The concept of shape plays a major role in such broad areas as artificial intelligence, computer vision, image processing and analysis, and pattern recognition. Some focus is needed because of the breadth of the subject, and the author provides it by choosing several criteria to distinguish his method of digital shape analysis. The distinctions are

• information-preserving versus non-information- preserving,

• contour-oriented versus region-oriented,

• scalar transform versus space domain,

• global shape analysis versus local shape analysis, and

• deterministic versus statistical.

Chapter 1, “Introduction,” provides the context and lays the foundations for the book. Only the shape of two-dimensional objects will be considered, and only in the form of boundary or external contour. Transformations under which shape is invariant are defined: translation, scaling, and rotation. Chapter 2, “Parametric Contour Representation, Similarity and Symmetry,” thoroughly discusses seven information-preserving curve representations--position function, tangent function, acceleration function, tangent angle function, cumulative angular function, periodic cumulative angular function, and curvature function. Rigorous mathematical definitions are given, and theorems related to representations and operators are proven. Geometric similarity and mirror-similarity are defined. The part on symmetry is the most interesting and illuminating. Both mirror-symmetry and rotational symmetry are expressed as conditions for the seven contour representations.

Chapter 3, “Fourier Series Expansions of Parametric Contour Representations and Their Relation to Similarity and Symmetry,” has almost the same outline as chapter 2, only it deals with information-preserving Fourier expansions instead of with direct curve representations. The mathematical aspects are also more difficult. One important result is that, except for the simplest shape, the circle, the Fourier series is always infinite; thus, in practice, issues related to truncation errors and aliasing must be considered.

The most important part of the book is chapter 4, “Measurement of Similarity, Mirror-similarity and Symmetry.” Here the terms “dissimilarity” and “dissymmetry” become important as the author tries to quantitatively express and measure things that are usually qualitative. Measures are described both for contour representations and for their Fourier expansions. Normalization is compared to optimization for dissimilarity measures. A couple of relevant experiments are described. The experiment for dissimilarity involves 18 shapes and 25 dissimilarity measures. Advanced cluster analysis is done on the resulting matrix. The other experiment, for dissymmetry, uses 12 different shapes.

Chapter 5, “Discussion,” considers “the merits and the limitations of the approach and indicates some routes to possible extensions to overcome these limitations.” The three appendices, “Some Mathematical Concepts and Properties,” “A Method for a Fast and Reliable Computation of Moments of Regions Bounded by Polygons,” and “Estimation of Contour Representations Using Polynomial Filters,” make the book almost completely self-contained by providing more mathematical background.

The book is research oriented but, like many valuable research works, is also a survey of the state of the art in the domain. Each of the five chapters and three appendices has its own “References” section, providing a total of 29 pages of references. The fact that almost all of these references are commented on briefly but in a relevant way in the text is enough to make the book a valuable research tool. Beyond that, the author’s view of the field and his own research and mathematical expertise contribute to making the book highly recommended for anyone pursuing serious research in the area of image analysis. Advanced graduate or postgraduate courses could be taught from this book, although no exercises are provided. Overall, the book is a gem of mathematical reasoning.