[Paper Review] Valuations and Boolean Models

Julia Hörrmann, Wolfgang Weil|arXiv (Cornell University)|Oct 27, 2015

Point processes and geometric inequalities19 references3 citations

TL;DR

This paper establishes mean value formulas for scalar- and tensor-valued valuations in stationary Boolean models using translative integral geometry. It introduces harmonic intrinsic volumes as a key tool, enabling the estimation of particle intensity in non-isotropic Boolean models through series expansions involving densities of these functionals, generalizing Miles and Davy's 1976 results to non-isotropic settings.

ABSTRACT

Valuations, as additive functionals, allow various applications in Stochastic Geometry, yielding mean value formulas for specific random closed sets and processes of convex or polyconvex particles. In particular, valuations are especially adapted to Boolean models, the latter being the union sets of Poisson particle processes. In this chapter, we collect mean value formulas for scalar- and tensor-valued valuations applied to Boolean models under quite general invariance assumptions.

Motivation & Objective

• To extend mean value formulas for Boolean models beyond isotropic cases, where rotational symmetry is often unrealistic.
• To develop a framework for estimating the intensity (mean number of particles per unit volume) in stationary, non-isotropic Boolean models.
• To apply harmonic intrinsic volumes—generalized, rotationally sensitive functionals—to capture directional anisotropy in random spatial structures.
• To derive explicit series representations for intensity in two and three dimensions using densities of harmonic intrinsic volumes.
• To provide a theoretical foundation for modeling non-isotropic random spatial processes in stochastic geometry and image analysis.

Proposed method

• Utilizes translative integral formulas for general valuations, particularly for curvature measures and mixed volumes, to derive mean value formulas.
• Applies the theory of spherical harmonics to define harmonic intrinsic volumes $ V_j^{l,p} $, which generalize the classical intrinsic volumes $ V_j $.
• Employs the rotation invariance property of harmonic intrinsic volumes: $ \int_{SO_n} V_j^{l,p}(\vartheta K) \, \nu(d\vartheta) = \delta_{(l,p),(0,1)} V_j(K) $, enabling separation of isotropic and anisotropic components.
• Derives iterated translative formulas for harmonic intrinsic volumes, leading to density formulas for Boolean models: $ \overline{V}^{l,p}_j(Z) = e^{-\overline{V}_n(X)} \sum_{\mathbf{m} \in \text{mix}(j)} \frac{(-1)^{|\mathbf{m}|-1}}{|\mathbf{m}|!} \overline{V}^{l,p}_{\mathbf{m}}(Y,\dots,Y) $.
• Introduces the concept of rotation regular grain distributions, where the grain distribution $ \mathbf{Q} $ decomposes into a rotation-invariant part $ \tilde{\mathbf{Q}} $ and a rotation-dependent function $ \eta $, enabling analysis of non-isotropic models.
• Derives series representations for intensity $ \gamma $ in terms of densities of harmonic intrinsic volumes, with $ \rho = 1 / (1 - \overline{V}_d(Z)) $, valid in 2D and 3D.

Experimental results

Research questions

• RQ1 How can the intensity of a stationary, non-isotropic Boolean model be estimated from geometric characteristics of the union set?
• RQ2 What role do harmonic intrinsic volumes play in capturing directional anisotropy in random spatial structures?
• RQ3 Can the classical Miles and Davy (1976) formulas for isotropic Boolean models be generalized to non-isotropic settings?
• RQ4 What is the structure of translative integral formulas for harmonic intrinsic volumes in higher dimensions?
• RQ5 How do the densities of harmonic intrinsic volumes relate to the underlying grain distribution in a Boolean model?

Key findings

• In two dimensions, the intensity $ \gamma $ of a stationary Boolean model with rotation regular grain distribution admits a series representation: $ \gamma = \rho \overline{V}_0(Z) + \rho^2 \sum_{l,m=0}^\infty \sum_{p=1}^{D(2,l)} \sum_{q=1}^{D(2,m)} c_{l,m}^{p,q} \overline{V}_1^{l,p}(Z) \overline{V}_1^{m,q}(Z) $, with $ \rho = 1 / (1 - \overline{V}_2(Z)) $.
• In three dimensions, the intensity $ \gamma $ is given by a more complex series: $ \gamma = \rho \overline{V}_0(Z) + \rho^2 \sum_{l,m} \sum_{p,q} d_{l,m}^{p,q} \overline{V}_1^{l,p}(Z) \overline{V}_2^{m,q}(Z) + \rho^3 \sum_{l,m,o} \sum_{p,q,s} e_{l,m,o}^{p,q,s} \overline{V}_2^{l,p}(Z) \overline{V}_2^{m,q}(Z) \overline{V}_2^{o,s}(Z) $, with $ \rho = 1 / (1 - \overline{V}_3(Z)) $.
• Harmonic intrinsic volumes $ V_j^{l,p} $ are real-valued functionals that generalize the classical intrinsic volumes $ V_j $, with $ V_j^{0,1} = V_j $, and they vanish under rotation averaging unless $ (l,p) = (0,1) $.
• For isotropic Boolean models, $ \overline{V}_j^{l,p}(Z) = 0 $ unless $ (l,p) = (0,1) $, confirming that harmonic intrinsic volumes reduce to standard intrinsic volumes in the isotropic case.
• The derived formulas generalize the 1976 results of Miles and Davy by allowing non-isotropic particle distributions, enabling intensity estimation from measurements of specific harmonic intrinsic volume densities.
• The method relies on the decomposition of the grain distribution into a rotation-invariant part and a rotation-dependent component $ \eta $, which allows modeling of anisotropic structures while preserving mathematical tractability.