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Geometric computing : for wavelet transforms, robot vision, learning, control and action
Bayro-Corrochano E., Springer Publishing Company, Incorporated, New York, NY, 2010. 615 pp. Type: Book (978-1-848829-28-2)
Date Reviewed: Oct 5 2010

The theory and applications of geometric algebra (Clifford algebra), an advanced mathematical language, are addressed in this book. It aims to clarify the theory and fundamental aspects of the application of geometric algebra to problems in image processing in artificial intelligence and related fields.

Bayro-Corrochano introduces the theory and new techniques of geometric algebra by showing their application in various domains that range from neural computing and robotics to medical image processing. In particular, the author presents and demonstrates the importance of geometric computing for building autonomous systems and pushing forward advances in cognitive systems research. Finally, he concludes that geometric algebra greatly helps to express the ideas and concepts, and to develop algorithms, in the broad domain of robot physics.

In particular, chapters 1 to 6 contain background material about geometric algebra, such as the representation of geometric objects; the kinematics of points, lines, and planes; using three-dimensional (3D) geometric algebra; and the computational advantages of geometric algebra for modeling and solving problems in robotics, computer vision, artificial intelligence, neural computing, and medical image processing.

Chapters 7 to 12 present fundamental theoretical issues for the applications discussed in the chapters that follow, such as the main issues of the implementation of computer programs for geometric algebra, issues about geometric neurocomputing, a study of the Clifford wavelet and Fourier transforms, the use of geometric algebra techniques to formulate the N-view geometry of computer vision, and issues about robot kinematics and dynamics using a language based on points, lines, planes and spheres.

Chapters 13 to 22 present various applications of geometric algebra. Such applications include the use of Lie operators for key point detection, the use of the quaternion Fourier transform for speech recognition, the use of the quaternion wavelet transform for optical flow estimation, the use of projective invariants for 3D shape and motion reconstruction and robot navigation, the use of tensor voting and geometric algebra for estimation of nonrigid motion, and the use of a geometric self-organizing neural net for the segmentation of two-dimensional (2D) contours and 3D shapes. Furthermore, experiments are presented using real data for robot-object recognition and interpolation, the calibration of sensors with respect to a robot frame, visual-guided grasping tasks using representations and geometric constraints, a 3D map reconstruction and relocalization using conformal geometric entities exploiting the Hough space, and the application of marching spheres for 3D medical shape representation and registration.

Finally, an appendix presents an outline of Clifford algebra, various concepts and definitions related to classic Clifford algebra and related algebras, and various useful formulas for geometric algebra.

The book is aimed at a graduate-level audience, and PowerPoint presentations are available for lecturers (a Web site link is provided). However, it can also be well suited for teaching a course or for self-study at the postgraduate level. Each chapter is accompanied by numerous examples and figures--quite a few of which are color illustrations--that will prove quite useful to the reader. The exercises are formulated to teach the fundamentals of geometric algebra and to stimulate the development of readers’ skills in creative and efficient geometric computing. The book could also be useful to scientists and engineers who are working in various areas related to the development and building of intelligent machines. I really enjoyed reading it.

Reviewer:  George K. Adam Review #: CR138445 (1104-0364)
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