Computational Noncommutative Algebra and Applications: Proceedings of the NATO Advanced Study Institute, on Computatoinal Noncommutative Algebra and Applications, ... II: Mathematics, Physics and Chemistry)

435 Pages·2004·5.78 MB·English

by  Jim Byrnes

Computational Noncommutative Algebra and
Applications
NATO Science Series
A Series presenting the results of scientific meetings supported under the NATO Science
Programme.
The Series is published by IOS Press, Amsterdam, and Kluwer Academic Publishers in conjunction
with the NATO Scientific Affairs Division
Sub-Series
I. Life and Behavioural Sciences IOS Press
II. Mathematics,Physics and Chemistry Kluwer Academic Publishers
III.Computer and Systems Science IOS Press
IV.Earth and Environmental Sciences Kluwer Academic Publishers
V. Science and Technology Policy IOS Press
The NATO Science Series continues the series of books published formerly as the NATO ASI Series.
The NATO Science Programme offers support for collaboration in civil science between scientists of
countries of the Euro-Atlantic Partnership Council.The types of scientific meeting generally supported
are “Advanced Study Institutes” and “Advanced Research Workshops”, although other types of
meeting are supported from time to time.The NATO Science Series collects together the results of
these meetings.The meetings are co-organized bij scientists from NATO countries and scientists from
NATO’s Partner countries – countries of the CIS and Central and Eastern Europe.
Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances
in a field.
Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and
identification of directions for future action.
As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science
Series has been re-organised and there are currently Five Sub-series as noted above.Please consult
the following web sites for information on previous volumes published in the Series, as well as details of
earlier Sub-series.
http://www.nato.int/science
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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