Statistical tests of heterogeneity for anisotropic multifractional Brownian fields

Abstract

In this paper, we deal with some anisotropic extensions of the multifractional Brownian fields that account for spatial phenomena whose properties of regularity and directionality may both vary in space. Our aim is to set statistical tests to decide whether an observed field of this kind is heterogeneous or not. The statistical methodology relies upon a field analysis by quadratic variations, which are averages of square field increments. Specific to our approach, these variations are computed locally in several directions. We establish an asymptotic result showing a linear Gaussian relationship between these variations and parameters related to regularity and directional properties of the model. Using this result, we then design a test procedure based on Fisher statistics of linear Gaussian models. Eventually we evaluate this procedure on simulated data. Keywords isotropy • anisotropy • homogeneity • heterogeneity • increment • quadratic variations • statistical test • anisotropic fractional Brownian field • multifractional Brownian field

Surface Regularity via the Estimation of Fractional Brownian Motion Index

“…The fBs, itself, is an isotropic model that has the same properties whatever the orientation. For random fields, there has been a lot of research devoted to the definition of anisotropic models, as well as the characterization and estimation of their directional properties [12,6,10,13,37,35,36,34,38,11,44]. Defining and analyzing anisotropic fBs is another path of research that is opened by this work.…”

“…The Hurst index is a feature which characterizes the global regularity of a fractional Brownian object (random field or surface) assuming that it is homogeneous. Some heterogeneous random field models (indexed on R d ) known as multifractional Brownian fields [5,29,9,2,3,44] were defined to account for local variations of the regularity of a field. Some methods were also constructed to estimate the local regularity of these fields.…”

“…The introduction of the fractional Brownian motion (fBm) by Mandelbrot and Van Ness [25,27] has inspired a great deal of research towards the modeling of phenomena of an irregular nature. Initially defined in R, this random process has been extended to R d to give birth to a wide variety of random fields accounting for complex spatial phenomena (see, for instance, [12,36,5,29,44] or [15] for an exhaustive survey). Besides, there is still an active research to find appropriate definitions of fBm on non-Euclidean spaces such as surfaces [22,23,43,17,18].…”

“…There is a huge statistical literature about the estimation of the Hurst index of the original fBm (see [7,14,15] for review) and its extensions in R d (see, for instance, [4,37,44]), ranging from spectral methods to filtering methods such as quadratic variations or wavelet transforms. By contrast, to the best of our knowledge, there is not any method available for the estimation of the Hurst index of a fBm defined on a surface.…”

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

“…For the simulation of random fields, the library PyAFBF relies upon the turning-band method developed in (Biermé et al, 2015). This method was historically designed to facilitate research on the anisotropic fractional Brownian field (AFBF) (Bonami & Estrade, 2003) and related models (Benassi et al, 1997;Guyon & Perrin, 2000;Peltier & Levy Vehel, 1996;Polisano et al, 2014;Vu & Richard, The library is of interest for researchers in image processing where random fields can serve as texture or noise models for medical images (Biermé et al, 2009;Biermé & Richard, 2011;Richard, 2015Richard, , 2016aRichard & Biermé, 2010) or photographic films (Richard, 2017). It could also be interesting for machine learning researchers who could include the random field simulation in the design or the learning of image generative models such as Generative Adversarial Networks.…”