In a fascinating article on models for particulate microstructure, Hermann and Ohser argue that the superposition of stochastically independent Boolean models may be thought of as producing sets whose boundaries have unusual properties of dimension. Hermann and Ohser employ such superposition models to analyse the particulate structure of rust. In the present paper we provide a theoretical foundation for their work, with respect to both the definition of dimension and its statistical estimation. We argue that the dimension of set boundaries is not well defined in either the Minkowski or capacity senses. It can, however, be properly defined using a mathematical formalization of the practical notion of analysing a spatial pattern at different resolution levels. Adopting this as a basis, we consider properties of estimators of dimension.
"On the dimension of the boundary of clumps in a multi-type Boolean model." Ann. Statist. 24 (4) 1521 - 1535, August 1996. https://doi.org/10.1214/aos/1032298281