This book is to be an accessible book on patterns, their representation, and inference. There are a small number of ideas and techniques that, when mastered, make the subject more accessible. This book has arisen from ten years of a research program which the authors have embarked upon, building on the more abstract developments of metric pattern theory developed by one of the authors during the 1970s and 1980s. The material has been taught over multiple semesters as part of a second year graduate-level course in pattern theory, essentially an introduction for students interested in the representation of patterns which are observed in the natural world. The course has attracted students studying biomedical engineering, computer science, electrical engineering, and applied mathematics interested in speech recognition and computational linguistics, as well as areas of image analysis, and computer vision. Now the concept of patterns pervades the history of intellectual endeavor; it is one of the eternal followers in human thought. It appears again and again in science, taking on different forms in the various disciplines, and made rigorous through mathematical formalization. But the concept also lives in a less stringent form in the humanities, in novels and plays, even in everyday language. We use it all the time without attributing a formal meaning to it and yet with little risk of misunderstanding. So, what do we really mean by a pattern? Can we define it in strictly logical terms? And if we can, what use can we make of such a definition? These questions were answered by General Pattern Theory, a discipline initiated by Ulf Grenander in the late 1960s [1–5]. It has been an ambitious effort with the only original sketchy program having few if any practical applications, growing in mathematical maturity with a multitude of applications having appeared in biology/medicine and in computer vision, in language theory and object recognition, to mention but a few. Pattern theory attempts to provide an algebraic framework for describing patterns as structures regulated by rules, essentially a finite number of both local and global combinatory operations. Pattern theory takes a compositional view of the world, building more and more complex structures starting from simple ones. The basic rules for combining and building complex patterns from simpler ones are encoded via graphs and rules on transformation of these graphs.
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