Geometric Computing for Perception Action Systems

1. Front Matter

   Pages i-xvi

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2. Fundamental Concepts

   1. Front Matter

      Pages 1-1

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   2. Mathematical Preliminaries

      • Eduardo Bayro Corrochano

      Pages 3-17

   3. Kinematics of the 2D and 3D Spaces

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      Pages 19-37

   4. Lie Algebras and Algebra of Incidence Using the Null Cone and Affine Plane

      • Eduardo Bayro Corrochano

      Pages 39-66

   5. Geometric Algebra of Computer Vision

      • Eduardo Bayro Corrochano

      Pages 67-92

3. Practical Applications

   1. Front Matter

      Pages 93-93

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   2. Computing the Kinematics of Robot Manipulators

      • Eduardo Bayro Corrochano

      Pages 95-114

   3. Image Processing

      • Eduardo Bayro Corrochano

      Pages 115-136

   4. Applications in Computer Vision

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      Pages 137-168

   5. Rigid Motion Estimation Using Line Observations

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      Pages 169-200

   6. Geometric Neuralcomputing

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      Pages 201-224

4. Back Matter

   Pages 225-235

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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

• Algebra
• Processing
• Variable
• algorithms
• artificial intelligence
• complexity
• image processing
• robot

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)

• Computer Science Department, Centro de Investigaciòn y Estudios Avanzados del I.P.N., Unidad de Guadalajara, Guadalajara, Jalisco, Mexico

  Eduardo Bayro Corrochano

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