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Book review

Author(s) Stoyan, D.
Stoyan, H.
Title Fraktale, Formen, Punktfelder; Methoden der Geometrie-Statistik
Publisher Akademie Verlag
Year of publication 1992
Reviewed by Günter Vojta

Fractal geometry and stochastic geometry constitute important parts of modern geometry of growing interest for basic research in mathematics and theoretical physics as well as for numerous branches of applied science. The book under review presents a good introduction into these fields and an excellent survey of applications and useful methods covering wide areas of physics, materials research, geology, biology, and other disciplines. This book is not a work on mathematics, rather it is written for applied scientists stressing the methods of modern statistical inference. In its kind it is a unique contribution to the relevant literature.

The text is divided into three parts which can be read independently. The first part (50 pp), titled fractals and methods to determine fractal dimensions, contains an introduction into the fractal concept, Hausdorff measure and Hausdorff dimension, other notions of dimension, and a lot of examples of deterministic and stochastic fractals together with practicable methods for dimension calculations.

The second part (140 pp) covers fundamentals and applications of shape statistics in two dimensions. Important topics treated include the description of contours, Fourier analysis, figures disturbed by noise, set-theoretic shape analysis, stochastic polygons, and the description of shapes by so-called landmarks (characteristic points within the shapes).

The third part (145 pp) is devoted to the theory and applications of random point fields (stochastic point processes) in two dimensions, with chapters on finite point fields, Poisson fields, the general theory of point fields, statistics of homogeneous point fields, and point field models including Gibbs fields. The newly developed theory of correlations in marked point fields is taken into account.

Several special mathematical topics are treated in short appendices (25 pp). The text is supplemented by 138 very instructive figures and 23 tables. Extended references with full titles (19 pp) are given separately for the three parts. There is a list of symbols and an index.

The volume is well written and well produced. Certainly theoreticians will sometimes miss mathematical rigour or detailed proofs, and occasionally the text goes as a collection of recipes. However, the special feature of this work is a readable systematic representation of a huge material ready for applications and enriched by a wealth of important or very interesting background information, useful instructions, practical hints and warnings. The authors draw indeed on plentiful resources; the first writer - professor of mathematics at the Bergakademie Freiberg (Saxonia) - is well-known for numerous original papers and author or co-author of several successful books on related topics. The book, written in German, deserves a translation into English.

This work can be offered to anyone who is interested in the fields of fractals, stochastic geometry or point processes, particularly research workers and all libraries for pure and applied sciences including engineering sciences.