Hermine Biermé

Statistical Tests of Anisotropy for Fractional Brownian Textures. Application to Full-field Digital Mammography

In this paper, we propose a new and generic methodology for the analysis of texture anisotropy. The methodology is based on the stochastic modeling of textures by anisotropic fractional Brownian fields. It includes original statistical tests that permit to determine whether a texture is anisotropic or not. These tests are based on the estimation of directional parameters of the fields by generalized quadratic variations. Their construction is founded on a new theoretical result about the convergence of test statistics, which is proved in the paper. The methodology is applied to simulated data and discussed. We show that on a database composed of 116 full-field digital mammograms, about 60 percent of textures can be considered as anisotropic with a high level of confidence. These empirical results strongly suggest that anisotropic fractional Brownian fields are better-suited than the commonly used fractional Brownian fields to the modeling of mammogram textures.

Crossings of smooth shot noise processes

In this paper, we consider smooth shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the mean number of crossings function is obtained through an integral formula. Moreover, as the intensity increases, or equivalently as the number of shots becomes larger, a normal convergence to the classical Rices formula for Gaussian processes is obtained. The Gaussian kernel function, that corresponds to many applications in Physics, is studied in detail and two different regimes are exhibited.

On the perimeter of excursion sets of shot noise random fields

In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets of shot noise random fields in dimension n≥1. This will then allow us to derive the asymptotic behavior of these mean perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity. In particular, we show that two cases occur: we have a Gaussian asymptotic behavior when the kernel function of the shot noise has no jump part, whereas the asymptotic is non-Gaussian when there are jumps.

A Turning-Band Method for the Simulation of Anisotropic Fractional Brownian Fields

In this article, we propose a method for simulating realizations of two-dimensional anisotropic fractional Brownian fields (AFBF) introduced by Bonami and Estrade. The method is adapted from a generic simulation method called the turning-band method (TBM) due to Matheron. The TBM reduces the problem of simulating a field in two dimensions by combining independent processes simulated on oriented bands. In the AFBF context, the simulation fields are constructed by discretizing an integral equation arising from the application of the TBM to nonstationary anisotropic fields. This guarantees the convergence of simulations as the step of discretization is decreased. The construction is followed by a theoretical study of the convergence rate (the detailed proofs are available in the online supplementary materials). Another key feature of this work is the simulation of band processes. Using self-similarity properties, processes are simulated exactly on bands with a circulant embedding method, so that simulation errors are exclusively due to the field approximation. Moreover, we design a dynamic programming algorithm that selects band orientations achieving the optimal trade-off between computational cost and precision. Finally, we conduct a numerical study showing that the approximation error does not significantly depend on the regularity of the fields to be simulated, nor on their degree of anisotropy. Experiments also suggest that simulations preserve the statistical properties of the original field.

Analysis of Texture Anisotropy Based on Some Gaussian Fields with Spectral Density

In this paper, we describe a statistical framework for the analysis of anisotropy of image texture. This framework is based on the modeling of the image by two kinds of non-stationary anisotropic Gaussian field with stationary increments and spectral density: the extended fractional Brownian field (EFBF) and a specific Gaussian operator scaling field (GOSF), which both correspond to a generalization of the fractional Brownian field. In this framework, we tackle anisotropy analysis using some directional processes that are either defined as a restriction of the image on an oriented line or as a projection of the image along a direction. In the context of EFBF and GOSF, we specify links between the regularity of line and projection processes and model parameters, and explain how field anisotropy can be apprehended from the analysis of process regularity. Adapting generalized quadratic variations, we also define some estimators of the regularity of line and projection processes, and study their convergence to field model parameters. Estimators are also evaluated on simulated data, and applied for illustration to medical images of the breast and the bone.

Modulus of continuity of some conditionally sub-Gaussian fields, application to stable random fields

In this paper, we study modulus of continuity and rate of convergence of series of conditionally sub-Gaussian random fields. This framework includes both classical series representations of Gaussian fields and LePage series representations of stable fields. We enlighten their anisotropic properties by using an adapted quasi-metric instead of the classical Euclidean norm. We specify our assumptions in the case of shot noise series where arrival times of a Poisson process are involved. This allows us to state unified results for harmonizable (multi)operator scaling stable random fields through their LePage series representation, as well as to study sample path properties of their multistable analogous.

Non-rigid registration of magnetic resonance imaging of brain

This document presents a non-rigid registration framework for the use of brain magnetic resonance (MR) images comparison. More precisely, we want to compare pre-operative and post-operative MR images in order to assess the deformation due to a surgical removal. Consequently, we propose an application of the theory developed in [3], associated with a new matching criterion based on the representation of the gradient in the dual of a RKHS (Reproducing Kernel Hilbert Space). Moreover, all objects are defined from a periodic point of view, allowing the construction of an efficient algorithm. Numerical results are presented.

Invariance principles for operator-scaling Gaussian random fields

Recently, Hammond and Sheffield introduced a model of correlated random walks that scale to fractional Brownian motions with long-range dependence. In this paper, we consider a natural generalization of this model to dimension

Operator scaling stable random fields

d\geq 2

Self-similar Random Fields and Rescaled Random Balls Models

. We define a