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

Abstract

The characterization and estimation of the Hölder regularity of random elds 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 rarely sucient to characterize textures as it does not 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 Hölder regularity of random elds. Using the spectral density, we dene a notion of asymptotic topothesy which quanties directional contributions of eld high-frequencies to the Hölder regularity. This notion is related to the topothesy function of the so-called anisotropic fractional Brownian elds, but is dened in a more generic framework of intrinsic random elds. 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.

Statistical tests of heterogeneity for anisotropic multifractional Brownian fields

“…Random fields derived from the fractional Brownian motion has proved to be useful models for the analysis of rough textures in the domain of image processing. In particular, anisotropic fractional Brownian fields could be applied to analyze textures appearing in images as various as bone radiographs [10], mammographs [15,36,32,33] , or photographic films [35]. Accounting for directional properties of textures, they enabled to achieve various image classification tasks.…”

“…Following an approach that is similar to the one developed in [32,33,34] for AFBF, estimates involved in our field analysis could serve for defining some measures of the heterogeneity of AMFBF. Moreover, the approach proposed in [35] to estimate the so-called asymptotic topothesy of AFBF could also be extended to the one of tangent fields of AMFBF. This would path the way to the local estimation of the whole topothesy and Hurst functions of AMFBF.…”

Semi-parametric estimation of the variogram scale parameter of a Gaussian process with stationary increments

“…For instance, they enable to interpolate observations by Kriging, notably in computer experiment designs to build a metamodel [44,54]. A second type of application of Gaussian processes is the analysis of local characteristics of images [45] and one dimensional signals (e.g. in finance, see [26,61] and the references therein).…”

“…To conclude, our method of estimation row by row and column by column is perfectly coherent with the separable model (5.2), which is widely used in Kriging applications. Of course this method does not permit to estimate anisotropy in other directions as in [46] or topothesy in the sense of [45] that correspond to more complex models including variations of the Hölder exponent.…”

Surface Regularity via the Estimation of Fractional Brownian Motion Index

“…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.…”