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The fusion of algebra, analysis and geometry, and their application to real world problems, have been dominant themes underlying mathematics for over a century. Geometric algebras, introduced and classified by Clifford in the late 19th century, have played a prominent role in this effort, as seen in the mathematical work of Cartan, Brauer, Weyl, Chevelley, Atiyah, and Bott, and in applications to physics in the work of Pauli, Dirac and others. One of the most important applications of geometric algebras to geometry is to the representation of groups of Euclidean and Minkowski rotations. This aspect and its direct relation to robotics and vision will be discussed in several chapters of this multi-authored textbook, which resulted from the ASI meeting.
Moreover, group theory, beginning with the work of Burnside, Frobenius and Schur, has been influenced by even more general problems. As a result, general group actions have provided the setting for powerful methods within group theory and for the use of groups in applications to physics, chemistry, molecular biology, and signal processing. These aspects, too, will be covered in detail.
With the rapidly growing importance of, and ever expanding conceptual and computational demands on signal and image processing in remote sensing, computer vision, medical image processing, and biological signal processing, and on neural and quantum computing, geometric algebras, and computational group harmonic analysis, the topics of the book have emerged as key tools. The list of authors includes many of the world's leading experts in the development of new algebraic modeling and signal representation methodologies, novel Fourier-based and geometric transforms, and computational algorithms required for realizing the potential of these new application fields.
Dedication. Preface. Acknowledgments. Clifford Geometric Algebras in Multilinear Algebra and Non-Euclidean Geometries; Garret Sobczyk.- Geometric algebra Projective Geometries;Affine and other geometries; Affine Geometry of pseudo-euclidean space; Conformal Geometry and the Horosphere; References.
Content-Based Information Retrieval by Group Theoretical Methods Michael Clausen, Frank Kurth.- Introduction; Motivating Examples; General Concept;
Fault Tolerance.- Applications, Prototypes, and Test Results; Related Work and Future Research; References.- Four Problems in Radar; Michael C. Wicks and Braham Himed.-
Introduction; Radar Fundamentals; Radar Waveforms; Signal Processing; Space-Time Adaptive Processing; Four Problems in Radar; Conclusions.
Introduction to Generalized Classical and Quantum Signal and System Theories on Groups and Hypergroups; Valeriy Labunets.-Generalized classical signal/system theory on hypergroups; Generalized quantum signal/system theory on hypergroups; Conclusion; References. Lie Groups and Lie Algebras in Robotics; J.M. Selig.- Introduction—Rigid Body Motions; Lie Groups; Finite Screw Motions; Mechanical Joints; Invisible Motion and Gripping; Forward Kinematics; Lie Algebra; The Adjoint Representation; The Exponential Map Derivatives of Exponentials; Jacobians; Concluding Remarks; References. Quantum/Classical Interface: a Geometric Approach from the Classical Side; William E. Baylis.- Introduction Paravector Space as Spacetime; Eigenspinors; Spin; Dirac Equation; Bell’s Theorem; Qubits and Entanglement; Conclusions; References. PONS, Reed-Muller Codes, and Group Algebras; Myoung An et al.- Introduction; Analytic Theory of One-Dimensional PONS (Welti);Shapiro Sequences, Reed-Muller Codes, and Functional Equations;Group Algebras;
Reformulation of Classical PONS; Group Algebra of Classical PONS; Group Algebra Convolution; Splitting Sequences; Historical Appendix on PONS; References.
Clifford Algebras as a Unified Language; Valeriy Labunets.- Introduction; Clifford algebras as models of physical spaces; Clifford Algebras as Models of Perceptual Multicolor Spaces;
Hypercomplex-Valued invariants of nD multicolor images; Conclusions; Acknowledgments; References. Recent Progress and Applications in Group FFTs; Daniel N. Rockmore.-
Introduction; Finite group FFTs; FFTs for compact groups; Noncompact groups; References. Group Filters and Image Processing; Richard Tolimieri and Myoung An.- Introduction: Classical Digital Signal Processing; Abelian Group DSP; Nonabelian Groups; Examples; Group Transforms; Group Filters; Line-like Images; Acknowledgments; References. A Geometric Algebra Approach to Some Problems of Robot Vision; Gerald Sommer.- Introduction; Local Analysis of Multi-dimensional Signals; Knowledge Based Neural Computing; Acknowledgments; References. Group Theory in Radar and Signal Processing; William Moran, Jonathan H. Manton.- Introduction; How a Radar Works;Representations; Representations and Radar; Ambiguity Functions;The Wide Band Case; References. Geometry of Paravector Space with Applications to Relativistic Physics; William E. Baylis.- Clifford Algebras in Physics; Paravector Space as Spacetime; Interpretation; Eigenspinors; Maxwell’s Equation; Conclusions; References. A Unified Approach to Fourier-Clifford-Prometheus Transforms Ekaterima L.-Rundbla et al.- Introduction; New construction of classical and multiparametric Prometheus transforms; PONS associated with Abelian groups; Fast Fourier-Prometheus Transforms; Conclusions; Acknowledgments; References. Fast Color Wavelet Transforms; Ekaterina L.-Rundblad et al.- Introduction; Color images; Color Wavelet-Haar-Prometheus transforms;Edge detection and compression of color images; Conclusion; Acknowledgments; References. Selected Problems; Various Authors.- Transformations of Euclidean Space and Clifford Geometric; Algebra ;References; On the Distribution of Kloosterman Sums on Polynomials over Quaternions; References; Harmonic Sliding Analysis Problems; References;
Spectral Analysis under Conditions of Uncertainty; A Canonical Basis for Maximal Tori of the Reductive Centrizer of a Nilpotent Element; References;6 The Quantum Chaos Conjecture References; Four Problems in Radar; Topic Index; Author Index