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TitleGeometric Computing: for Wavelet Transforms, Robot Vision, Learning, Control and Action
Author
LanguageEnglish
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Total Pages625
Table of Contents
                            Cover
Geometric Computing
	ISBN 1848829280
	Foreword
	Preface
Part I Fundamentals of Geometric Algebra
	1 Introduction to Geometric Algebra
		1.1 History of Geometric Algebra
		1.2 What Is Geometric Algebra?
			1.2.1 Basic Definitions
			1.2.2 Nonorthonormal Frames and Reciprocal Frames
			1.2.3 Some Useful Formulas
			1.2.4 Multivector Products
			1.2.5 Further Properties of the Geometric Product
			1.2.6 Dual Blades and Duality in the Geometric Product
			1.2.7 Multivector Operations
		1.3 Linear Algebra
		1.4 Simplexes
		1.5 Geometric Calculus
			1.5.1 Multivector-Valued Functions and the Inner Product
			1.5.2 The Multivector Integral
			1.5.3 The Vector Derivative
			1.5.4 Grad, Div, and Curl
			1.5.5 Multivector Fields
			1.5.6 Convolution and Correlation of Scalar Fields
			1.5.7 Clifford Convolution and Correlation
			1.5.8 Linear Algebra Derivations
			1.5.9 Reciprocal Frames with Curvilinear Coordinates
			1.5.10 Geometric Calculus in 2D
			1.5.11 Electromagnetism: The Maxwell Equations
			1.5.12 Spinors, Shrödinger Pauli, and Dirac Equations
			1.5.13 Spinor Operators
		1.6 Exercises
	2 Geometric Algebra for Modeling in Robot Physics
		2.1 The Roots of Geometry and Algebra
		2.2 Geometric Algebra: A Unified Mathematical Language
		2.3 What Does Geometric Algebra Offer for GeometricComputing?
			2.3.1 Coordinate-Free Mathematical System
			2.3.2 Models for Euclidean and Pseudo-Euclidean Geometry
			2.3.3 Subspaces as Computing Elements
			2.3.4 Representation of Orthogonal Transformations
			2.3.5 Objects and Operators
			2.3.6 Extension of Linear Transformations
			2.3.7 Signals and Wavelets in the Geometric Algebra Framework
			2.3.8 Kinematics and Dynamics
		2.4 Solving Problems in Perception and Action Systems
Part II Euclidean, Pseudo-Euclidean, Lie and Incidence Algebras,and Conformal Geometries
	3 2D, 3D, and 4D Geometric Algebras
		3.1 Complex, Double, and Dual Numbers
		3.2 2D Geometric Algebras of the Plane
		3.3 3D Geometric Algebra for the Euclidean 3D Space
			3.3.1 The Algebra of Rotors
			3.3.2 Orthogonal Rotors
			3.3.3 Recovering a Rotor
		3.4 Quaternion Algebra
		3.5 Lie Algebras and Bivector Algebras
			3.5.1 Lie Group of Rotors
			3.5.2 Bivector Lie Algebra
			3.5.3 Complex Structures and Unitary Groups
			3.5.4 Hermitian Inner Product and Unitary Groups
		3.6 4D Geometric Algebra for 3D Kinematics
			3.6.1 Motor Algebra
			3.6.2 Motors, Rotors, and Translators in G+3,0,1
			3.6.3 Properties of Motors
		3.7 4D Geometric Algebra for Projective 3D Space
		3.8 Conclusion
		3.9 Exercises
	4 Kinematics of the 2D and 3D Spaces
		4.1 Introduction
		4.2 Representation of Points, Lines, and Planes Using 3D Geometric Algebra
		4.3 Representation of Points, Lines, and PlanesUsing Motor Algebra
		4.4 Representation of Points, Lines, and Planes Using 4D Geometric Algebra
		4.5 Motion of Points, Lines, and Planes in 3D Geometric Algebra
		4.6 Motion of Points, Lines, and Planes Using Motor Algebra
		4.7 Motion of Points, Lines, and Planes Using 4D Geometric Algebra
		4.8 Spatial Velocity of Points, Lines, and Planes
			4.8.1 Rigid-Body Spatial Velocity Using Matrices
			4.8.2 Angular Velocity Using Rotors
			4.8.3 Rigid-Body Spatial Velocity Using Motor Algebra
			4.8.4 Point, Line, and Plane Spatial Velocities Using Motor Algebra
		4.9 Incidence Relations Between Points, Lines, and Planes
			4.9.1 Flags of Points, Lines, and Planes
		4.10 Conclusion
		4.11 Exercises
	5 Lie Algebras and the Algebra of Incidence Using the Null Cone and Affine Plane
		5.1 Introduction
		5.2 Geometric Algebra of Reciprocal Null Cones
			5.2.1 Reciprocal Null Cones
			5.2.2 The Universal Geometric Algebra Gn,n
			5.2.3 The Lie Algebra of Null Spaces
			5.2.4 The Standard Bases of Gn,n
			5.2.5 Representations and Operations Using Bivector Matrices
			5.2.6 Bivector Representation of Linear Operators
		5.3 Horosphere and n-Dimensional Affine Plane
		5.4 The General Linear Group
			5.4.1 The General Linear Algebra gl(N) of the General Linear Lie Group GL(N)
			5.4.2 The Orthogonal Groups
		5.5 Computing Rigid Motion in the Affine Plane
		5.6 The Lie Algebra of the Affine Plane
		5.7 The Algebra of Incidence
			5.7.1 Incidence Relations in the Affine n-Plane
			5.7.2 Directed Distances
			5.7.3 Incidence Relations in the Affine 3-Plane
			5.7.4 Geometric Constraints as Flags
		5.8 Conclusion
		5.9 Exercises
	6 Conformal Geometric Algebra
		6.1 Introduction
			6.1.1 Conformal Split
			6.1.2 Conformal Splits for Points and Simplexes
			6.1.3 Euclidean and Conformal Spaces
			6.1.4 Stereographic Projection
			6.1.5 Inner- and Outer-Product Null Spaces
			6.1.6 Spheres and Planes
			6.1.7 Geometric Identities, Meet and Join Operations, Duals, and Flats
			6.1.8 Meet, Pair of Points, and Plunge
			6.1.9 Simplexes and Spheres
		6.2 The 3D Affine Plane
			6.2.1 Lines and Planes
			6.2.2 Directed Distance
		6.3 The Lie Algebra
		6.4 Conformal Transformations
			6.4.1 Inversion
			6.4.2 Reflection
			6.4.3 Translation
			6.4.4 Transversion
			6.4.5 Rotation
			6.4.6 Rigid Motion Using Flags
			6.4.7 Dilation
			6.4.8 Involution
			6.4.9 Conformal Transformation
		6.5 Ruled Surfaces
			6.5.1 Cone and Conics
			6.5.2 Cycloidal Curves
			6.5.3 Helicoid
			6.5.4 Sphere and Cone
			6.5.5 Hyperboloid, Ellipsoids, and Conoid
		6.6 Exercises
	7 Programming Issues
		7.1 Main Issues for an Efficient Implementation
			7.1.1 Specific Aspects for the Implementation
		7.2 Implementation Practicalities
			7.2.1 Specification of the Geometric Algebra, Gp,q
			7.2.2 The General Multivector Class
			7.2.3 Optimization of Multivector Functions
			7.2.4 Factorization
			7.2.5 Speeding Up Geometric Algebra Expressions
			7.2.6 Multivector Software Packets
Part III Geometric Computing for Image Processing, Computer Vision, and Neurocomputing
	8 Clifford–Fourier and Wavelet Transforms
		8.1 Introduction
		8.2 Image Analysis in the Frequency Domain
			8.2.1 The One-Dimensional Fourier Transform
			8.2.2 The Two-Dimensional Fourier Transform
			8.2.3 Quaternionic Fourier Transform
			8.2.4 2D Analytic Signals
			8.2.5 Properties of the QFT
			8.2.6 Discrete QFT
		8.3 Image Analysis Using the Phase Concept
			8.3.1 2D Gabor Filters
			8.3.2 The Phase Concept
		8.4 Clifford–Fourier Transforms
			8.4.1 Tri-Dimensional Clifford–Fourier Transform
			8.4.2 Space and Time Geometric AlgebraFourier Transform
			8.4.3 n-Dimensional Clifford–Fourier Transform
		8.5 From Real to Clifford Wavelet Transforms for Multiresolution Analysis
			8.5.1 Real Wavelet Transform
			8.5.2 Discrete Wavelets
			8.5.3 Wavelet Pyramid
			8.5.4 Complex Wavelet Transform
			8.5.5 Quaternion Wavelet Transform
			8.5.6 Quaternionic Wavelet Pyramid
			8.5.7 The Tridimensional Clifford Wavelet Transform
			8.5.8 The Continuous Conformal Geometric Algebra Wavelet Transform
			8.5.9 The n-Dimensional Clifford Wavelet Transform
		8.6 Conclusion
	9 Geometric Algebra of Computer Vision
		9.1 Introduction
		9.2 The Geometric Algebras of 3D and 4D Spaces
			9.2.1 3D Space and the 2D Image Plane
			9.2.2 The Geometric Algebra of 3D Euclidean Space
			9.2.3 A 4D Geometric Algebra for Projective Space
			9.2.4 Projective Transformations
			9.2.5 The Projective Split
		9.3 The Algebra of Incidence
			9.3.1 The Bracket
			9.3.2 The Duality Principle and Meet and Join Operations
		9.4 Algebra in Projective Space
			9.4.1 Intersection of a Line and a Plane
			9.4.2 Intersection of Two Planes
			9.4.3 Intersection of Two Lines
			9.4.4 Implementation of the Algebra
		9.5 Projective Invariants
			9.5.1 The 1D Cross-Ratio
			9.5.2 2D Generalization of the Cross-Ratio
			9.5.3 3D Generalization of the Cross-Ratio
		9.6 Visual Geometry of n-Uncalibrated Cameras
			9.6.1 Geometry of One View
			9.6.2 Geometry of Two Views
			9.6.3 Geometry of Three Views
			9.6.4 Geometry of n-Views
		9.7 Omnidirectional Vision
			9.7.1 Omnidirectional Vision and Geometric Algebra
			9.7.2 Point Projection
			9.7.3 Inverse Point Projection
		9.8 Invariants in the Conformal Space
			9.8.1 Invariants and Omnidirectional Vision
			9.8.2 Projective and Permutation p2-Invariants
		9.9 Conclusion
		9.10 Exercises
	10 Geometric Neuralcomputing
		10.1 Introduction
		10.2 Real-Valued Neural Networks
		10.3 Complex MLP and Quaternionic MLP
		10.4 Geometric Algebra Neural Networks
			10.4.1 The Activation Function
			10.4.2 The Geometric Neuron
			10.4.3 Feedforward Geometric Neural Networks
			10.4.4 Generalized Geometric Neural Networks
			10.4.5 The Learning Rule
			10.4.6 Multidimensional Back-Propagation Training Rule
			10.4.7 Simplification of the Learning Rule Using the Density Theorem
			10.4.8 Learning Using the Appropriate Geometric Algebras
		10.5 Support Vector Machines in Geometric Algebra
		10.6 Linear Clifford Support Vector Machinesfor Classification
		10.7 Nonlinear Clifford Support Vector Machines For Classification
		10.8 Clifford SVM for Regression
		10.9 Conclusion
Part IV Geometric Computing of Robot Kinematics and Dynamics
	11 Kinematics
		11.1 Introduction
		11.2 Elementary Transformations of Robot Manipulators
			11.2.1 The Denavit–Hartenberg Parameterization
			11.2.2 Representations of Prismatic and Revolute Transformations
			11.2.3 Grasping by Using Constraint Equations
		11.3 Direct Kinematics of Robot Manipulators
			11.3.1 MAPLE Program for Motor Algebra Computations
		11.4 Inverse Kinematics of Robot ManipulatorsUsing Motor Algebra
			11.4.1 The Rendezvous Method
			11.4.2 Computing 1, 2, and d3 Using a Point
			11.4.3 Computing 4 and 5 Using a Line
			11.4.4 Computing 6 Using a Plane Representation
		11.5 Inverse Kinematics Using the 3D Affine Plane
		11.6 Inverse Kinematic Using Conformal Geometric Algebra
		11.7 Conclusion
	12 Dynamics
		12.1 Introduction
		12.2 Differential Kinematics
		12.3 Dynamics
			12.3.1 Kinetic Energy
			12.3.2 Potential Energy
			12.3.3  Lagrange's Equations
		12.4 Complexity Analysis
			12.4.1 Computing  M
			12.4.2 Computing G
		12.5 Conclusion
Part V Applications I: Image Processing, Computer Vision,and Neurocomputing
	13 Applications of Lie Filters, and Quaternion Fourier and Wavelet Transforms
		13.1 Lie Filters in the Affine Plane
			13.1.1 The Design of an Image Filter
			13.1.2 Recognition of Hand Gestures
		13.2 Representation of Speech as 2D Signals
		13.3 Preprocessing of Speech 2D Representations Using the QFT and Quaternionic Gabor Filter
			13.3.1 Method 1
			13.3.2 Method 2
		13.4 Recognition of French Phonemes Using Neurocomputing
		13.5 Application of QWT
			13.5.1 Estimation of the Quaternionic Phase
			13.5.2 Confidence Interval
			13.5.3 Discussion on Similarity Distance and the Phase Concept
			13.5.4 Optical Flow Estimation
		13.6 Conclusion
	14 Invariants Theory in Computer Vision and Omnidirectional Vision
		14.1 Introduction
		14.2 Conics and Pascal's Theorem
		14.3 Computing Intrinsic Camera Parameters
		14.4 Projective Invariants
			14.4.1 The 1D Cross-Ratio
			14.4.2 2D Generalization of the Cross-Ratio
			14.4.3 3D Generalization of the Cross-Ratio
			14.4.4 Generation of 3D Projective Invariants
		14.5 3D Projective Invariants from Multiple Views
			14.5.1 Projective Invariants Using Two Views
			14.5.2 Projective Invariant of Points Using Three Uncalibrated Cameras
			14.5.3 Comparison of the Projective Invariants
		14.6 Visually Guided Grasping
			14.6.1 Parallel Orienting
			14.6.2 Centering
			14.6.3 Grasping
			14.6.4 Holding the Object
		14.7 Camera Self-Localization
		14.8 Projective Depth
		14.9 Shape and Motion
			14.9.1 The Join-Image
			14.9.2 The SVD Method
			14.9.3 Completion of the 3D Shape Using Invariants
		14.10 Omnidirectional Vision Landmark Identification Using Projective Invariants
			14.10.1 Learning Phase
			14.10.2 Recognition Phase
			14.10.3 Omnidirectional Vision and Invariants for Robot Navigation
			14.10.4 Learning Phase
			14.10.5 Recognition Phase
			14.10.6 Quantitative Results
		14.11 Conclusions
	15 Registration of 3D Points Using GA and Tensor Voting
		15.1 Problem Formulation
			15.1.1 The Geometric Constraint
		15.2 Tensor Voting
			15.2.1 Tensor Representation in 3D
			15.2.2 Voting Fields in 3D
			15.2.3 Detection of 3D Surfaces
			15.2.4 Estimation of 3D Correspondences
		15.3 Experimental Analysis
			15.3.1 Correspondences Between 3D Pointsby Rigid Motion
			15.3.2 Multiple Overlapping Motions and Nonrigid Motion
			15.3.3 Extension to Nonrigid Motion
		15.4 Conclusions
	16 Applications in Neuralcomputing
		16.1 Experiments Using Geometric Feedforward Neural Networks
			16.1.1 Learning a High Nonlinear Mapping
			16.1.2 Encoder–Decoder Problem
			16.1.3 Prediction
		16.2 Experiments Using Clifford Support Vector Machines
			16.2.1 3D Spiral: Nonlinear Classification Problem
			16.2.2 Object Recognition
			16.2.3 Multi-Case Interpolation
		16.3 Conclusion
	17 Neural Computing for 2D Contour and 3D Surface Reconstruction
		17.1 Determining the Shape of an Object
			17.1.1 Automatic Sample Selection Using GGVF
			17.1.2 Learning the Shape Using Versors
		17.2 Experiments
		17.3 Conclusion
Part VI Applications II: Robotics and Medical Robotics
	18 Rigid Motion Estimation Using Line Observations
		18.1 Introduction
		18.2 Batch Estimation Using SVD Techniques
			18.2.1 Solving AX = XB Using Motor Algebra
			18.2.2 Estimation of the Hand–Eye Motor Using SVD
		18.3 Experimental Results
		18.4 Discussion
		18.5 Recursive Estimation Using Kalman Filter Techniques
			18.5.1 The Kalman Filter
			18.5.2 The Extended Kalman Filter
			18.5.3 The Rotor-Extended Kalman Filter
		18.6 The Motor-Extended Kalman Filter
			18.6.1 Representation of the Line Motion Model in Linear Algebra
			18.6.2 Linearization of the Measurement Model
			18.6.3 Enforcing a Geometric Constraint
			18.6.4 Operation of the MEKF Algorithm
			18.6.5 Estimation of the Relative Positioning of a Robot End-Effector
		18.7 Conclusion
	19 Tracker Endoscope Calibration and Body-Sensors' Calibration
		19.1 Camera Device Calibration
			19.1.1 Rigid Body Motion in CGA
			19.1.2 Hand–Eye Calibration in CGA
			19.1.3 Tracker Endoscope Calibration
		19.2 Body-Sensor Calibration
			19.2.1 Body–Eye Calibration
			19.2.2 Algorithm Simplification
		19.3 Conclusions
	20 Tracking, Grasping, and Object Manipulation
		20.1 Tracking
			20.1.1 Exact Linearization via Feedback
			20.1.2 Visual Jacobian
			20.1.3 Exact Linearization via Feedback
			20.1.4 Experimental Results
		20.2 Barrett Hand Direct Kinematics
		20.3 Pose Estimation
			20.3.1 Segmentation
			20.3.2 Object Projection
		20.4 Grasping Objects
			20.4.1 First Style of Grasping
			20.4.2 Second Style of Grasping
			20.4.3 Third Style of Grasping
		20.5 Target Pose
			20.5.1 Object Pose
		20.6 Visually Guided Grasping
			20.6.1 Results
		20.7 Fuzzy Logic and Conformal Geometric Algebra for Grasping
			20.7.1 Mandami Fuzzy System
			20.7.2 Direct Kinematics of the Barrett Hand
			20.7.3 Fuzzy Grasping of Objects
		20.8 Conclusion
	21 3D Maps, Navigation, and Relocalization
		21.1 Map Building
			21.1.1 Matching Laser Readings
			21.1.2 Map Building
			21.1.3 Line Map
			21.1.4 3D Map Building
		21.2 Navigation
			21.2.1 Localization
			21.2.2 Adding Objects to the 3D Map
			21.2.3 Path Following
		21.3 3D Map Building Using Laser and Stereo Vision
			21.3.1 Laser Rangefinder
			21.3.2 Stereo Camera System with Pan-Tilt Unit
		21.4 Relocation Using Lines and the Hough Transform
		21.5 Experiments
		21.6 Conclusions
	22 Modeling and Registration of Medical Data
		22.1 Background
			22.1.1 Union of Spheres
			22.1.2 The Marching Cubes Algorithm
		22.2 Segmentation
		22.3 Marching Spheres
			22.3.1 Experimental Results for Modeling
		22.4 Registration of Two Models
			22.4.1 Sphere Matching
			22.4.2 Experimental Results for Registration
		22.5 Conclusions
Part VII Appendix
	23 Clifford Algebras and Related Algebras
		23.1 Clifford Algebras
			23.1.1 Basic Properties
			23.1.2 Definitions and Existence
			23.1.3 Real and Complex Clifford Algebras
			23.1.4 Involutions
			23.1.5 Structure and Classification of Clifford Algebras
			23.1.6 Clifford Groups, Pin and Spin Groups, and Spinors
		23.2 Related Algebras
			23.2.1 Gibbs' Vector Algebra
			23.2.2 Exterior Algebras
			23.2.3 Grassmann–Cayley Algebras
	24 Notation
	25 Useful Formulas for Geometric Algebra
References
Index
                        
Document Text Contents
Page 1







Page 2




GeometricComputing



Page 312




10.8CliffordSVMforRegression
295wke2e3DlXjD1.˛ke2e3/j.N˛ke2e3/j.xke2e3/j;:::;
wkIDlXjD1.˛kI/j.N˛kI/j.xkI/j:(10.56)Wecannowrede˚netheentriesofthevector
aofEq.
10.40foravectorof
Dmultivectorsasfollows:
aDhŒOa1s;Oa1e1;Oa1e2;:::;
Oa1I;:::;Œ
Oaks;Oake1;Oake2;:::;
OakI:::;Œ
OaDs;OaDe1;OaDe2;:::;
OaDIi:(10.57)andtheentriesforthe
kelementarecomputedusing
lsamplesasfollows:
OaTksDŒ.˛
ks1˛ks1/;.˛
ks2˛ks2/;:::;.˛
ksl˛ksl;
OaT
ke1DŒ.˛
ke11˛ke11/;:::;.˛
ke1l˛ke1l;
:::;
OaTkIDŒ.˛
kI1˛kI1/;.˛
kI2˛kI2/;:::;.˛
kIl˛kIl:
(10.58)Now,wecanrewritetheCliffordproduct
wTw,aswedidin(
10.44)Œ(10.45)andrewritetheprimalproblemasfollows:
min
12aTHaCC.CN/subjectto
.ywxb/jCj.wxCby/jCNjij>D0;Nij>D0forall
i;j;
(10.59)Thereafter,wewritestraightforwardlythedualof(
10.59)forsolvingtheregression
problem:
max˛T.NCy/˛T.y/12aTHasubjectto
lXjD1.˛sj˛sj/D0;lXjD1.˛e1j˛e1j/D0;:::;
lXjD1.˛Ij˛Ij/D0;


Page 313




29610GeometricNeuralcomputing
0.˛s/jC;0
.˛e1/jC;:::;
0.˛e1e2/jC;:::;0
.˛I/jC;j
D1;:::;l;
0.N˛s/jC;0
.N˛e1/jC;:::;
0.N˛e1e2/jC;:::;0
.N˛I/jC;j
D1;:::;l:
(10.60)AsexplainedinSect.
10.7,fornonlinearregressionweutilizeaparticularkernel
forcomputing
k.xm;xn/D˚.xm/˚.
xn/.Wecanusethekernelsdescribedin
Sect.10.7.Bytheuseofotherlossfunctions,liketheLaplace,complex,orpolyno-
mial,onecanextendEq.
10.60toincludeextraconstraints.
10.9Conclusion
Accordingtotheliterature,therearebasicallytwomathematicalsystemsusedin
neuralcomputing:tensoralgebraandmatrixalgebra.Incontrast,theauthorhas
chosentousethecoordinate-freesystemofCliffordorgeometricalgebraforthe

analysisanddesignoffeedforwardneura
lnetworks.Ourworkshowsthatreal-,
complex-,andquaternion-valuedneuralnetworksaresimplyparticularcasesofge-

ometricalgebramultidimensionalneuralnetworks,andthatsomecanbegenerated

usingsupportmultivectormachines.
Also,inthischapterthereal-valuedSVMisgeneralizedtoClifford-valuedSVM
andisusedforclassi˚cation,regression,andinterpolation.Inparticular,thegener-
ationofRBFnetworksingeometricalgebraiseasierusinganSVM,whichallows
oneto˚ndtheoptimalparametersautomatically.TheCSVMacceptsmultiplemul-

tivectorinputsandmultivectoroutputs,likeaMIMOarchitecture,thatallowsus
tohavemulticlassapplications.WecanuseCSVMovercomplex,quaternion,or
hypercomplexnumbersaccordingtoourneeds.
Theuseoffeedforwardneuralnetwo
rksandtheSVmachineswithinthe
geometricalgebraframeworkwidenstheirsphereofapplicabilityand,further-

more,expandsourunderstandi
ngoftheiruseformultidimensionallearning.The
experimentalanalysiscon˚rmsthepotentialofgeometricneuralnetworksand

Clifford-valuedSVMsforavarietyofrealapplicationsusingmultidimensional

representations,suchasingraphics,augmentedreality,machinelearning,computer
vision,medicalimageprocessing,androbotics.



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Index
621TTait
ŒPeterGuthrie,
585tangentoperators,
129Taylorexpansion,
76,78Taylorseries,
63,84tensor

Œ,ellipsoid-type,
414Œ,stick,
414tensorcalculus,
35TensorVoting,
405Œmethodology,
409tensors
Œ,second-ordersymmetric,
409test

Œcollinearity,
398Œconvexhull,
399Œcoplanarity,
399theorem
Œ,Apollonius,
186Œ,ArtinŒWedderburn,
580Œ,Desargues,
146,187,274Œ,Mercer,
292Œ,Pascal,
187Œ,inprojectiveandEuclideangeometry,
594Œ,secondfundamentalofinvarianttheory,
591ŒCartanŒDieudonn´
e,584TimeDelayNeuralNetwork,
357touchingcondition,
305tracker

Œendoscope,
491Œendoscopecalibration,
494ŒPolaris,
496transform
Œ,Hartley,
203Œ,Hilbert,
205,207transformation,
64,65
,69Œ,Euclidean,
80,81
Œ,backward,
303,313Œ,forward,
303,313Œ,groupontheplane,
64Œ,prismatic,
301,303Œ,revolute,
301,303Œ,screw,
301Œ,shear,
65Œofpoints,lines,planes,
301translation,
173translator,
82,302transversor,
176trifocaltensor,
262,384trilinearconstraint,
261trilinearconstraint,
262trivector,
10,66twist,
82Œ,coupled,
180Œ,exponential,
104U
union,
139UnionofSpheres,
557unitpseudoscalar,
240unitsphere,
595universalalgebra
Clp;q
.V/
,582universalapproximators,
279universalgeometricalgebra,
119universaltableofgeometricalgebras,
287Vvector
Œ,basesorthogonalnull,
122Œ,basesreciprocal,
118Œ˚eld,
137vectorderivative,
27velocity
Œ,linearandangular,
105versor,
133,173,443Œ,fordilation,
179Œ,forinversion,
175Œ,fortransversion,
176Œ,representation,
173vertex,
180visualinvariants,
134visuallandmarkconstruction,
399voting
Œ,ball˚eld,
414Œ,sparsestick,
416voting˚elds,
411Voting˚eldsin3D,
410Wwavelet
Œ,3-DimensionalCliffordTransform,
233Œ,Cliffordtransform,
233Œ,Euclidean,
235Œ,TridimensionalCliffordTransform,
232Œ,complextransform,
223Œ,continuousconformalgeometricalgebra
transform,
234Œ,mother,
220Œ,n-DimensionalCliffordTransform,
235Œ,quaterniontransform,
225Œ,quaternionicanalysis,
231Œ,quaternionicpyramid,
228,230Œ,spherical,
235


Page 625




622Index
waveletpyramid,
223wavelettransform
Œ,continuous,
220wedgeproduct,
9Weylrepresentations,
583WilliamK.Clifford,
575Wittebasis,
123Wolfedualprogramming,
290X
XORproblem,
425

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