Cover
Title
Copyright
Preface
Table of Contents
List of Contributors
Part I. A Unified Algebraic Approach for Classical Geometries
1. New Algebraic Tools for Classical Geometry. David Hestenes, Hongbo Li, and Alyn Rockwood
1.1 Introduction
1.2 Geometric Algebra of a Vector Space
1.3 Linear Transformations
1.4 Vectors as Geometrical Points
1.5 Linearizing the Euclidean Group
2. Generalized Homogeneous Coordinates for Computational Geometry. Hongbo Li, David Hestenes, and Alyn Rockwood
2.1 Introduction
2.2 Minkowski Space with Conformal and Projective Splits
2.3 Homogeneous Model of Euclidean Space
2.4 Euclidean Spheres and Hyperspheres
2.5 Multi-dimensional Spheres, Planes, and Simplexes
2.6 Relation among Spheres and Hyperplanes
2.7 Conformal Transformations
3. Spherical Conformal Geometry with Geometric Algebra. Hongbo Li, David Hestenes, and Alyn Rockwood
3.1 Introduction
3.2 Homogeneous Model of Spherical Space
3.3 Relation between Two Spheres or Hyperplanes
3.4 Spheres and Planes of Dimension r
3.5 Stereographic Projection
3.6 Conformal Transformations
3.6.1 Inversions and Reflections
3.6.2 Other Typical Conformal Transformations
4. A Universal Model for Conformal Geometries of Euclidean, Spherical and Double-Hyperbolic Spaces. Hongbo Li, David Hestenes, and Alyn Rockwood
4.1 Introduction
4.2 The Hyperboloid Model
4.2.1 Generalized Points
4.2.2 Total Spheres
4.3 The Homogeneous Model
4.3.1 Generalized Points
4.3.2 Total Spheres
4.3.3 Total Spheres of Dimensional r
4.4 Stereographic Projection
4.5 The Conformal Ball Model
4.6 The Hemisphere Model
4.7 The Half-Space Model
4.8 The Klein Ball Model
4.9 A Universal Model for Euclidean, Spherical, and Hyperbolic Spaces
5. Geo-MAP Unification. Ambj¨orn Naeve and Lars Svensson
5.1 Introduction
5.2 Historical Background
5.3 Geometric Background
5.3.1 Affine Space
5.3.2 Projective Space
5.4 The Unified Geo-MAP Computational Framework
5.4.1 Geo-MAP Unification
5.4.2 A Simple Example
5.4.3 Expressing Euclidean Operations in the Surrounding Geometric Algebra
5.5 Applying the Geo-MAP Technique to Geometrical Optics
5.5.1 Some Geometric-Optical Background
5.5.2 Determining the Second Order Law of Reflection for Planar Light Rays
5.5.3 Interpreting the Second Order Law of Reflection Geometrically
5.6 Summary and Conclusions
5.6.1 The Geo-MAP Unification Technique
5.6.2 Algebraic and Combinatorial Construction of a Geometric Algebra
5.7 Acknowledgements
5.8 Appendix: Construction of a Geometric Algebra
6. Honing Geometric Algebra for Its Use in the Computer Sciences. Leo Dorst
6.1 Introduction
6.2 The Internal Structure of Geometric Algebra
6.3 The Contraction: An Alternative Inner Product
6.4 The Design of Theorems and ‘Filters’
6.4.1 Proving Theorems
6.4.2 Example: Proof of a Duality
6.4.3 Filter Design to Specification
6.4.4 Example: The Design of the Meet Operation
6.5 Splitting Algebras Explicitly
6.6 The Rich Semantics of the Meet Operation
6.6.1 Meeting Blades
6.6.2 Meets of Affine Subspaces
6.6.3 Scalar Meets Yield Distances between Subspaces
6.6.4 The Euclidean Distance between Affine Subspaces
6.7 The Use and Interpretation of Geometric Algebra
6.8 Geometrical Models of Multivectors
6.9 Conclusions
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
7.1 Introduction
7.2 Groups of Transformations and Invariants
7.3 Pattern Recognition
7.4 Clifford Algebras as Unified Language for Pattern Recognition
7.4.1 Clifford Algebras as Models of Geometrical and Perceptual Spaces
7.4.2 Clifford Algebra of Motion and Affine Groups of Metric Spaces
7.4.3 Algebraic Models of Perceptual Color Spaces
7.5 Hypercomplex–Valued Moments and Invariants
7.5.1 Classical ℝ–Valued Moments and Invariants
7.5.2 Generalized Complex Moments and Invariants
7.5.3 Triplet Moments and Invariants of Color Images
7.5.4 Quaternionic Moments and Invariants of 3–D Images
7.5.5 Hypercomplex–Valued Invariants of n–D Images
7.6 Conclusion
8. Non-commutative Hypercomplex Fourier Transforms of Multidimensional Signals. Thomas B¨ulow, Michael Felsberg, and Gerald Sommer
8.1 Introduction
8.2 1-D Harmonic Transforms
8.3 2-D Harmonic Transforms
8.3.1 Real and Complex Harmonic Transforms
8.3.2 The Quaternionic Fourier Transform (QFT)
8.4 Some Properties of the QFT
8.4.1 The Hierarchy of Harmonic Transforms
8.4.2 The Main QFT-Theorems
8.5 The Clifford Fourier Transform
8.6 Historical Remarks
8.7 Conclusion
9. Commutative Hypercomplex Fourier Transforms of Multidimensional Signals. Michael Felsberg, Thomas B¨ulow, and Gerald Sommer
9.1 Introduction
9.2 Hypercomplex Algebras
9.2.1 Basic Definitions
9.2.2 The Commutative Algebra ℋ2
9.3 The Two-Dimensional Hypercomplex Fourier Analysis
9.3.1 The Two-Dimensional Hypercomplex Fourier Transform
9.3.2 Main Theorems of the HFT2
9.3.3 The Affine Theorem of the HFT2
9.4 The n-Dimensional Hypercomplex Fourier Analysis
9.4.1 The Isomorphism between ℋn and the 2n−1-Fold Cartesian Product of ℂ
9.4.2 The n-Dimensional Hypercomplex Fourier Transform
9.5 Conclusion
10. Fast Algorithms of Hypercomplex Fourier Transforms. Michael Felsberg, Thomas B¨ulow, Gerald Sommer, and Vladimir M. Chernov
10.1 Introduction
10.2 Discrete Quaternionic Fourier Transform and Fast Quaternionic Fourier Transform
10.2.1 Derivation of DQFT and FQFT
10.2.2 Optimizations by Hermite Symmetry
10.2.3 Complexities
10.3 Discrete and Fast n-Dimensional Transforms
10.3.1 Discrete Commutative Hypercomplex Fourier Transform and Fast Commutative Hypercomplex Fourier Transform
10.3.2 Optimizations and Complexities
10.4 Fast Algorithms by FFT
10.4.1 Cascading 1-D FFTs
10.4.2 HFT by Complex Fourier Transform
10.4.3 Complexities
10.5 Conclusion and Summary
11. Local Hypercomplex Signal Representations and Applications. Thomas B¨ulow and Gerald Sommer
11.1 Introduction
11.2 The Analytic Signal
11.2.1 The One-Dimensional Analytic Signal
11.2.2 Complex Approaches to the Two-Dimensional Analytic Signal
11.2.3 The 2-D Quaternionic Analytic Signal
11.2.4 Instantaneous Amplitude
11.2.5 The n-Dimensional Analytic Signal
11.3 Local Phase in Image Processing
11.3.1 Local Complex Phase
11.3.2 Quaternionic Gabor Filters
11.3.3 Local Quaternionic Phase
11.3.4 Relations between Complex and Quaternionic Gabor Filters
11.3.5 Algorithmic Complexity of Gabor Filtering
11.4 Texture Segmentation Using the Quaternionic Phase
11.4.1 The Gabor Filter Approach
11.4.2 Quaternionic Extension of Bovik’s Approach
11.4.3 Experimental Results
11.4.4 Detection of Defects in Woven Materials
11.5 Conclusion
12. Introduction to Neural Computation in Clifford Algebra. Sven Buchholz and Gerald Sommer
12.1 Introduction
12.2 An Outline of Clifford Algebra
12.3 The Clifford Neuron
12.3.1 The Real Neuron
12.3.2 The Clifford Neuron
12.4 Clifford Neurons as Linear Operators
12.4.1 The Clifford Group
12.4.2 Spinor Neurons
12.4.3 Simulations with Spinor Neurons
12.5 M¨obius Transformations
12.6 Summary
13. Clifford Algebra Multilayer Perceptrons. Sven Buchholz and Gerald Sommer
13.1 Introduction and Preliminaries
13.2 Universal Approximation by Clifford MLPs
13.3 Activation Functions
13.3.1 Real Activation Functions
13.3.2 Activation Function of Clifford MLPs
13.4 Clifford Back–Propagation Algorithm
13.5 Experimental Results
13.6 Conclusions and Outlook
Part III. Geometric Algebra for Computer Vision and Robotics
14. A Unified Description of Multiple View Geometry. Christian B.U. Perwass and Joan Lasenby
14.1 Introduction
14.2 Projective Geometry
14.3 The Fundamental Matrix
14.3.1 Derivation
14.3.2 Rank of F
14.3.3 Degrees of Freedom of F
14.3.4 Transferring Points with F
14.3.5 Epipoles of F
14.4 The Trifocal Tensor
14.4.1 Derivation
14.4.2 Transferring Lines
14.4.3 Transferring Points
14.4.4 Rank of T
14.4.5 Degrees of Freedom of T
14.4.6 Constraints on T
14.4.7 Relation between T and F
14.4.8 Second Order Constraints
14.4.9 Epipoles
14.5 The Quadfocal Tensor
14.5.1 Derivation
14.5.2 Transferring Lines
14.5.3 Rank of Q
14.5.4 Degrees of Freedom of Q
14.5.5 Constraints on Q
14.5.6 Relation between Q and T
14.6 Reconstruction and the Trifocal Tensor
14.7 Conclusion
15. 3D-Reconstruction from Vanishing Points. Christian B.U. Perwass and Joan Lasenby
15.1 Introduction
15.2 Image Plane Bases
15.3 Plane Collineation
15.4 The Plane at Infinity and Its Collineation
15.5 Vanishing Points and P∞
15.5.1 Calculating Vanishing Points
15.5.2 Vanishing Points from Multiple Parallel Lines
15.5.3 Ψ∞ from Vanishing Points
15.6 3D-Reconstruction of Image Points
15.6.1 The Geometry
15.6.2 The Minimization Function
15.7 Experimental Results
15.7.1 Synthetic Data
15.7.2 Real Data
15.8 Conclusions
16. Analysis and Computation of the Intrinsic Camera Parameters. Eduardo Bayro-Corrochano and Bodo Rosenhahn
16.1 Introduction
16.2 Conics and the Theorem of Pascal
16.3 Computing the Kruppa Equations in the Geometric Algebra
16.3.1 The Scenario
16.3.2 Standard Kruppa Equations
16.3.3 Kruppa’s Equations Using Brackets
16.4 Camera Calibration Using Pascal’s Theorem
16.4.1 Computing Stationary Intrinsic Parameters
16.4.2 Computing Non–stationary Intrinsic Parameters
16.5 Experimental Analysis
16.5.1 Experiments with Simulated Images
16.5.2 Experiments with Real Images
16.6 Conclusions
17. Coordinate-Free Projective Geometry for Computer Vision. Hongbo Li and Gerald Sommer
17.1 Introduction
17.2 Preparatory Mathematics
17.2.1 Dual Bases
17.2.2 Projective and Affine Spaces
17.2.3 Projective Splits
17.3 Camera Modeling and Calibration
17.3.1 Pinhole Cameras
17.3.2 Camera Constraints
17.3.3 Camera Calibration
17.4 Epipolar and Trifocal Geometries
17.4.1 Epipolar Geometry
17.4.2 Trifocal Geometry
17.5 Relations among Epipoles, Epipolar Tensors, and Trifocal Tensors of Three Cameras
17.5.1 Relations on Epipolar Tensors
17.5.2 Relations on Trifocal Tensors I
17.5.3 Relations on Trifocal Tensors II
17.5.4 Relations on Trifocal Tensors III
17.5.5 Relations on Trifocal Tensors IV
17.5.6 Relations on Trifocal Tensors V
17.5.7 Relations on Trifocal Tensors VI
17.5.8 A Unified Treatment of Degree-six Constraints
17.6 Conclusion
18. The Geometry and Algebra of Kinematics. Eduardo Bayro-Corrochano
18.1 Introduction
18.2 The Euclidean 3D Geometric Algebra
18.2.1 3D Rotors
18.3 The 4D Geometric Algebra for 3D Kinematics
18.3.1 The Motor Algebra
18.3.2 Motors, Rotors, and Translators
18.3.3 Properties of Motors
18.4 Representation of Points, Lines, and Planes Using 3D and 4D Geometric Algebras
18.4.1 Representation of Points, Lines, and Planes in the 3D GA
18.4.2 Representation of Points, Lines, and Planes in the 4D GA
18.5 Modeling the Motion of Points, Lines, and Planes Using 3D and 4D Geometric Algebras
18.5.1 Motion of Points, Lines, and Planes in the 3D GA
18.5.2 Motion of Points, Lines, and Planes in the 4D GA
18.6 Conclusion
19. Kinematics of Robot Manipulators in the Motor Algebra. Eduardo Bayro-Corrochano and Detlef K¨ahler
19.1 Introduction
19.2 Motor Algebra for the Kinematics of Robot Manipulators
19.2.1 The Denavit–Hartenberg Parameterization
19.2.2 Representations of Prismatic and Revolute Transformations
19.2.3 Grasping by Using Constraint Equations
19.3 Direct Kinematics of Robot Manipulators
19.3.1 Maple Program for Motor Algebra Computations
19.4 Inverse Kinematics of Robot Manipulators
19.4.1 The Rendezvous Method
19.4.2 Computing θ1, θ2 and d3 Using a Point Representation
19.4.3 Computing θ4 and θ5 Using a Line Representation
19.4.4 Computing θ6 Using a Plane Representation
19.5 Conclusion
20. Using the Algebra of Dual Quaternions for Motion Alignment. Kostas Daniilidis
20.1 Introduction
20.2 Even Subalgebras of Non-degenerate ℛp,q,r
20.3 Even Subalgebras of Degenerate ℛp,q,r
20.4 Line Transformation
20.5 Motion Estimation from 3D-Line Matches
20.6 The Principle of Transference
20.7 Relating Coordinate Systems to Each Other
20.8 Conclusion
21. The Motor Extended Kalman Filter for Dynamic Rigid Motion Estimation from Line Observations. Yiwen Zhang, Gerald Sommer, and Eduardo Bayro-Corrochano
21.1 Introduction
21.2 Kalman Filter Techniques
21.2.1 The Kalman Filter
21.2.2 The Extended Kalman Filter
21.3 3-D Line Motion Model
21.3.1 Geometric Algebra 𝒢+ 3,0,1 and Pl¨ucker Line Model
21.3.2 Pl¨ucker Line Motion Model in 𝒢+ 3,0,1
21.3.3 Interpretation of the Pl¨ucker Line Motion Model in Linear Algebra
21.4 The Motor Extended Kalman Filter
21.4.1 Discrete Dynamic Model Using Motor State
21.4.2 Linearization of the Measurement Model
21.4.3 Constraints Problem
21.4.4 The MEKF Algorithm
21.4.5 A Batch Method of Analytical Solution
21.5 Experimental Analysis of the MEKF
21.5.1 Simulation
21.5.2 Real Experiment
21.6 Conclusion
References
Author Index
Subject Index