A rotational integral formula for intrinsic volumes
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 \(X\), \(\int_{{\mathcal F}_j^d}V_k(X\cap F_j)\, dF_j^d= c_{d,j,k}\, V_{d-j+k}(X)\), \(j=0,\dots,d\), \(k=0,1,\dots,j\), \(c_{d,j,k}\) is a known constant. Here \({\mathcal F}_j^d\) is the set of \(j\)-dimensional affine subspaces in \({\mathbb R}^d\) and \(F_j=x+L_j\), where \(L_j\) is a \(j\)-subspace and \(x\in L_j^{\perp}\). Furthermore, \(dF_j^d=dx^{d-j}\, dL_j^d\) is the element of the motion invariant measure on \(j\)-dimensional affine subspaces, where \(dL_j^d\) is the element of the rotation invariant measure on the set of \(j\)-spaces, and \(dx^{d-j}\) is the element of the Lebesgue measure in \(L_j^{\perp}\). The paper is organized as follows. In Section 2, basic concepts from geometric measure theory are shortly summarized. The rotational integral formula for intrinsic volumes is presented in Section 3. In Section 4, the authors show how certain weight factors appearing in the rotational integral formula can be expressed in terms of hypergeometric functions. Modified sectional intrinsic volumes with a clearer relation to geometric properties of the original set are introduced in Section 5. Applications to stereological particle analysis is shortly discussed in Section 6. Section 7 is devoted to a geometric measure theoretic proof of the main theorem. In Section 8, extensions of the main theorem are discussed. The presented results are illustrated by 6 simple examples.
scientific article; zbMATH DE number 5363184
scientific article