the solution pair. Moreover, the explicit expression for both components Y and Z of the solution pair is given.

PyAFBF: a Python library for sampling image textures from the anisotropic fractional Brownian field.

Richard, F. J., (2022). PyAFBF: a Python library for sampling image textures from the anisotropic fractional Brownian field.. Journal of Open Source Software, 7(75), 3821, https://doi.org/10.21105/joss.03821

The multiplicative chaos of H=0 fractional Brownian fields

We consider a family of fractional Brownian fields {BH}H∈(0,1) on Rd, where H denotes their Hurst parameter. We first define a rich class of normalizing kernels ψ and we rescale the normalised field by the square-root of the gamma function Γ(H), such that the covariance of XH(x)=Γ(H)12(BH(x)−∫RdBH(u)ψ(u,x)du), converges to the covariance of a log-correlated Gaussian field when H↓0. We then use Berestycki’s “good points” approach (Electron. Commun. Probab. 22 (2017) Paper No. 27) in order to derive the convergence of the exponential measure of the fractional Brownian field MγH(dx)=eγXH(x)−γ22E[XH(x)2]dx, towards a Gaussian multiplicative chaos, as H↓0 for all γ∈(0,γ∗(d)), where γ∗(d)> 7 4d. As a corollary we establish the L2 convergence of MγH over the sets of “good points”, where the field XH has a typical behaviour. As a by-product of the convergence result, we prove that for log-normal rough volatility models with small Hurst parameter, the volatility process is supported on the sets of “good points” with probability close to 1. Moreover, on these sets the volatility converges in L2 to the volatility of multifractal random walks.

Tangent fields, intrinsic stationarity, and self similarity

This paper studies the local structure of continuous random fields on Rd taking values in a complete separable linear metric space V. Extending seminal work of Falconer, we show that the generalized (1+k)-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron. These results motivate the further study of the structure of V-valued intrinsic random functions of order k (IRFk, k=0,1,…). To this end, we focus on the special case where V is a Hilbert space. Building on the work of Sasvari and Berschneider, we establish the spectral characterization of all second order V-valued IRFk’s, extending the classical Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar V-valued IRFk’s, generalizing results of Dobrushin and Didier, Meerschaert and Pipiras, among others. These processes are the Hilbert-space-valued versions of the general k-th order operator fractional Brownian fields and are characterized by their self-similarity operator exponent as well as a finite trace class operator valued spectral measure. We conclude with several examples motivating future applications to probability and statistics.

Fault classification via energy based features of two-dimensional image data

Automated anomaly detection is the prerequisite to minimize human errors and costs caused by manual inspection. Recently, image-based anomaly detections have gained more attention by widely adopting machine vision systems and computer-aided detections. We propose a classification method using spectral features based on 2D discrete wavelet packet transform under the hierarchical structure of wavelet energies. By capturing the self-similar and long-range dependent characteristics of 2D fractional Brownian field (fBf), wavelet packet spectra are derived to construct a linear model representing the relationship between wavelet energies and resolution levels. 2D DWPT-based energy features effectively preserve irregular oscillations in original images at high-frequency domains as well as at low-frequency domains under a pyramidal structure. In comparison with the existing 2D discrete wavelet transform method, the proposed method shows a potential in efficiently classifying normal and abnormal image data in a numerical example and a real industrial application.

Log-periodogram regression of two-dimensional intrinsically stationary random fields

We propose a new semiparametric model for two-dimensional intrinsically stationary random fields and an estimator for the long memory parameter of the model. The model includes a fractional Brownian field, which is isotropic and has been used to model many physical processes in space, as a special case but also allows some anisotropicity. The estimator is based on tapered discrete Fourier transforms and periodograms of observations. Then we apply a log-periodogram regression, which is originally proposed to estimate a long-memory parameter of semiparametric models for time series data. We prove that for our model, the estimator is still consistent and has the limiting normal distribution as the sample size goes to infinity. Furthermore, it is robust to model misspecification. We conduct a computational simulation to compare the performance of it with those of different estimators proposed by other authors and apply our model to an empirical analysis of real data.

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, ) self-similar processes.

Uniform Convergence of the Series Expansion of the Multifractional Brownian Motion

In this paper we define the multifractional Brownian motion and we give some properties. we study the uniform Convergence of the Serie expansion. After having determined the covariance function, we give in proposition 2 another proof of almost sure uniform convergence on compact K of the series. We will finish by showing that the m.B.f is locally astymptotically self-similar, with field or fractional Brownian field with Hurst exposant H. One of the problem, for application of multifractional Brownian motion, is the regularity of the function. In the filtered white noise model the increments are no more homogeneous as in fractional Brownian field case. It is obvious when we consider the tangent field associated with a function. Still the multifractional function in the previous model is constant and it is not convient for many applications. We show the uniform convergence of the series on K. We deduce from the previous questions the almost sure uniform convergence of the series to a mBm.

From directed polymers in spatial-correlated environment to stochastic heat equations driven by fractional noise in [formula omitted] dimensions

We consider the limit behavior of partition function of directed polymers in random environment, which is represented by a linear model instead of a family of i.i.d.variables in 1+1 dimensions. Under the assumption on the environment that its spatial correlation decays algebraically, using the method developed in Alberts et al. (2014), we show that the scaled partition function, as a process defined on [0,1]×R, converges weakly to the solution to some stochastic heat equations driven by fractional Brownian field. The fractional Hurst parameter is determined by the correlation exponent of the random environment. Here multiple Itô integral with respect to fractional Gaussian field and spectral representation of stationary process are heavily involved.

Three-dimensional characterization method of pile–rock interface roughness based on fractal geometry

Shaft friction is an important component of the pile foundation bearing capacity in construction projects, whereas the roughness of a pile–rock interface is one of the key components affecting shaft friction. An accurate characterization of the roughness of a pile–rock interface can help in determining the lateral friction of the pile–rock interface. In the present study, a new fractal geometry characterization parameter was proposed based on traditional roughness characterization methods and the structural function of a fractional Brownian field, namely, the three-dimensional (3D) characteristic roughness parameter l∗. Its definition, geometric meaning, and calculation method are described herein. In addition, photogrammetry was used to obtain grayscale images of pile–rock interfaces from an actual construction project. The results verified through a 3D reconstruction method reveal that the grayscale values can be used to represent the changes in the interface profile. Consequently, the 3D fractal dimension D, the surface root mean square deviation Sq, and the 3D characteristic roughness l∗ were calculated based on the grayscale values of the processed images. A comparison and verification of the results demonstrate that the 3D characteristic roughness parameter is more objective and advantageous than the other parameters. The 3D characteristic roughness is not only suitable for characterizing the roughness of pile–rock interfaces, but also applicable to characterize the roughness of other surface types, including rock discontinuities, rock joints, and concrete and road surfaces.

Nonexistence of fractional Brownian fields indexed by cylinders

We show in this article that there exists no $H$-fractional Brownian field indexed by the cylinder $\mathbb{S} ^{1} \times ]0,\varepsilon [$ endowed with its product distance $d$ for any $\varepsilon >0$ and $H>0$. This is equivalent to say that $d^{2H}$ is not a negative definite kernel, which also leaves us without a proof that many classical stationary kernels, such that the Gaussian and exponential kernels, are positive definite kernels – or valid covariances – on the cylinder. We generalise this result from the cylinder to any Riemannian Cartesian product with a minimal closed geodesic. We also investigate the case of the cylinder endowed with a distance asymptotically close to the product distance in the neighbourhood of a circle. As a consequence of our result, we show that the set of $H$ such that $d^{2H}$ is negative definite behaves in a discontinuous way with respect to the Gromov-Hausdorff convergence on compact metric spaces. These results extend our comprehension of kernel construction on metric spaces, and in particular call for alternatives to classical kernels to allow for Gaussian modelling and kernel method learning on cylinders.

Fast and exact synthesis of some operator scaling Gaussian random fields

Abstract Operator scaling Gaussian random fields, as anisotropic generalizations of self-similar fields, know an increasing interest for theoretical studies in the literature. However, up to now, they were only defined through stochastic integrals, without explicit covariance functions. In this paper we exhibit explicit covariance functions, as anisotropic generalizations of fractional Brownian fields ones, and define corresponding Operator scaling Gaussian random fields. This allows us to propose a fast and exact method of simulation in dimension 2 based on the circulant embedding matrix method, following ideas of Stein [34] for fractional Brownian surfaces syntheses. This is a first piece of work to popularize these models in anisotropic spatial data modeling.

A fast algorithm for multifield representation and multiscale simulation of high‐quality 3D stochastic aggregate microstructures by concurrent coupling of stationary Gaussian and fractional Brownian random fields

SummaryThis study presents a multiscale computational framework for the representation and generation of concrete aggregate microstructures on the basis of the multifield theory, which couples the stationary Gaussian random field with the fractional Brownian random field. Specifically, the stationary Gaussian field is utilized to simulate the morphological shape of an aggregate on the coarse scale, whereas the surface topography of the aggregate on the fine scale is represented by the fractional Brownian field. To bridge the 2 scales, a concurrent coupling formula is proposed. This coupled technique allows for smooth transition between the coarse and fine scales and permits the rapid generation of highly realistic concrete aggregates that can be tailored to the desired quality and requirements, making the algorithm computationally appealing. In the generation of the random fields on the 2 scales, the Fourier representation of block circulant covariance matrices with circulant blocks is exploited, which yields substantial efficiency advantages over the conventional Cholesky decomposition approach in factorizing covariance matrices as well as simulating random fields. Meanwhile, a microsurface postprocessing and reconstruction procedure is also developed to convert the generated random fields into realistic 3D shapes. The numerical methodology proposed in this study offers tremendous potential for a plethora of applications in cement‐based materials.

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.

Anisotropy of Hölder Gaussian random fields: characterization, estimation, and application to image textures

The characterization and estimation of the Holder regularity of random fields has long been an important topic of Probability theory and Statistics. This notion of regularity has also been widely used in Image Analysis to measure the roughness of textures. However, such a measure is often not sufficient to characterize textures as it does account for their directional properties (e.g. isotropy and anisotropy). In this paper, we present an approach to further characterize directional properties associated to the Holder regularity of random fields. Using the spectral density, we define a notion of asymptotic topothesy which quantifies directional contributions of field high-frequencies to the Holder regularity. This notion is related to the topothesy function of the so-called anisotropic fractional Brownian fields, but is defined in a more generic framework of intrinsic random fields. We then propose a method based on multi-oriented quadratic variations to estimate this asymptotic topothesy. Eventually, we evaluate this method on synthetic data and apply it for the characterization of historical photographic papers.

Domain and range symmetries of operator fractional Brownian fields

An operator fractional Brownian field (OFBF) is a Gaussian, stationary increment Rn-valued random field on Rm that satisfies the operator self-similarity property {X(cEt)}t∈Rm=L{cHX(t)}t∈Rm, c>0, for two matrix exponents (E,H). In this paper, we characterize the domain and range symmetries of OFBF, respectively, as maximal groups with respect to equivalence classes generated by orbits and, based on a new anisotropic polar-harmonizable representation of OFBF, as intersections of centralizers. We also describe the sets of possible pairs of domain and range symmetry groups in dimensions (m,1) and (2,2).

Wavelet-based estimator for the hurst parameters of fractional brownian sheet

It is proposed a class of statistical estimators Ĥ = (Ĥ1, … Ĥ) for the Hurst parameters H =(H1, …, Hd) of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are asymptotically normal. These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals, which is important in texture classification and improvement of diffusion tensor imaging (DTI) of nuclear magnetic resonance (NMR). Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators. We find that when Hi ≥ ½, the estimators are accurate, and when Hi < ½, there are some bias.

Some limit results on supremum of Shepp statistics for fractional Brownian motion

Define the incremental fractional Brownian field ZH(τ, s) = BH(s + τ) − BH(s), where BH(s) is a standard fractional Brownian motion with Hurst parameter H ∈ (0, 1). In this paper, we first derive an exact asymptotic of distribution of the maximum \({M_H}\left( {{T_u}} \right) = {\sup _{r \in [0,1],s \in [0,x{T_u}]}}{Z_H}\left( {\tau ,s} \right)\), which holds uniformly for x ∈ [A,B] with A,B two positive constants. We apply the findings to analyse the tail asymptotic and limit theorem of MH(T) with a random index T. In the end, we also prove an almost sure limit theorem for the maximum M1/2(T) with non-random index T.

Brownian Integrated Covariance Functions for Gaussian Process Modeling: Sigmoidal Versus Localized Basis Functions

ABSTRACTGaussian process modeling, or kriging, is a popular method for modeling data from deterministic computer simulations, and the most common choices of covariance function are Gaussian, power exponential, and Matérn. A characteristic of these covariance functions is that the basis functions associated with their corresponding response predictors are localized, in the sense that they decay to zero as the input location moves away from the simulated input sites. As a result, the predictors tend to revert to the prior mean, which can result in a bumpy fitted response surface. In contrast, a fractional Brownian field model results in a predictor with basis functions that are nonlocalized and more sigmoidal in shape, although it suffers from drawbacks such as inability to represent smooth response surfaces. We propose a class of Brownian integrated covariance functions obtained by incorporating an integrator (as in the white noise integral representation of a fractional Brownian field) into any stationary covariance function. Brownian integrated covariance models result in predictor basis functions that are nonlocalized and sigmoidal, but they are capable of modeling smooth response surfaces. We discuss fundamental differences between Brownian integrated and other covariance functions, and we illustrate by comparing Brownian integrated power exponential with regular power exponential kriging models in a number of examples. Supplementary materials for this article are available online.