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Tables of Contents for Geometric Computing With Clifford Algebra
Chapter/Section Title
Page #
Page Count
Part I. A Unified Algebraic Approach for Classical Geometries
New Algebraic Tools for Classical Geometry
3
24
David Hestenes
Hongbo Li
Alyn Rockwood
Introduction
3
1
Geometric Algebra of a Vector Space
4
9
Linear Transformations
13
6
Vectors as Geometrical Points
19
4
Linearizing the Euclidean Group
23
4
Generalized Homogeneous Coordinates for Computational Geometry
27
34
Hongbo Li
David Hestenes
Alyn Rockwood
Introduction
27
2
Minkowski Space with Conformal and Projective Splits
29
4
Homogeneous Model of Euclidean Space
33
7
Euclidean Spheres and Hyperspheres
40
1
Multi-dimensional Spheres, Planes, and Simplexes
41
5
Relation among Spheres and Hyperplanes
46
6
Conformal Transformations
52
9
Spherical Conformal Geometry with Geometric Algebra
61
16
Hongbo Li
David Hestenes
Alyn Rockwood
Introduction
61
1
Homogeneous Model of Spherical Space
62
4
Relation between Two Spheres or Hyperplanes
66
2
Spheres and Planes of Dimension r
68
2
Stereographic Projection
70
2
Conformal Transformations
72
5
A Universal Model for Conformal Geometries of Euclidean, Spherical and Double-Hyperbolic Spaces
77
28
Hongbo Li
David Hestenes
Alyn Rockwood
Introduction
77
2
The Hyperboloid Model
79
4
The Homogeneous Model
83
7
Stereographic Projection
90
2
The Conformal Ball Model
92
1
The Hemisphere Model
93
1
The Half-Space Model
94
3
The Klein Ball Model
97
2
A Universal Model for Euclidean, Spherical, and Hyperbolic Spaces
99
6
Geo-MAP Unification
105
22
Ambjorn Naeve
Lars Svensson
Introduction
105
1
Historical Background
106
2
Geometric Background
108
2
The Unified Geo-MAP Computational Framework
110
4
Applying the Geo-MAP Technique to Geometrical Optics
114
8
Summary and Conclusions
122
1
Acknowledgements
123
1
Appendix: Construction of a Geometric Algebra
123
4
Honing Geometric Algebra for Its Use in the Computer Sciences
127
28
Leo Dorst
Introduction
127
1
The Internal Structure of Geometric Algebra
128
6
The Contraction: An Alternative Inner Product
134
2
The Design of Theorems and `Filters'
136
7
Splitting Algebras Explicitly
143
2
The Rich Semantics of the Meet Operation
145
5
The Use and Interpretation of Geometric Algebra
150
1
Geometrical Models of Multivectors
151
1
Conclusions
151
4
Part II. Algebraic Embedding of Signal Theory and Neural Computation
Spatial --- Color Clifford Algebras for Invarian Image Recognition
155
32
Ekaterina Rundblad-Labunets
Valeri Labunets
Introduction
155
2
Groups of Transformations and Invariants
157
1
Pattern Recognition
157
3
Clifford Algebras as Unified Language for Pattern Recognition
160
9
Hypercomplex-Valued Moments and Invariants
169
16
Conclusion
185
2
Non-commutative Hypercomplex Fourier Transforms of Multidimensional Signals
187
22
Thomas Bulow
Michael Felsberg
Gerald Sommer
Introduction
187
1
1-D Harmonic Transforms
188
3
2-D Harmonic Transforms
191
3
Some Properties of the QFT
194
11
The Clifford Fourier Transform
205
1
Historical Remarks
206
1
Conclusion
207
2
Commutative Hypercomplex Fourier Transforms of Multidimensional Signals
209
22
Michael Felsberg
Thomas Bulow
Gerald Sommer
Introduction
209
1
Hypercomplex Algebras
210
3
The Two-Dimensional Hypercomplex Fourier Analysis
213
8
The n-Dimensional Hypercomplex Fourier Analysis
221
8
Conclusion
229
2
Fast Algorithms of Hypercomplex Fourier Transforms
231
24
Michael Felsberg
Thomas Bulow
Gerald Sommer
Vladimir M. Chernov
Introduction
231
1
Discrete Quaternionic Fourier Transform and Fast Quaternionic Fourier Transform
232
10
Discrete and Fast n-Dimensional Transforms
242
5
Fast Algorithms by FFT
247
7
Conclusion and Summary
254
1
Local Hypercomplex Signal Representations and Applications
255
36
Thomas Bulow
Gerald Sommer
Introduction
255
1
The Analytic Signal
256
14
Local Phase in Image Processing
270
9
Texture Segmentation Using the Quaternionic Phase
279
10
Conclusion
289
2
Introduction to Neural Computation in Clifford Algebra
291
24
Sven Buchholz
Gerald Sommer
Introduction
291
1
An Outline of Clifford Algebra
292
3
The Clifford Neuron
295
4
Clifford Neurons as Linear Operators
299
10
Mobius Transformations
309
5
Summary
314
1
Clifford Algebra Multilayer Perceptrons
315
22
Sven Buchholz
Gerald Sommer
Introduction and Preliminaries
315
2
Universal Approximation by Clifford MLPs
317
3
Activation Functions
320
4
Clifford Back --- Propagation Algorithm
324
3
Experimental Results
327
7
Conclusions and Outlook
334
3
Part III Geometric Algebra for Computer Vision and Robotics
A Unified Description of Multiple View Geometry
337
34
Christian B.U. Perwass
Joan Lasenby
Introduction
337
1
Projective Geometry
338
3
The Fundamental Matrix
341
5
The Trifocal Tensor
346
12
The Quadfocal Tensor
358
6
Reconstruction and the Trifocal Tensor
364
5
Conclusion
369
2
3D-Reconstruction from Vanishing Points
371
22
Christian B.U. Perwass
Joan Lasenby
Introduction
371
1
Image Plane Bases
372
3
Plane Collineation
375
3
The Plane at Infinity and Its Collineation
378
2
Vanishing Points and P∞
380
2
3D-Reconstruction of Image Points
382
4
Experimental Results
386
6
Conclusions
392
1
Analysis and Computation of the Intrinsic Camera Parameters
393
22
Eduardo Bayro-Corrochano
Bodo Rosenhahn
Introduction
393
1
Conics and the Theorem of Pascal
394
2
Computing the Kruppa Equations in the Geometric Algebra
396
8
Camera Calibration Using Pascal's Theorem
404
7
Experimental Analysis
411
3
Conclusions
414
1
Coordinate-Free Projective Geometry for Computer Vision
415
40
Hongbo Li
Gerald Sommer
Introduction
415
2
Preparatory Mathematics
417
4
Camera Modeling and Calibration
421
3
Epipolar and Trifocal Geometrics
424
5
Relations among Epipoles, Epipolar Tensors, and Trifocal Tensors of Three Cameras
429
24
Conclusion
453
2
The Geometry and Algebra of Kinematics
455
16
Eduardo Bayro-Corrochano
Introduction
455
2
The Euclidean 3D Geometric Algebra
457
1
The 4D Geometric Algebra for 3D Kinematics
458
7
Representation of Points, Lines, and Planes Using 3D and 4D Geometric Algebras
465
2
Modeling the Motion of Points, Lines, and Planes Using 3D and 4D Geometric Algebras
467
3
Conclusion
470
1
Kinematics of Robot Manipulators in the Motor Algebra
471
18
Eduardo Bayro-Corrochano
Detlef Kahler
Introduction
471
1
Motor Algebra for the Kinematics of Robot Manipulators
472
6
Direct Kinematics of Robot Manipulators
478
3
Inverse Kinematics of Robot Manipulators
481
7
Conclusion
488
1
Using the Algebra of Dual Quaternions for Motion Alignment
489
12
Kostas Daniilidis
Introduction
489
1
Even Subalgebras of Non-degenerate Rp,q,r
490
1
Even Subalgebras of Degenerate Rp,q,r
491
2
Line Transformation
493
1
Motion Estimation from 3D-Line Matches
494
2
The Principle of Transference
496
2
Relating Coordinate Systems to Each Other
498
1
Conclusion
499
2
The Motor Extended Kalman Filter for Dynamic Rigid Motion Estimation from Line Observations
501
30
Yiwen Zhang
Gerald Sommer
Eduardo Bayro-Corrochano
Introduction
501
3
Kalman Filter Techniques
504
3
3-D Line Motion Model
507
8
The Motor Extended Kalman Filter
515
6
Experimental Analysis of the MEKF
521
7
Conclusion
528
3
References
531
12
Author Index
543
1
Subject Index
544