One of the most crucial, yet most elusive, problems of machine perception is that of pattern representation. The central issue addressed over the four decades of research in this area is which, if any, mathematical or statistical model can adequately represent and synthetically generate the countless varieties of patterns produced by all the distinct types of objects, phenomena and environments encountered, especially when enriched by their biological and natural variability, and by the observation noise.

The model should be compact without compromising the representational capacity and be defined in terms of primitives that are computable from the sensory data.

One of the earliest attempts at developing a comprehensive pattern theory was based on grammatical models in which the representational capacity was achieved by means of a hierarchy of pattern primitives and primitive composition rules. Different sets of rules would generate different classes of patterns. One serious limiting factor of the grammatical (otherwise known as syntactic) approach to pattern representation is its restricted ability to cope with measurement noise. In this respect, statistical models, which to date have been most instrumental in the development of real pattern analysis applications (optical character recognition, speech recognition and so on), can cope with much richer families of pattern classes that would normally be observed in the presence of noise. However, neither the syntactic nor the statistical approach can inherently model the greatest source of pattern variability, namely that arising from transformations to which patterns are subjected, due to, for instance, changing sensing conditions (for example, the viewing geometry in the case of vision).

Grenander's General Pattern Theory is the result of some 25 years of research that puts forward a unified representational model for sensory patterns. The representation is an algebraic structure with topological, probabilistic and statistical aspects. Its basic building blocks are generators equipped with bonds allowing their spatial interconnection. The representational capacity of the structure lies in the combinatorial nature of the configuration spaces of these generators. Invariance to transformations and deformations is handled by equivalence classes.

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This book is divided into seven parts. The first introduces the algebraic structures adopted for pattern representation. The second part discusses topological properties of such regular structures. The third part is motivated by biological growth patterns. It extends the basic algebraic representation developed in the earlier parts of the book to facilitate the modelling of the dynamics of growth processes at the phenomenological level.

In the next part of the book the concept of probability measure over the configuration space of the algebraic representation is introduced. The direct probability dependencies in the chosen family of probability measures are restricted to extend only to immediate neighbours (the neighbours in the Markovian sense). The fifth part discusses how the pattern theoretic constructs developed are related to what can be observed. The largest section of the book, part six, deals with the issues of pattern inference. The final part indicates how the pattern theoretic structures can be created.

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Grenander's book is neither light reading, nor necessarily the final answer to the pattern representation issue. It is, however, a scholarly work of seminal importance. It defines the state of the art in pattern representation and no doubt will influence the direction of future research on this topic for years to come. In this respect it will be an essential volume in the library of any research group or individual working in the field of machine perception and seriously interested in solving the pattern representation problem.

Josef Kittler is professor of machine intelligence, University of Surrey.