Synopses & Reviews
This book presents a unified mathematical treatment of diverse problems in the fields of cognitive systems using Clifford, or geometric, algebra. Geometric algebra provides a rich general mathematical framework for the development of the ideas of multilinear algebra, projective and affine geometry, calculus on manifolds, the representation of Lie groups and Lie algebras, and many other areas of applications. By treating a wide spectrum of problems in a common geometric language, the book offers both new insights and new solutions that should be useful to scientists and engineers working in different but related areas of artificial intelligence. It looks at building intelligence systems through the construction of Perception Action Cycles; critical to this concept is incorporating representation and learning in a flexible geometric system. Each chapter is written in accessible terms accompanied by numerous examples and figures that clarify the application of geometric algebra to problems in geometric computing, image processing, computer vision, robotics, neural computing and engineering. Topics and features: *Introduces a nonspecialist to Clifford, or geometric, algebra and it shows applications in artificial intelligence *Thorough discussion of several tasks of signal and image processing, computer vision, robotics, neurocomputing and engineering using the geometric algebra framework *Features the computing frameworks of the linear model n-dimensional affine plane and the nonlinear model of Euclidean space known as the horosphere, and addresses the relationship of these models to conformal, affine and projective geometries *Applications of geometric algebra to other related areas: aeronautics, mechatronics, graphics engineering, and speech processing *Exercises and hints for the development of future computer software packages for extensive calculations in geometric algebra The book is an essential resource for computer scientists, AI researchers, and electrical engineers and includes computer programs to clarify and demonstrate the importance of geometric computing for cognitive systems and artificial autonomous systems research.
Review
From the reviews: MATHEMATICAL REVIEWS "We are sure that the mathematicians, computer scientists, engineers and physicists will enjoy reading this book." "For the case of perception action cycles the author of this nice book shows that the Clifford algebra ... of multivectors of an n-dimensional vector space is indeed superior to previous mathematical structures used to deal with this subject. ... We are sure that mathematicians, computer scientists, engineers and physicists will enjoy reading this book." (Waldyr Alves Rodrigues, Jr., Mathematical Reviews, Issue 2003 d)
Synopsis
All the efforts to build an intelligent machine have not yet produced a satisfactory autonomous system despite the great progress that has been made in developing computer hardware over the last three decades. The complexity of the tasks that a cognitive system must perform is still not understood well enough. Let us call the endeavor of building intelligent systems as the construction of Perception Action Cycles (PAC). The key idea is to incorporate representation and learning in a flexible geometric system. Until now this issue has always been a matter of neurocomputing. The most frequently used algebraic system for neurocomputation is matrix algebra. However, calculations in geometric algebra often reveal a geometric structure which remains obscure in the equivalent matrix computations. The development of PAC in a unified comprehensive mathematical system is urgently needed to bring unity and coherance to the problems of artificial intelligence. Accordingly, we are motivated by the challenge of applying geometric algebra to the development of PAC systems. Geometric algebra provides the general mathematical framework for the development of the ideas of multi-linear algebra, multi-variable analysis, and the representation of LIE groups and LIE algebras. There is strong evidence that geobetric albegra can be used to carry out efficient computations at all levels in the cognitive system. Geometric algebra reduces the complexity of algebraic expressions and as a result, it improves algorithms both in speed and accuracy. Thus, our goal is to construct PAC systems solely in the geometric algebra language. The preliminary chapters of this book introduce the reader to geometric algebra and the necessary mathematical concepts that will be needed. The latter chapters deal with a variety of applications in the field of cognitive systems in
Synopsis
After an introduction to geometric algebra, and the necessary math concepts that are needed, the book examines a variety of applications in the field of cognitive systems using geometric algebra as the mathematical system. There is strong evidence that geobetric albegra can be used to carry out efficient computations at all levels in the cognitive system. Geometric algebra reduces the complexity of algebraic expressions and as a result, it improves algorithms both in speed and accuracy. The book is addressed to a broad audience of computer scientists, cyberneticists, and engineers. It contains computer programs to clarify and demonstrate the importance of geometric algebra in cognitive systems.
Description
Includes bibliographical references (p. [225]-230) and index.
Table of Contents
Mathematical Preliminaries * Lie Algebras and Geometric Algebra for Robotics and Image Analysis * Kinematics of 2-Space and 3-Space * Mathematics of the Human Eye * Image Analysis and Low Level Operations * Theory of Extended Kalman Filter * Geometric Algebra of Computer Vision * Analysis and Computation of Projective Invariants * Geometric Computing of Intrinsic Camera Parameters * Geometric Approach for Computing Shape and Motion * Geometric Neural Computing * Geometric Computing in Robotics