Computers & Geosciences Vol. 19, No. 8, pp. 1209-1210, 1993 Pergamon Press Ltd. Printed in Great Britain

REVIEW

Spatial Tessellations: Concepts and Applications of Voronoi Diagrams by Atsuyuki Okabe, Barry Boots, and Kokichi Sugihara, 1992, John Wiley & Sons, New York, 532 p., ISBN 0 471 93430 5, US $I 12.00

Territory and travel are primitive spatial concepts which are familiar to all of us from earliest childhood. We played within our own yard or went along a path to a friend's yard. In a broader sense, we have come to see the whole world in terms of locations and connections.

Computer geologists use these concepts in their profession, although they are concerned with more elaborate forms. The relationships between object and event, location and occurrence, size and shape, form and function, position and direction, sequence and adjacency, are examples of spatial ideas. So in practice space can be treated as a set of regions and actions which occur within and between those regions.

A specific and rigorous approach to such concepts is provided by the Voronoi diagram, and this allows the computation of quantitative results. The interiors of the regions are collectively exhaustive and mutu- ally exclusive, and the boundaries of the regions are shared with contiguous regions. Each region has a center and a connection between centers is implied by a shared boundary. Space is tessellated by a Voronoi diagram.

The authors have provided the first extensive and comprehensive survey of Voronoi diagrams and the ways in which they can be applied. Because their book describes the extent of knowledge about the partitioning of space into regions and the connec- tivity of those regions, it provides a key to ways in which the Voronoi diagram can be used by computer geologists.

The book flows smoothly through nine chapters arranged so that the technical reader, even if unfamil- iar with this subject, is led on to increasingly signifi- cant aspects. But because so much research is being applied to this subject and because applications are so varied, even scientists who have studied and worked with Voronoi and Delaunay structures will be re- warded with additional insights.

The book begins with a brief background on the subject; many geologists will be aware of the long history of polygonal and triangular prism techniques in ore reserve estimation but possibly few will be aware that Descartes used Voronoi diagrams in his discussion of the solar system over 300 years ago. This is followed by a thumbnail review of several useful concepts and techniques

for manipulating Voronoi and Delaunay sets. Practical applications are mentioned early in the book, and this offers a motivation to a wide range of scientists to look more deeply into this subject. The role of these constructions in computational geometry algorithms, such as those for finding the convex hull of a point set or various network graphs, is fundamental.

In the second chapter, the subject is defined more fully and its significant theoretical qualities are ex- plained in specific terms. For example, a Delaunay triangulation on the plane is the set of all possible triangles such that the circumcircles of the triangles are empty; no datum lies within any of these circum- spheres. This has pertinence for spatial interpolation as explained in chapter six.

Although Voronoi and Delaunay diagrams do have prosaic origins, similar to other rich and fruitful ideas, they have been extended and generalized to a considerable degree. Chapter three discusses vari- ations such as the weighted Voronoi diagrams. These use a set of weights for each center with respect to the other centers so that the shape and boundary of a Voronoi region depends on the weighted distances to other centers. When the Voronoi diagram is applied to regions where there are obstacles that force a discontinuity, shortest-path diagrams can be used. Network Voronoi node-diagrams are applied to grid systems.

The next chapter describes some general algor- ithms used to compute Voronoi diagrams. The wide- spread usefulness of these diagrams, combined with their numerical complexity, has led to prodigious efforts and prolific results in terms of implemen- tations. For an example, the incremental approach builds a diagram by incorporating each datum in turn.

Another technique, the divide-and-conquer methods, subdivide and resubdivide the data into atomic subsets with trivial diagrams and then as- semble the resulting partial diagrams to form the final diagram. The widely applied plane sweep algor- ithms pass a computational front through the data, generating a Voronoi diagram in their wake.

Chapter five concerns the situation where the data are known to be related to, or produced by, a Poisson point process. In this situation, Voronoi diagrams and Delaunay tessellations provide a window for observing and measuring Poisson point process be- havior. A vast amount of work has been done in this direction, and the authors provide a succinct and perceptive overview of the many conclusions that have been the fruit of these labors.

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When a set of data can be treated as samples of a continuous bivariate function, interpolation may be used to produce a representative surface which fits or approximates the data. Chapter six is concerned with a variety of interpolation techniques such as nearest neighbor, natural neighbor, and triangle-based. The concepts of spatial ordering as derived from Voronoi diagrams are discussed. Such ordering provides the aggregate of adjacency relationships among the data which have important uses such as approximating missing data, identifying outliers, and estimating sampling sufficiency.

Voronoi diagrams can be used as effective models of spatial processes, and chapter seven discusses examples of these. Growth models exemplify the behavior of processes such as recrystallization which may be considered to grow from nucleation sites. Spatial-temporal models exemplify processes whose centers are moving. Two-species models describe prey-predator or immiscible fluids processes.

Chapter eight concerns pattern analysis, pattern recognition, and density estimation. Applications oc- cur in many scientific disciplines, and studying these patterns leads to improved understanding of the phenomena of interest. Clustering and categorizing can be done with Voronoi diagrams because they emphasize the distinctions or homogeneity in subsets of the data. The polygon method of ore reserve estimation is a venerable example.

Finally, chapter nine may be thought to be the most important section of the book as it treats Iocational optimization. Where is the best place to drill one more million dollar hole to get the most information? Where can the least number of fire halls be efficaciously located? What arrange- ment of seismic traverses can be used to locate targets of a minimum size? Spatial optimization would seem to have immense potential and compu- tational techniques to achieve this will continue to improve.

Undoubtedly, when you dip into this book you will soon agree that it provides an outstanding synopsis of current knowledge and developments on a subject which has increasing relevance to scientists, and especially to those natural scientists who are con- cerned with spatial information. Particular aspects can be explored in greater depth through the bibli- ography which lists some 700 books and technical papers which the authors have considered. Unfamil- iar concepts and terms can be approached through the comprehensive index. And there are surprisingly few typographical errors.

This book is a reference that all spatial analyst/ programmers will want to have handy.

Department of Mathematics The University of Western Australia Nedlands, WA 6009 Australia

D A ~ WAT~N