A rotational integral formula for intrinsic volumes
type: Item
instance of: scholarly article
title: A rotational integral formula for intrinsic volumes
zbMATH Open document ID: 1154.60010
author: Eva B. Vedel Jensen
author: Jan Rataj
published in: Advances in Applied Mathematics
publication date: 2008-11-06T00:00:00Z
review text: In local stereology, geometric identities involving sections through a fixed point are used. A geometric identity has the following general form \(\int \alpha(X\cap L)\, dL = \beta(X)\), where \(\alpha\) and \(\beta\) are geometrical quantities (volume, surface area or, more generally, intrinsic volumes), \(X\) is the spatial object of interest, \(L\) is the probe (line, plane, grid of parallel lines, linear subspace, affine subspace) and \(dL\) is `uniform integration' over positions of \(L\) (integration with respect to a measure invariant under a certain group action).ln geometric tomography, \(\alpha(X\cap L)\) is called a section function for particular choices of \(\alpha\). The mathematical foundation of local stereology has been developed by \textit{E. B. Vedel Jensen} [Local Stereology, World Scientific, New York (1998; Zbl 0909.62087)]. A number of geometric identities have been developed in local stereology, including a generalized Blaschke-Petkantschin formula, a slice formula, a geometric identity for surface area and a vertical section formula. A review of these geometric identities has recently been given by \textit{S. Kötzer} [``Geometric identities in stereological particle analysis'', Image Anal. Stereol. 25, 63--74 (2006)]. Recall that for a subset \(X\) of \({\mathbb R}^d\), satisfying certain regularity, we can define \(d+1\) intrinsic volumes \(V_k(X)\), \(k=0,\dots,d\). The formula to be derived in the present paper shows how the rotational average of intrinsic volumes of sections of a spatial structure relates to principal curvatures and their corresponding principal directions of the spatial structure itself. The formula can be regarded as a rotational version of the classical Crofton formula, relating integrals of intrinsic volumes defined on \(j\)-dimensional affine subspaces to intrinsic volumes of the original set \(X\), \(\int_{{\mathcal F}_j^d}V_k(X\cap F_j)\, dF_j^d= c_{d,j,k}\, V_{d-j+k}(X)\), \(j=0,\dots,d\), \(k=0,1,\dots,j\), \(c_…
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 5363184
zbMATH Keywords: geometric measure theory
zbMATH Keywords: integral geometry
zbMATH Keywords: Grassmann manifold
zbMATH Keywords: set with positive reach
zbMATH Keywords: stereology
reviewed by: Viktor Ohanyan
MaRDI profile type: Publication
full work available at URL: j.aam.2008.03.004
OpenAlex ID: W2053598185
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DOI: 10.1016/J.AAM.2008.03.004
description: scientific article; zbMATH DE number 5363184