Minkowski tensor density formulas for Boolean models

TL;DR

This paper introduces Minkowski tensor densities as global shape descriptors for Boolean models, relating local and global properties, and demonstrating their use in characterizing anisotropy and estimating model parameters.

Contribution

It develops density formulas for Minkowski tensors in Boolean models, extending intrinsic volume formulas, and shows their application in anisotropy analysis and parameter estimation.

Findings

Minkowski tensor densities are proportional to intrinsic volume densities in isotropic models.

Density formulas are validated through simulations with elliptical and rectangular grains.

Tensor densities effectively characterize grain orientation and anisotropy.

Abstract

A stationary Boolean model is the union set of random compact particles which are attached to the points of a stationary Poisson point process. For a stationary Boolean model with convex grains we consider a recently developed collection of shape descriptors, the so called Minkowski tensors. By combining spatial and probabilistic averaging we define Minkowski tensor densities of a Boolean model. These densities are global characteristics of the union set which can be estimated from observations. In contrast local characteristics like the mean Minkowski tensor of a single random particle cannot be observed directly, since the particles overlap. We relate the global to the local properties by density formulas for the Minkowski tensors. These density formulas generalize the well known formulas for intrinsic volume densities and are obtained by applying results from translative integral geometry. For an isotropic Boolean model we observe that the Minkowski tensor densities are proportional to the intrinsic volume densities, whereas for a non-isotropic Boolean model this is usually not the case. Our results support the idea that the degree of anisotropy of a Boolean model may be expressed in terms of the Minkowski tensor densities. Furthermore we observe that for smooth grains the mean curvature radius function of a particle can be reconstructed from the Minkowski tensor densities. In a simulation study we determine numerically Minkowski tensor densities for non-isotropic Boolean models based on ellipses and on rectangles in two dimensions and find excellent agreement with the derived analytic density formulas. The tensor densities can be used to characterize the orientational distribution of the grains and to estimate model parameters for non-isotropic distributions.