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Reviewer: Vladik Ya. Kreinovich

Many objects have rigid shapes?they can be moved and rotated, but not reshaped. Such are nuts, bolts, and other to-be-assembled pieces that are usually manipulated by industrial robots; such are obstacles that mobile robots are trying to avoid; and such are the components of the robots themselves?for example, a robot manipulator is usually assembled from rigid shape joints. Many algorithms of computational geometry are designed for processing rigid shapes. However, in practice, many shapes are nonrigid: in computer vision, we need to present a human body; in medical applications, we need to deal with the nonrigid shapes of organs and vessels. To characterize the shape of a rigid body, it is sufficient to describe six parameters: three describe the location of the center of its mass, and three describe its orientation. To describe the shape of a nonrigid body, we potentially need infinitely more parameters. As a result, the computational geometry of nonrigid bodies is much more difficult. This book provides an introduction to this geometry. The book starts with the basic mathematical notions and tools that mathematicians use to describe nonrigid shapes: surfaces, intrinsic metric (length of the shortest path along the surface), curvature, manifolds, and metric spaces in general (chapter 2). To process the shapes, we need to represent them in a computer. As mentioned before, a perfect description of a shape requires an infinite number of parameters; therefore, in order to represent shapes in a computer, we must approximate them by using shapes from a finite-parametric family. A natural way to do this is to discretize the shape (that is, to represent each surface by a finite number of points). Chapter 3 describes the optimal way of selecting these points, and chapter 4 describes how to compute the intrinsic metric, based on this selection. Selecting the optimal points and finding the shortest path are examples of optimization. There are many other applications of computational geometry where we want to find the best (optimal) solution. To help with these applications, the authors review numerical optimization techniques in chapter 5. Chapter 6 describes the use of these optimization techniques to estimate the degree of similarity between two rigid surfaces, as the smallest possible distance attainable by appropriate rotation and shift. When one of the surfaces is flexible, more transformations are possible and the computation of the distance gets more complicated. From a geometric viewpoint, the two surfaces are similar if they have the same intrinsic metric. Therefore, one way to compare them is to embed each intrinsic metric d ( a , b ) into a higher-dimensional space, in which d ( a , b ) is equal to (or close to) the Euclidean distance between the corresponding points f ( a ) and f ( b ); the two surfaces are similar if and only if their embeddings are similar to rigid bodies. The corresponding embedding algorithms are described in chapter 7. An alternative approach, based on comparing the corresponding spectra, is described in chapter 8. Embedding into a higher-dimensional Euclidean space often requires high dimensions and, thus, time-consuming computations. For many sphere-like surfaces, it is computationally more efficient to embed into a higher-dimensional sphere; such methods are described in chapter 9. A more general idea, explored in chapter 10, is to embed both surfaces into a single space that is, in some reasonable sense, most appropriate for comparing the given surfaces. In many practical situations, it is important to compare only parts of the objects' surfaces. For example, in two differently clothed images of the same person, only the face and hands may be similar; related algorithms are covered in chapter 11. Until now, the emphasis was on finding a mapping between the shapes. In some applications, it is desirable to supplement this mapping with a continuous transition between the shapes¿for example, to fill in intermediate images in a video sequence or in a cartoon, or to morph two images (chapter 12). Chapter 13 presents another successful application of nonrigid geometry: an expression-invariant and reasonably reliable face recognition system, based on a three-dimensional (3D) facial image. The success of this system relies on good representation of texture¿the fine-level structure of the corresponding surfaces. Overall, the book is not always easy to read, but it does explain relevant mathematical notions, such as Gromov's metric geometry ideas, in a very understandable and entertaining way, with numerous images and exercises. It describes a vast variety of computational geometry applications. I highly recommend it to both computer scientists interested in learning more about the latest advances in computational geometry and to geometers looking for applications. This unique book can serve as an excellent textbook for many related courses, for self-study, or as a reference. Online Computing Reviews Service