Table Of ContentComputational Noncommutative Algebra and
Applications
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Series II:Mathematics,Physics and Chemistry – Vol.136
Computational Noncommutative
Algebra and Applications
edited by
Jim Byrnes
Prometheus Inc.,
Newport, Rhode Island, U.S.A.
KLUWER ACADEMIC PUBLISHERS
NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 1-4020-2307-3
Print ISBN: 1-4020-1982-3
©2005 Springer Science + Business Media, Inc.
Print ©2004 Kluwer Academic Publishers
Dordrecht
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Dedication
This book is dedicated to the cherished memory of Edward Joseph Sanchas, a
most decent and trusted human being whom we always counted on to take
charge of any and all problems. He was a great man. We miss him deeply.
Contents
Preface xi
Acknowledgments xiv
CliffordGeometricAlgebrasinMultilinearAlgebraandNon-EuclideanGe-
ometries 1
Garret Sobczyk
1 Geometric algebra 2
2 Projective Geometries 11
3 Affine and other geometries 17
4 Affine Geometry of pseudo-euclidean space 19
5 Conformal Geometry and the Horosphere 21
References 27
Content-Based Information Retrieval by Group Theoretical Methods 29
Michael Clausen, Frank Kurth
1 Introduction 29
2 Motivating Examples 32
3 General Concept 39
4 Fault Tolerance 45
5 Applications, Prototypes, and Test Results 48
6 Related Work and Future Research 51
References 54
Four Problems in Radar 57
Michael C. Wicks and Braham Himed
1 Introduction 57
2 Radar Fundamentals 58
3 Radar Waveforms 60
4 Signal Processing 63
5 Space-Time Adaptive Processing 67
6 Four Problems in Radar 71
7 Conclusions 72
Introduction to Generalized Classical and Quantum Signal and System
Theories on Groups and Hypergroups 75
Valeriy Labunets
1 Generalized classical signal/system theory on hypergroups 77
2 Generalized quantum signal/system theory on hypergroups 91
3 Conclusion 99
References 99
vi COMPUTATIONAL NONCOMMUTATIVE ALGEBRA
Lie Groups and Lie Algebras in Robotics 101
J.M. Selig
1 Introduction—Rigid Body Motions 102
2 Lie Groups 104
3 Finite Screw Motions 106
4 Mechanical Joints 108
5 Invisible Motion and Gripping 110
6 Forward Kinematics 111
7 Lie Algebra 112
8 The Adjoint Representation 113
9 The Exponential Map 115
10 Derivatives of Exponentials 119
11 Jacobians 122
12 Concluding Remarks 124
References 125
Quantum/Classical Interface: a Geometric Approach from the Classical
Side 127
William E. Baylis
1 Introduction 127
2 ParavectorSpace as Spacetime 128
3 Eigenspinors 129
4 Spin 132
5 Dirac Equation 137
6 Bell’s Theorem 145
7 Qubits and Entanglement 146
8 Conclusions 153
References 154
PONS, Reed-Muller Codes, and Group Algebras 155
Myoung An, Jim Byrnes, William Moran, B. Saffari, Harold S. Shapiro, and
Richard Tolimieri
1 Introduction 156
2 Analytic theory of one-dimensional PONS 156
3 Shapiro sequences, Reed-Muller codes, and functional equations165
4 Group Algebras 174
5 Reformulation of classical PONS 178
6 Group Algebra of Classical PONS 179
7 Group Algebra Convolution 182
8 Splitting Sequences 188
References 195
Contents vii
Clifford Algebras as a Unified Language for Image Processing and Pattern
Recognition 197
Valeriy Labunets
1 Introduction 197
2 Clifford algebras as models of physical spaces 200
3 Clifford Algebras as Models of Perceptual Multicolor Spaces 205
4 Hypercomplex-valued invariants of nD multicolor images 217
5 Conclusions 223
Acknowledgments 224
References 224
Recent Progress and Applications in Group FFTs 227
Daniel N. Rockmore
1 Introduction 227
2 Finite group FFTs 233
3 FFTs for compact groups 244
4 Noncompact groups 249
References 251
Group Filters and Image Processing 255
Richard Tolimieri and Myoung An
1 Introduction: Classical Digital Signal Processing 256
2 Abelian Group DSP 258
3 Nonabelian Groups 265
4 Examples 285
5 Group Transforms 291
6 Group Filters 295
7 Line-like Images 304
References 308
A Geometric Algebra Approach to Some Problems of Robot Vision 309
Gerald Sommer
1 Introduction 309
2 Local Analysis of Multi-dimensional Signals 312
3 Knowledge Based Neural Computing 324
Acknowledgments 335
References 335
viii COMPUTATIONAL NONCOMMUTATIVE ALGEBRA
Group Theory in Radar and Signal Processing 339
William Moran, Jonathan H. Manton
1 Introduction 339
2 How a Radar Works 341
3 Representations 343
4 Representations and Radar 347
5 Ambiguity Functions 357
6 The Wide Band Case 360
References 362
Geometry of ParavectorSpace with Applications to Relativistic Physics 363
William E. Baylis
1 Clifford Algebras in Physics 364
2 ParavectorSpace as Spacetime 368
3 Interpretation 375
4 Eigenspinors 381
5 Maxwell’s Equation 382
6 Conclusions 386
References 387
A Unified Approach to Fourier-Clifford-PrometheusSequences,Transforms
and Filter Banks 389
Ekaterina L.-Rundblad, Valeriy Labunets, and Ilya Nikitin
1 Introduction 389
2 NewconstructionofclassicalandmultiparametricPrometheus
transforms 391
3 PONS associated with Abelian groups 393
4 Fast Fourier-Prometheus Transforms 397
5 Conclusions 399
Acknowledgments 399
References 400
FastColorWavelet-Haar-Hartley-PrometheusTransformsforImageProcessing 401
Ekaterina L.-Rundblad, Alexei Maidan, Peter Novak, Valeriy Labunets
1 Introduction
2 Color images 402
3 Color Wavelet-Haar-Prometheustransforms 404
4 Edge detection and compression of color images 406
5 Conclusion 408
Acknowledgments 410
References 410
