Conditions are derived for the asymptotic normality of a general class of vector-valued functionals of stationary Boolean models in the d-dimensional Euclidean space, where a Lindeberg-type central limit theorem for m-dependent random fields, m ∈ N, is applied. These functionals can be used to construct joint estimators for the vector of specific intrinsic volumes of the underlying Boolean model. Extensions to functionals of more general germ–grain models satisfying some mixing and integrability conditions are also discussed.

A theoretical approach to estimate the Minkowski functionals, i.e., area fraction, specifc boundary length and specifc Euler number in 2D, and their asymptotic covariance matrix proposed by Spodarev and Schmidt (2005) and Pantle et al. (2006a;b) is applied to real image data. These two-dimensional images show mammary gland tissue and should be classifed automatically as tumor-free or mammary cancer, respectively. The estimation procedure is illustrated step-by-step and the calculations are described in detail. To reduce dependencies from chosen parameters, a least-squares approach is considered as recommended by Klenk et al. (2006). Emphasis is placed on the detailed description of the estimation procedure and the application of the theory to real image data

For stationary vector-valued random fields on the asymptotic covariance matrix for estimators of the mean vector can be given by integrated covariance functions. To construct asymptotic confidence intervals and significance tests for the mean vector, non-parametric estimators of these integrated covariance functions are required. Integrability conditions are derived under which the estimators of the covariance matrix are mean-square consistent. For random fields induced by stationary Boolean models with convex grains, these conditions are expressed by sufficient assumptions on the grain distribution. Performance issues are discussed by means of numerical examples for Gaussian random fields and the intrinsic volume densities of planar Boolean models with uniformly bounded grains. Copyright (c) 2009 Board of the Foundation of the Scandinavian Journal of Statistics.

Conditions are derived for the asymptotic normality of a general class of vector-valued functionals of stationary Boolean models in the d-dimensional Euclidean space, where a Lindeberg-type central limit theorem for m-dependent random fields, m ∈ N, is applied. These functionals can be used to construct joint estimators for the vector of specific intrinsic volumes of the underlying Boolean model. Extensions to functionals of more general germ-grain models satisfying some mixing and integrability conditions are also discussed.

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