Central limit theorems for functionals of stationary germ-grain models

Abstract

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.

Algorithms for the computation of the Minkowski functionals of deterministic and random polyconvex sets

“…It means that EY i (x) = l i does not depend on x 2 R d and the covariance CovðY i ðxÞ; Y i ðx þ hÞÞ ¼ Cov Y i ðhÞ depends only on h 2 R d for all i. Furthermore, we assume that their mean values l i can be represented as l i ¼ P d j¼0 a ij V j ðNÞ, where the matrix A = (a ij ) of constant coefficients a ij is regular for n = d. Then, it holds It follows from (17) that random fields Y i must be somehow connected with the random set N. A sufficiently large family of such fields is given in [20]. There, one puts Y i (x) = f i ((N À x) \ K i ) for a conditionally bounded additive set functional f i and a small scanning window K i .…”

“…There, one puts Y i (x) = f i ((N À x) \ K i ) for a conditionally bounded additive set functional f i and a small scanning window K i . In what follows, concrete examples of such fields will be given; see also [20]. One can expect that such examples are constructed from set functionals F i of the form (5).…”

“…By the method of moments, b V ðNÞ ¼ A À1 b l is an unbiased estimator for V ðNÞ in the case n = d. If a sequence of monotonously increasing observation windows W k is used, the above estimator is L 2 -consistent and asymptotically normally distributed as k fi 1. This fact is used to construct asymptotic Gaussian tests for the estimated specific intrinsic volumes, see [20]. They enable us to compare the geometrical structure of two homogeneous binary images by detecting significant difference in their estimated specific intrinsic volumes.…”

“…For a rectangular observation window W, the edge correction (20) can be used alternatively. Then, the vector (21) must be replaced by…”

“…the very efficient algorithms given in [9]. This is the price one has to pay for the nice statistical properties of these methods that allow their use in image comparison based on asymptotical Gauss tests (see [20] for details).…”