Non-parametric estimation of the directional distribution of stationary line and fibre processes

Abstract

Two non-parametric methods for the estimation of the directional measure of stationary line and fibre processes in d-dimensional space are presented. The input data for both methods are intersection counts with finitely many test windows situated in hyperplanes. The first estimator is a measure valued maximum likelihood estimator, if applied to Poisson line processes. The second estimator uses an approximation of the associated zonoid (the Steiner compact) by zonotopes. Consistency of both estimators is proved (without use of the Poisson assumption). The estimation methods are compared The analysis of systems of random fibres is a frequent problem in biology, metallography and other applied sciences. Mathematically, these systems are modelled as stationary fibre processes Y in Rd, d > 2. The anisotropy of such processes can be characterized quantitatively using the directional distribution P0. This distribution indicates, for example, the existence of preferred directions of the fibres. The distribution Po, which is sometimes called the rose of directions, is an even probability measure on the unit sphere Sd-1 in Rd. It can be interpreted as the distribution of the normalized tangential vector (and its antipodal) of a fibre of Y at a 'typical' point. The estimation of Po from samples of Y is often difficult, as tangents of fibres in numerous fibre points have to be determined. Therefore, intersections of Y with hyperplanes are considered, and enumeration of the section process is used to gain information about P0. To be more precise, the intersection Y n u1 of Y with the hyperplane u1 (with unit normal vector u) is almost surely a process of points. The intersection Y nu? is stationary in u1. Its intensity y (u) (mean number of points per unit (d - 1)-volume in u-) gives rise to an even continuous function y on the unit sphere, which is often called the rose of intersection. It determines P0 uniquely: y is the cosine transform of P0. (Sometimes this integral transform is called Buffon transform in the stereological context.) The cosine transform is injective on the space of even finite Borel measures on Sd-1. The inversion of the cosine transform is an ill-posed problem which is deteriorated by the fact that in most applications only finitely many (estimated) values