On the Specific Area of Inhomogeneous Boolean Models. Existence Results and Applications

Abstract

The problem of the evaluation of the so-called specific area of a random closed set, in connection with its mean boundary measure, is mentioned in the classical book by Matheron on random closed sets (Matheron, 1975, p. 50); it is still an open problem, in general. We offer here an overview of some recent results concerning the existence of the specific area of inhomogeneous Boolean models, unifying results from geometric measure theory and from stochastic geometry. A discussion of possible applications to image analysis concerning the estimation of the mean surface density of random closed sets, and, in particular, to material science concerning birth-and-growth processes, is also provided.

On the local approximation of mean densities of random closed sets

“…The problem of the evaluation and the estimation of the mean density of lower dimensional random closed sets (i.e., with Hausdorff dimension less than d), and in particular of the mean surface density λ ∂Θ for full dimensional random sets, is of great interest in several real applications. We mention, for instance, applications in image analysis (e.g., [17] and reference therein), in medicine (e.g., in studying tumor growth [4]), and in material science in phase-transition models (e.g., [27]). (See also [1,8,10] and references therein.…”

“…In many examples and applications the random sets Z i are uniquely determined by suitable random parameters S ∈ K. For instance, in the very simple case of random balls, K = R + and S is the radius of a ball centred in the origin; in applications to birth-and-growth processes, in some models K = R d and S is the spatial location of the nucleus (e.g., [1], Example 2); in segment processes in R 2 , K = R + × [0, 2π] and S = (L, α) where L and α are the random length and orientation of the segment through the origin, respectively (e.g., [26], Example 2); etc. So, in order to use similar notation to previous works (e.g., [26,27]), we shall consider random sets Θ described by marked point processes Φ = {(X i , S i )} in R d with marks in a suitable mark space K so that Z i = Z(S i ) is a random set containing the origin:…”

“…We refer to [26] and [27] (and reference therein) for a more detailed discussion on σ Θ ; we point out that only results for Boolean models with position-independent grains has been given there, whereas in [19] general germ-grains models are considered assuming that the grains are convex, so that results and techniques from convex and integral geometry can be applied. In this last mentioned paper, some formulae for contact distributions and mean densities of inhomogeneous germ-grain models are to be taken in weak form (e.g., [19], Theorem 4.1), unless to add further suitable integrability assumptions (e.g., in [19], Remark 4.4, the existence of a dominating integrable function is assumed).…”

A general formula for the anisotropic outer Minkowski content of a set

“…Possible applications of this are the study of the evolution equations of the mean density of the surface measure of random sets evolving in time and modelling grain growth in recrystallization processes in materials science (see, for example, [17] and the references therein for a more exhaustive treatment). Note that isotropic growth may be modelled by the Minkowski enlargement of the involved crystals, and so the role played by the outer Minkowski content in the study of the surface measure of the crystallized region is evident.…”

On Modelling Recrystallization Processes With Random Growth Velocities of the Grains in Materials Science

“…Models of volume growth have been studied extensively, since the pioneering work by Kolmogorov (1937). We consider here a simple case of the so-called normal growth model (see also, e.g., Capasso and Villa, 2007b;Villa, 2010b and references therein); namely, we shall consider the case in which all the grains develop with random velocity G constant in time or time dependent, so that for any time t all the grains have spherical shape (this is due to the fact that G is not space-dependent).…”