Geometric computing with Clifford algebras: theoretical foundations and applications in computer vision and robotics ; with 16 tables
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2001
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 551 S. graph. Darst. |
| ISBN: | 3540411984 |
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| 245 | 1 | 0 | |a Geometric computing with Clifford algebras |b theoretical foundations and applications in computer vision and robotics ; with 16 tables |c Gerald Sommer (ed.) |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2001 | |
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GERALD SOMMER (ED.) GEOMETRIC COMPUTING WITH CLIFFORD ALGEBRAS
THEORETICAL FOUNDATIONS AND APPLICATIONS IN COMPUTER VISION AND ROBOTICS
WITH 89 FIGURES AND 16 TABLES SPRINGER GERALD SOMMER (ED.) GEOMETRIC
COMPUTING WITH CLIFFORD ALGEBRAS THEORETICAL FOUNDATIONS AND
APPLICATIONS IN COMPUTER VISION AND ROBOTICS WITH 89 FIGURES AND 16
TABLES SPRINGER TABLE OF CONTENTS PART I. A UNIFIED ALGEBRAIC APPROACH
FOR CLASSICAL GEOMETRIES 1. NEW ALGEBRAIC TOOLS FOR CLASSICAL GEOMETRY
DAVID HESTENES. HONGBO LI, AND ALYN ROCKWOOD 3 1.1 . INTRODUCTION 3 1.2
GEOMETRIC ALGEBRA OF A VECTOR SPACE 4 1.3 LINEAR TRANSFORMATIONS 13 1.4
VECTORS AS GEOMETRICAL POINTS 19 1.5 LINEARIZING THE EUCLIDEAN GROUP 23
2. GENERALIZED HOMOGENEOUS COORDINATES FOR COMPUTATIONAL GEOMETRY HONGBO
LI. DAVID HESTENES, AND ALYN ROCKWOOD 27 2.1 INTRODUCTION 27 2.2
MINKOWSKI SPACE WITH CONFORMAL AND PROJECTIVE SPLITS 29 2.3 HOMOGENEOUS
MODEL OF EUCLIDEAN SPACE 33 2.4 EUCLIDEAN SPHERES AND HYPERSPHERES 40
2.5 MULTI-DIMENSIONAL SPHERES. PLANES, AND SIMPLEXES 41 2.6 RELATION
AMONG SPHERES AND HYPERPLANES 46 2.7 CONFORMAL TRANSFORMATIONS 52 3.
SPHERICAL CONFORMAL GEOMETRY WITH GEOMETRIC ALGEBRA HONGBO LI, DAVID
HESTENES, AND ALYN ROCKWOOD 61 3.1 INTRODUCTION 61 3.2 HOMOGENEOUS MODEL
OF SPHERICAL SPACE 62 3.3 RELATION BETWEEN TWO SPHERES OR HYPERPLANES 66
3.4 SPHERES AND PLANES OF DIMENSION R 68 3.5 STEREOGRAPHIC PROJECTION 70
3.6 CONFORMAL TRANSFORMATIONS 72 XII TABLE OF CONTENTS 4. A UNIVERSAL
MODEL FOR CONFORMAL GEOMETRIES OF EUCLIDEAN, SPHERICAL AND
DOUBLE-HYPERBOLIC SPACES HONGBO LI, DAVID HESTENES, AND ALYN ROCKWOOD 77
4.1 INTRODUCTION 77 4.2 THE HYPERBOLOID MODEL 79 4.3 THE HOMOGENEOUS
MODEL 83 4.4 STEREOGRAPHIC PROJECTION 90 4.5 THE CONFORMAL BALL MODEL 92
4.6 THE HEMISPHERE MODEL 93 4.7 THE HALF-SPACE MODEL 94 4.8 THE KLEIN
BALL MODEL 97 4.9 A UNIVERSAL MODEL FOR EUCLIDEAN, SPHERICAL, AND
HYPERBOLIC SPACES 99 5. GEO-MAP UNIFICATION AMBJORN NAEVE AND LARS
SVENSSON 105 5.1 INTRODUCTION 105 5.2 HISTORICAL BACKGROUND 106 5.3
GEOMETRIC BACKGROUND 108 5.4 THE UNIFIED GEO-MAP COMPUTATIONAL FRAMEWORK
110 5.5 APPLYING THE GEO-MAP TECHNIQUE TO GEOMETRICAL OPTICS. . . . 114
5.6 SUMMARY AND CONCLUSIONS 122 5.7 ACKNOWLEDGEMENTS 123 5.8 APPENDIX:
CONSTRUCTION OF A GEOMETRIC ALGEBRA 123 6. HONING GEOMETRIC ALGEBRA FOR
ITS USE IN THE COMPUTER SCIENCES LEO DORST 127 6.1 INTRODUCTION 127 6.2
THE INTERNAL STRUCTURE OF GEOMETRIC ALGEBRA 128 6.3 THE CONTRACTION: AN
ALTERNATIVE INNER PRODUCT 134 6.4 THE DESIGN OF THEOREMS AND 'FILTERS'
136 6.5 SPLITTING ALGEBRAS EXPLICITLY 143 6.6 THE RICH SEMANTICS OF THE
MEET OPERATION 145 6.7 THE USE AND INTERPRETATION OF GEOMETRIC ALGEBRA
150 6.8 GEOMETRICAL MODELS OF MULTIVECTORS 151 6.9 CONCLUSIONS 151 TABLE
OF CONTENTS XIII PART II. ALGEBRAIC EMBEDDING OF SIGNAL THEORY AND
NEURAL COMPUTATION 7. SPATIAL*COLOR CLIFFORD ALGEBRAS FOR INVARIANT
IMAGE RECOGNITION EKATERINA RUNDBLAD-LABUNETS AND VALERI LABUNETS 155
7.1 INTRODUCTION 155 7.2 GROUPS OF TRANSFORMATIONS AND INVARIANTS 157
7.3 PATTERN RECOGNITION 157 7.4 CLIFFORD ALGEBRAS AS UNIFIED LANGUAGE
FOR PATTERN RECOGNITION 160 7.5 HYPERCOMPLEX-VALUED MOMENTS AND
INVARIANTS 169 7.6 CONCLUSION 185 8. NON-COMMUTATIVE HYPERCOMPLEX
FOURIER TRANSFORMS OF MULTIDIMENSIONAL SIGNALS THOMAS BIILOW, MICHAEL
FELSBERG, AND GERALD SOMMER 187 8.1 INTRODUCTION 187 8.2 1-D HARMONIC
TRANSFORMS 188 8.3 2-D HARMONIC TRANSFORMS 191 8.4 SOME PROPERTIES OF
THE QFT 194 8.5 THE CLIFFORD FOURIER TRANSFORM 205 8.6 HISTORICAL
REMARKS 206 8.7 CONCLUSION 207 9. COMMUTATIVE HYPERCOMPLEX FOURIER
TRANSFORMS OF MULTIDIMENSIONAL SIGNALS MICHAEL FELSBERG, THOMAS BIILOW,
AND GERALD SOMMER 209 9.1 INTRODUCTION '. 209 9.2 HYPERCOMPLEX ALGEBRAS
210 9.3 THE TWO-DIMENSIONAL HYPERCOMPLEX FOURIER ANALYSIS 213 9.4 THE
N-DIMENSIONAL HYPERCOMPLEX FOURIER ANALYSIS 221 9.5 CONCLUSION 229 10.
FAST ALGORITHMS OF HYPERCOMPLEX FOURIER TRANSFORMS MICHAEL FELSBERG,
THOMAS BIILOW, GERALD SOMMER, AND VLADIMIR M. CHERNOV 231 10.1
INTRODUCTION 231 10.2 DISCRETE QUATERNIONIC FOURIER TRANSFORM AND FAST
QUATERNIONIC FOURIER TRANSFORM 232 10.3 DISCRETE AND FAST N-DIMENSIONAL
TRANSFORMS 242 XIV TABLE OF CONTENTS 10.4 FAST ALGORITHMS BY FFT 247
10.5 CONCLUSION AND SUMMARY 254 11. LOCAL HYPERCOMPLEX SIGNAL
REPRESENTATIONS AND APPLICATIONS THOMAS BIILOW AND GERALD SOMMER 255
11.1 INTRODUCTION 255 11.2 THE ANALYTIC SIGNAL 256 11.3 LOCAL PHASE IN
IMAGE PROCESSING 270 11.4 TEXTURE SEGMENTATION USING THE QUATERNIONIC
PHASE 279 11.5 CONCLUSION 289 12. INTRODUCTION TO NEURAL COMPUTATION IN
CLIFFORD ALGEBRA SVEN BUCHHOLZ AND GERALD SOMMER 291 12.1 INTRODUCTION
291 12.2 AN OUTLINE OF CLIFFORD ALGEBRA 292 12.3 THE CLIFFORD NEURON 295
12.4 CLIFFORD NEURONS AS LINEAR OPERATORS 299 12.5 MOBIUS
TRANSFORMATIONS 309 12.6 SUMMARY 314 13. CLIFFORD ALGEBRA MULTILAYER
PERCEPTRONS SVEN BUCHHOLZ AND GERALD SOMMER 315 13.1 INTRODUCTION AND
PRELIMINARIES 315 13.2 UNIVERSAL APPROXIMATION BY CLIFFORD MLPS 317 13.3
ACTIVATION FUNCTIONS 320 13.4 CLIFFORD BACK-PROPAGATION ALGORITHM 324
13.5 EXPERIMENTAL RESULTS 327 13.6 CONCLUSIONS AND OUTLOOK 334 PART III.
GEOMETRIC ALGEBRA FOR COMPUTER VISION AND ROBOTICS 14. A UNIFIED
DESCRIPTION OF MULTIPLE VIEW GEOMETRY CHRISTIAN B.U. PERWASS AND JOAN
LASENBY 337 14.1 INTRODUCTION 337 14.2 PROJECTIVE GEOMETRY 338 14.3 THE
FUNDAMENTAL MATRIX 341 14.4 THE TRIFOCAL TENSOR 346 14.5 THE QUADFOCAL
TENSOR 358 14.6 RECONSTRUCTION AND THE TRIFOCAL TENSOR 364 14.7
CONCLUSION 369 XVI TABLE OF CONTENTS 19. KINEMATICS OF ROBOT
MANIPULATORS IN THE MOTOR ALGEBRA EDUARDO BAYRO-CORROCHANO AND DETLEF
KAHLER 471 19.1 INTRODUCTION 471 19.2 MOTOR ALGEBRA FOR THE KINEMATICS
OF ROBOT MANIPULATORS . . . 472 19.3 DIRECT KINEMATICS OF ROBOT
MANIPULATORS 478 19.4 INVERSE KINEMATICS OF ROBOT MANIPULATORS 481 19.5
CONCLUSION 488 20. USING THE ALGEBRA OF DUAL QUATERNIONS FOR MOTION
ALIGNMENT KOSTAS DANIILIDIS 489 20.1 INTRODUCTION 489 20.2 EVEN
SUBALGEBRAS OF NON-DEGENERATE W- Q R 490 20.3 EVEN SUBALGEBRAS OF
DEGENERATE W Y ' Q - R 491 20.4 LINE TRANSFORMATION 493 20.5 MOTION
ESTIMATION FROM 3D-LINE MATCHES 494 20.6 THE PRINCIPLE OF TRANSFERENCE
496 20.7 RELATING COORDINATE SYSTEMS TO EACH OTHER 498 20.8 CONCLUSION
499 21. THE MOTOR EXTENDED KALMAN FILTER FOR DYNAMIC RIGID MOTION
ESTIMATION FROM LINE OBSERVATIONS YIWEN ZHANG, GERALD SOMMER, AND
EDUARDO BAYRO-CORROCHANO . . . 501 21.1 INTRODUCTION 501 21.2 KALMAN
FILTER TECHNIQUES 504 21.3 3-D LINE MOTION MODEL 507 21.4 THE MOTOR
EXTENDED KALMAN FILTER 515 21.5 EXPERIMENTAL ANALYSIS OF THE MEKF 521
21.6 CONCLUSION 528 REFERENCES 531 AUTHOR INDEX 543 SUBJECT INDEX 544 |
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| isbn | 3540411984 |
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| spelling | Geometric computing with Clifford algebras theoretical foundations and applications in computer vision and robotics ; with 16 tables Gerald Sommer (ed.) Berlin [u.a.] Springer 2001 XVIII, 551 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Clifford-Algebra Clifford-Algebra - Anwendung Clifford-Algebra (DE-588)4199958-7 gnd rswk-swf Anwendung (DE-588)4196864-5 gnd rswk-swf Clifford-Algebra (DE-588)4199958-7 s DE-604 Anwendung (DE-588)4196864-5 s Sommer, Gerald Sonstige oth HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009352867&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Geometric computing with Clifford algebras theoretical foundations and applications in computer vision and robotics ; with 16 tables Clifford-Algebra Clifford-Algebra - Anwendung Clifford-Algebra (DE-588)4199958-7 gnd Anwendung (DE-588)4196864-5 gnd |
| subject_GND | (DE-588)4199958-7 (DE-588)4196864-5 |
| title | Geometric computing with Clifford algebras theoretical foundations and applications in computer vision and robotics ; with 16 tables |
| title_auth | Geometric computing with Clifford algebras theoretical foundations and applications in computer vision and robotics ; with 16 tables |
| title_exact_search | Geometric computing with Clifford algebras theoretical foundations and applications in computer vision and robotics ; with 16 tables |
| title_full | Geometric computing with Clifford algebras theoretical foundations and applications in computer vision and robotics ; with 16 tables Gerald Sommer (ed.) |
| title_fullStr | Geometric computing with Clifford algebras theoretical foundations and applications in computer vision and robotics ; with 16 tables Gerald Sommer (ed.) |
| title_full_unstemmed | Geometric computing with Clifford algebras theoretical foundations and applications in computer vision and robotics ; with 16 tables Gerald Sommer (ed.) |
| title_short | Geometric computing with Clifford algebras |
| title_sort | geometric computing with clifford algebras theoretical foundations and applications in computer vision and robotics with 16 tables |
| title_sub | theoretical foundations and applications in computer vision and robotics ; with 16 tables |
| topic | Clifford-Algebra Clifford-Algebra - Anwendung Clifford-Algebra (DE-588)4199958-7 gnd Anwendung (DE-588)4196864-5 gnd |
| topic_facet | Clifford-Algebra Clifford-Algebra - Anwendung Anwendung |
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