Stereology and stochastic geometry (Q558525)

Mathematical stereology aims to make inference about the properties of random sets in Euclidean space using numerical characteristics of their lower-dimensional sections or projections. The main tools are based on application of methods from stochastic geometry (in particular point processes), convex geometry and geometric measure theory. The first three chapters concern with mathematical preliminaries (Gamma-function, integral transforms, probability distributions, measures and sets), which in many places begin with really basic tools like Venn diagrams for unions and intersections of sets. The corresponding appendices recall the Dedekind construction of real numbers and the axiom of choice (whose relevance to the main theme and the level of rigour of the rest of material seem to be greatly exaggerated). Chapter~4 discusses the construction of random probes using only elementary methods. Chapter~5 discusses mean value stereological formulae for spatial averages and lower-dimensional probes. Chapter~6 concerns the Wicksell problem and related issues concerning intersection of a fiber process with a stripe. Chapter~7 deals with projections and thick sections. Chapter~8 highlights relationships to statistical estimation (but without discussing edge effects). Some multidimensional generalisations form the subject of Chapter~9. Chapter~10 provides some patchy information about kriging, fractals and tessellations. Each chapter ends up with several problems. Quite inconveniently, the bibliography is split between individual chapters. The subject index is reasonably complete. The book under review stems from lecture notes of John E. Hilliard that have been written more than 20 years ago. Consequently, the book covers the most classical area of stereology: stereological identities for mean values, Wicksell problem and other similar problems involving integral equations, thick projections. Modern techniques based on point processes and those from the local stereology are not described at all. The issue of edge effects has neither found a place in the monograph. The interpretation of stochastic geometry in Chapter 10 is very narrow and patchy. The terminology and notation are very unusual and very out-of-date in places, e.g. the notation for intersection of sets, reflected sets, the authors call the fiber process a ``lineal array'', etc. The symbol for mathematical expectation has never been used! The book is written in an entertaining, but rather wordy and in many places not precise style. For instance, the authors only require finite additivity for probability measure and also for sequences of mutually (not pairwise!) disjoint events (p. 80), they believe that ``a space containing all its limit points is complete'' (p. 94) and ``ordinarily closed spaces are Polish'' (p. 94), they speak about the ``ring of convex compact sets'' (p. 95), they apparently think that the property \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) is something else than additivity and call it c-additivity (p. 81), etc. In the latter case they surely have misinterpreted this property from convex geometry, which is imposed only for convex \(A\) and \(B\). The book under review aims to provide an introduction to stereology for non-mathematicians. Indeed, mathematicians should rather read [\textit{E. B. Vedel Jensen}, ``Local stereology'' (1998; Zbl 0909.62087)] or the corresponding chapters from [\textit{D. Stoyan, W. S. Kendall} and \textit{J. Mecke}, ``Stochastic geometry and its applications''. 2nd ed. (1995; Zbl 0838.60002)] and statisticans would be better off with \textit{A. Baddeley} and \textit{E. B. Vedel Jensen} [``Stereology for statisticans'' (2005; Zbl 1086.62108)]. The applied mathematicians should read this book with lots of extra care, knowing that the book does not reflect the current research and contains some mathematical inaccuracies.