The discipline of digital geometry studies the geometric properties of models of objects that are conceptually defined in Euclidean space and discretized onto integer grid subsets of the space. While digital geometry is generally considered a branch of discrete geometry, it presents some unique characteristics and addresses certain problems that are often encountered in computer graphics, image processing, and related fields. This book is designed to lay out the theoretical foundations of digital geometry as a discipline and describe its fundamental algorithms and methods.

The book begins with an introduction to relevant basic geometry concepts, distributed over the first three chapters. This is followed by three chapters on the basics of digital geometry, digital curves, surfaces, and manifolds, progressing from definitional material to descriptions of basic algorithms for digital surfaces. The next section progresses to somewhat more advanced theory and practical methods, the former consisting of the theoretical aspects of discrete spaces and digital topology, and the latter covering mainly methods for the common problems of mesh generation and digitization. Computational methods are addressed next, beginning with a brief overview of measures, metrics, and computational geometry algorithms for well-known problems, followed by brief summaries of principles and computational methods for a wide variety of problems, ranging from curve and surface fitting and construction to spatial data structures, classification and clustering, and other methods for data analysis. The final three chapters discuss advanced theory and algorithms, starting with differential geometry of surfaces and progressing to digital topology and image analysis. A closing chapter addresses important proofs and future research directions.

This is an informative text covering a surprisingly wide range of topics. The author has succeeded in finding the appropriate (though highly variable) mix of mathematical theory, practical problems, computational approaches, and algorithms. The writing and production quality are generally good in spite of the occasional distracting minor error. The book is suitable for an upper-level undergraduate course and a follow-on graduate course. Researchers and practitioners will find it a reasonably adequate introduction (more details would have been useful in several places, especially for readers not enrolled in a college course). Given the considerable mathematical content in this book, it is more readable than might be expected, especially for readers familiar with principles and problems from related domains, especially computer graphics, image processing, and the theory of algorithms. Since the author explains basic concepts (though often rather briefly) before moving on to more advanced ideas, even readers new to much of the background material should be able to make fair headway.

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