On the dimension of the boundary of clumps in a multi-type Boolean model

• 成果类型: Article
• 署名作者: Hall, P; Polzehl, J
• 署名单位: Commonwealth Scientific & Industrial Research Organisation (CSIRO); Zuse Institute Berlin
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1996
• 页码: 1521-1535
• 关键词: surfaces; distributions; fractals

摘要

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.