Eduardo Bayro Corrochano - Böcker

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

All of 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. Recent progress in signal theory, computer vision and robotics shows that they are somehow separated from each other. The use of linear algebra over the real and complex numbers has not been adequate for the modeling of higher order geometric objects such as lines, planes and volumes. The aim of this book is to present a unified mathematical treatment of diverse problems in mathematics, physics, computer science and engineering using geometric algebra.

This book presents new results on applications of geometric algebra. The time when researchers and engineers were starting to realize the potential of quaternions for - plications in electrical, mechanic, and control engineering passed a long time ago. Since the publication of Space-Time Algebra by David Hestenes (1966) and Clifford Algebra to Geometric Calculus: A Uni?ed Language for Mathematics and Physics by David Hestenes and Garret Sobczyk (1984), consistent progress in the app- cations of geometric algebra has taken place. Particularly due to the great dev- opments in computer technology and the Internet, researchers have proposed new ideas and algorithms to tackle a variety of problems in the areas of computer science and engineering using the powerful language of geometric algebra. In this process, pioneer groups started the conference series entitled "Applications of Geometric Algebra in Computer Science and Engineering" (AGACSE) in order to promote the research activity in the domain of the application of geometric algebra. The ?rst conference, AGACSE'1999, organized by Eduardo Bayro-Corrochano and Garret Sobczyk, took place in Ixtapa-Zihuatanejo, Mexico, in July 1999.The contri- tions were published in Geometric Algebra with Applications in Science and En- neering, Birkhauser, 2001. The second conference, ACACSE'2001, was held in the Engineering Department of the Cambridge University on 9-13 July 2001 and was organizedbyLeoDorst,ChrisDoran,andJoanLasenby. Thebestconferencecont- butions appeared as a book entitled Applications of Geometric Algebra in Computer Science and Engineering, Birkhauser, 2002. The third conference, AGACSE'2008, took place in August 2008 in Grimma, Leipzig, Germany.

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

The goal of this book is to present a unified mathematical treatment of diverse problems in mathematics, physics, computer science, and engineer­ ing using geometric algebra. Geometric algebra was invented by William Kingdon Clifford in 1878 as a unification and generalization of the works of Grassmann and Hamilton, which came more than a quarter of a century before. Whereas the algebras of Clifford and Grassmann are well known in advanced mathematics and physics, they have never made an impact in elementary textbooks where the vector algebra of Gibbs-Heaviside still predominates. The approach to Clifford algebra adopted in most of the ar­ ticles here was pioneered in the 1960s by David Hestenes. Later, together with Garret Sobczyk, he developed it into a unified language for math­ ematics and physics. Sobczyk first learned about the power of geometric algebra in classes in electrodynamics and relativity taught by Hestenes at Arizona State University from 1966 to 1967. He still vividly remembers a feeling of disbelief that the fundamental geometric product of vectors could have been left out of his undergraduate mathematics education. Geometric algebra provides a rich, general mathematical framework for the develop­ ment of multilinear algebra, projective and affine geometry, calculus on a manifold, the representation of Lie groups and Lie algebras, the use of the horosphere and many other areas. This book is addressed to a broad audience of applied mathematicians, physicists, computer scientists, and engineers.

This book presents new results on applications of geometric algebra. The time when researchers and engineers were starting to realize the potential of quaternions for - plications in electrical, mechanic, and control engineering passed a long time ago. Since the publication of Space-Time Algebra by David Hestenes (1966) and Clifford Algebra to Geometric Calculus: A Uni?ed Language for Mathematics and Physics by David Hestenes and Garret Sobczyk (1984), consistent progress in the app- cations of geometric algebra has taken place. Particularly due to the great dev- opments in computer technology and the Internet, researchers have proposed new ideas and algorithms to tackle a variety of problems in the areas of computer science and engineering using the powerful language of geometric algebra. In this process, pioneer groups started the conference series entitled "Applications of Geometric Algebra in Computer Science and Engineering" (AGACSE) in order to promote the research activity in the domain of the application of geometric algebra. The ?rst conference, AGACSE'1999, organized by Eduardo Bayro-Corrochano and Garret Sobczyk, took place in Ixtapa-Zihuatanejo, Mexico, in July 1999.The contri- tions were published in Geometric Algebra with Applications in Science and En- neering, Birkhauser, 2001. The second conference, ACACSE'2001, was held in the Engineering Department of the Cambridge University on 9-13 July 2001 and was organizedbyLeoDorst,ChrisDoran,andJoanLasenby. Thebestconferencecont- butions appeared as a book entitled Applications of Geometric Algebra in Computer Science and Engineering, Birkhauser, 2002. The third conference, AGACSE'2008, took place in August 2008 in Grimma, Leipzig, Germany.

The goal of the Volume I Geometric Algebra for Computer Vision, Graphics  and Neural Computing is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra.Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry.By treating a wide spectrum of problems in a common language, this Volume I offers both new insights and new solutions that should be useful to scientists, and engineers working in different areas related with the development and building of intelligent machines. Each chapter is written in accessible terms accompanied by numerous examples, figures and a complementary appendix on Clifford algebras, all to clarify the theory and the crucial aspects of the application of geometric algebra to problems in graphics engineering, image processing, pattern recognition, computer vision, machine learning, neural computing and cognitive systems.

This book presents a unified mathematical treatment of diverse problems in thegeneral domain of robotics and associated fields using Clifford or geometric alge-bra. By addressing a wide spectrum of problems in a common language, it offersboth fresh insights and new solutions that are useful to scientists and engineersworking in areas related with robotics. It introduces non-specialists to Clifford and geometric algebra, and provides ex-amples to help readers learn how to compute using geometric entities and geomet-ric formulations. It also includes an in-depth study of applications of Lie grouptheory, Lie algebra, spinors and versors and the algebra of incidence using theuniversal geometric algebra generated by reciprocal null cones.Featuring a detailed study of kinematics, differential kinematics and dynamicsusing geometric algebra, the book also develops Euler Lagrangeand Hamiltoni-ans equations for dynamics using conformal geometric algebra, and the recursiveNewton-Euler using screw theory in the motor algebra framework. Further, itcomprehensively explores robot modeling and nonlinear controllers, and discussesseveral applications in computer vision, graphics, neurocomputing, quantum com-puting, robotics and control engineering using the geometric algebra framework.The book also includes over 200 exercises and tips for the development of futurecomputer software packages for extensive calculations in geometric algebra, and aentire section focusing on how to write the subroutines in C++, Matlab and Mapleto carry out efficient geometric computations in the geometric algebra framework.Lastly, it shows how program code can be optimized for real-time computations.An essential resource for applied physicists, computer scientists, AI researchers,roboticists and mechanical and electrical engineers, the book clarifies and demon-strates the importance of geometric computing for building autonomous systemsto advance cognitive systems research.

The goal of Geometric Algebra Applications Vol. III: Integral Transforms, Machine Learning, and Quantum Computing is to present a unified mathematical treatment of diverse problems in the general domain like Clifford Fourier Transforms, Deep Learning and Geometric Algebra Convolutional Neural Networks, Quaternion Quantum Fourier Transform and Geometric Quantum Computing. Topics and features ·         Introduces nonspecialists to Clifford, or geometric algebra and by example encourages the reader to learn to compute using geometric entities and geometric formulations.·         A study in depth for applications of Lie group theory, Lie algebra, projective geometry, and the algebra of incidence using the conformal geometric algebra.  ·         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 relationships of these models to conformal, affine, and projective geometries.·          Includes a thorough study of Integral transforms:  Quaternion and Clifford Transforms, quaternion analytic signal, monogenic signals, Hilbert transform, Riesz transform, Clifford Fourier Transform, Quaternion Wavelet transforms, Quaternion Quantum Fourier Transform, 3D Radon Transform and   Hough-Transform in geometric algebra. ·          Color image processing using the color model HSV, Quaternion Split rotors and motors, and the space-time Lorentz transform.·          Geometric neural computing using Split Quaternions, Geometric Algebra neural networks, Clifford Support Vector Machine and Neuro Control.·          Thorough discussion of several tasks of computer vision, graphics, neurocomputing, and robotics. machine learning, Deep Learning and CNNs, and Geometric Quantum Computing using the geometric algebra framework.·         130  exercises and hints for the development of future computer software packages for extensive calculations in geometric algebra. An entire section is dedicated to explaining how one should write the subroutines in C++, Phyton, Matlab, and Maple to carry out efficient geometric computations in the geometric algebra framework. Furthermore, it is shown how program code can be optimized for real-time computations.The book is an essential resource for applied mathematicians, physicists, computer scientists, graphics engineering, AI and Machine Learning researchers, roboticists and mechanical and electrical engineers, neurocomputing researchers, neuroscientists, and quantum computing specialists. It clarifies and demonstrates the importance of geometric computing for building autonomous systems and pushes forward advances in geometric cybernetics research.

The goal of Geometric Algebra Applications Vol. III: Integral Transforms, Machine Learning, and Quantum Computing is to present a unified mathematical treatment of diverse problems in the general domain like Clifford Fourier Transforms, Deep Learning and Geometric Algebra Convolutional Neural Networks, Quaternion Quantum Fourier Transform and Geometric Quantum Computing. Topics and features ·         Introduces nonspecialists to Clifford, or geometric algebra and by example encourages the reader to learn to compute using geometric entities and geometric formulations.·         A study in depth for applications of Lie group theory, Lie algebra, projective geometry, and the algebra of incidence using the conformal geometric algebra.  ·         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 relationships of these models to conformal, affine, and projective geometries.·          Includes a thorough study of Integral transforms:  Quaternion and Clifford Transforms, quaternion analytic signal, monogenic signals, Hilbert transform, Riesz transform, Clifford Fourier Transform, Quaternion Wavelet transforms, Quaternion Quantum Fourier Transform, 3D Radon Transform and   Hough-Transform in geometric algebra. ·          Color image processing using the color model HSV, Quaternion Split rotors and motors, and the space-time Lorentz transform.·          Geometric neural computing using Split Quaternions, Geometric Algebra neural networks, Clifford Support Vector Machine and Neuro Control.·          Thorough discussion of several tasks of computer vision, graphics, neurocomputing, and robotics. machine learning, Deep Learning and CNNs, and Geometric Quantum Computing using the geometric algebra framework.·         130  exercises and hints for the development of future computer software packages for extensive calculations in geometric algebra. An entire section is dedicated to explaining how one should write the subroutines in C++, Phyton, Matlab, and Maple to carry out efficient geometric computations in the geometric algebra framework. Furthermore, it is shown how program code can be optimized for real-time computations.The book is an essential resource for applied mathematicians, physicists, computer scientists, graphics engineering, AI and Machine Learning researchers, roboticists and mechanical and electrical engineers, neurocomputing researchers, neuroscientists, and quantum computing specialists. It clarifies and demonstrates the importance of geometric computing for building autonomous systems and pushes forward advances in geometric cybernetics research.

This book constitutes the refereed proceedings of the 19th Iberoamerican Congress on Pattern Recognition, CIARP 2014, held in Puerto Vallarta, Jalisco, Mexico, in November 2014. The papers are organized in topical sections on image coding, processing and analysis; pattern recognition and machine learning; neural networks for pattern recognition;

The goal of the Volume I Geometric Algebra for Computer Vision, Graphics  and Neural Computing is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra.Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry.By treating a wide spectrum of problems in a common language, this Volume I offers both new insights and new solutions that should be useful to scientists, and engineers working in different areas related with the development and building of intelligent machines. Each chapter is written in accessible terms accompanied by numerous examples, figures and a complementary appendix on Clifford algebras, all to clarify the theory and the crucial aspects of the application of geometric algebra to problems in graphics engineering, image processing, pattern recognition, computer vision, machine learning, neural computing and cognitive systems.

This handbook addresses a broad audience of applied mathematicians, physicists, computer scientists, and engineers, bringing together under a single cover the most recent advances in the applications of geometric computing in the most important fields related to building perception action systems: computer vision, robotics, image processing and understanding, pattern recognition, computer graphics, quantum computers, brain theory and neural networks.Various kinds of problems in these fields have been tackled using promising geometric methods, but such efforts have been mostly confined to specific disciplines. In this book we introduce diverse, powerful geometric methods in a unified manner, covering geometry theory and geometric computing methods related to the design of perception and action systems, intelligent autonomous systems and intelligent machines.The book is suitable for postgraduate students and researchers working on the design of intelligent systems.

Many computer scientists, engineers, applied mathematicians, and physicists use geometry theory and geometric computing methods in the design of perception-action systems, intelligent autonomous systems, and man-machine interfaces. This handbook brings together the most recent advances in the application of geometric computing for building such systems, with contributions from leading experts in the important fields of neuroscience, neural networks, image processing, pattern recognition, computer vision, uncertainty in geometric computations, conformal computational geometry, computer graphics and visualization, medical imagery, geometry and robotics, and reaching and motion planning. For the first time, the various methods are presented in a comprehensive, unified manner.This handbook is highly recommended for postgraduate students and researchers working on applications such as automated learning; geometric and fuzzy reasoning; human-like artificial vision; tele-operation; space maneuvering; haptics; rescue robots; man-machine interfaces; tele-immersion; computer- and robotics-aided neurosurgery or orthopedics; the assembly and design of humanoids; and systems for metalevel reasoning.