In the present paper, statistical procedures for estimating shape and orientation of arbitrary three-dimensional particles are developed. The focus of this work is on the case where the particles cannot be observed directly, but only via sections. Volume tensors are used for describing particle shape and orientation, and stereological estimators of the tensors are derived. It is shown that these estimators can be combined to provide consistent estimators of the moments of the so-called particle cover density. The covariance structure associated with the particle cover density depends on the orientation and shape of the particles. For instance, if the distribution of the typical particle is invariant under all rotations, then the covariance matrix is proportional to the identity matrix. A non-parametric test for such isotropy is developed. A flexible Lévy-based particle model is proposed, which may be analysed using a generalized methods of moments in which the volume tensors enter. The developed methods are used to study the cell organization in the human brain cortex.

Keywords: Ellipsoidal approximation, Local stereology, Optical rotator, Particle cover density, Particle shape, Particle orientation, Volume tensors