Linux Pocket Guide: Essential Commands

Preface
Acknowledgments
How to Use This Book
	Chapter Organization
	Interdependence of the Book Chapters
	Audience
	Exercises
	Use of Computer Geometric Algebra Programs
	Use as a Textbook
Contents
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 Reciprocal Frames with Curvilinear Coordinates
		1.2.4 Some Useful Formulas
	1.3 Multivector Products
		1.3.1 Further Properties of the Geometric Product
		1.3.2 Projections and Rejections
		1.3.3 Projective Split
		1.3.4 Generalized Inner Product
		1.3.5 Geometric Product of Multivectors
		1.3.6 Contractions and the Derivation
		1.3.7 Hodge Dual
		1.3.8 Dual Blades and Duality in the Geometric Product
	1.4 Multivector Operations
		1.4.1 Involution, Reversion, and Conjugation Operations
		1.4.2 Join and Meet Operations
		1.4.3 Multivector-Valued Functions and the Inner Product
		1.4.4 The Multivector Integral
		1.4.5 Convolution and Correlation of Scalar Fields
		1.4.6 Clifford Convolution and Correlation
	1.5 Linear Algebra
		1.5.1 Linear Algebra Derivations
	1.6 Simplexes
	1.7 Exercises
	References
2 2D, 3D, and 4D Geometric Algebras
	2.1 Complex, Double, and Dual Numbers
	2.2 2D Geometric Algebras of the Plane
	2.3 3D Geometric Algebra for the Euclidean 3D Space
		2.3.1 The Algebra of Rotors
		2.3.2 Orthogonal Rotors
		2.3.3 Recovering a Rotor
	2.4 Quaternion Algebra
		2.4.1 Split Quaternion Algebra
	2.5 4D Geometric Algebra for 3D Kinematics
		2.5.1 Motor Algebra
		2.5.2 Motors, Rotors, and Translators in G+3,0,1
		2.5.3 Properties of Motors
		2.5.4 The Klein Manifold
		2.5.5 Reciprocal Screws
		2.5.6 The Study Manifold
	2.6 4D Geometric Algebra for Projective 3D Space
	2.7 Conclusion
	2.8 Exercises
	References
3 Kinematics of the 2D and 3D Spaces
	3.1 Introduction
	3.2 Representation of Points, Lines, and Planes Using 3D Geometric Algebra
	3.3 Representation of Points, Lines, and Planes Using Motor Algebra
	3.4 Representation of Points, Lines, and Planes Using 4D Geometric Algebra
	3.5 Motion of Points, Lines, and Planes in 3D Geometric Algebra
	3.6 Motion of Points, Lines, and Planes Using Motor Algebra
	3.7 Motion of Points, Lines, and Planes Using 4D Geometric Algebra
	3.8 Spatial Velocity of Points, Lines, and Planes
		3.8.1 Rigid-Body Spatial Velocity Using Matrices
		3.8.2 Angular Velocity Using Rotors
		3.8.3 Rigid-Body Spatial Velocity Using Motor Algebra
		3.8.4 Point, Line, and Plane Spatial Velocities Using Motor Algebra
	3.9 Differential Kinematics
	3.10 Incidence Relations Between Points, Lines, and Planes
		3.10.1 Flags of Points, Lines, and Planes
	3.11 Conclusion
	3.12 Exercises
	References
4 Conformal Geometric Algebra
	4.1 Introduction
		4.1.1 Conformal Split
		4.1.2 Conformal Splits for Points and Simplexes
		4.1.3 Euclidean and Conformal Spaces
		4.1.4 Stereographic Projection
		4.1.5 Inner and Outer Product Null Spaces
		4.1.6 Spheres and Planes
		4.1.7 Geometric Identities, Dual, Meet, and Join Operations
		4.1.8 Simplexes and Spheres
	4.2 The 3D Affine Plane
		4.2.1 Lines and Planes
		4.2.2 Directed Distance
	4.3 The Lie Algebra
	4.4 Conformal Transformations
		4.4.1 Inversion
		4.4.2 Reflection
		4.4.3 Translation
		4.4.4 Transversion
		4.4.5 Rotation
		4.4.6 Rigid Motion Using Flags
		4.4.7 Dilation
		4.4.8 Involution
		4.4.9 Conformal Transformation
	4.5 Ruled Surfaces
		4.5.1 Cone and Conics
		4.5.2 Cycloidal Curves
		4.5.3 Helicoid
		4.5.4 Sphere and Cone
		4.5.5 Hyperboloid, Ellipsoids, and Conoid
	4.6 Exercises
	References
5 Incidence Algebra Using Conformal Geometric Algebra
	5.1 Conformal Geometric Algebra
	5.2 Directed Distance
		5.2.1 Point to Point
		5.2.2 Point to Line: Method 1
		5.2.3 Point to Line: Method 2
		5.2.4 Point to Sphere
		5.2.5 Point to Plane
		5.2.6 Line to Line
		5.2.7 Line to Line: Geometric Method
		5.2.8 Line to Plane
		5.2.9 Sphere to Line
		5.2.10 Sphere to Plane: Method 1
		5.2.11 Sphere to Plane: Method 2
		5.2.12 Sphere to Sphere
		5.2.13 Plane to Plane
		5.2.14 Circle to Point
		5.2.15 Circle to Circle
		5.2.16 Circle to Plane
		5.2.17 Circle to Line
		5.2.18 Sphere to Circle
	5.3 Intersection of Geometric Entities
		5.3.1 Circle–Circle Intersection
		5.3.2 Circle–Line Intersection
		5.3.3 Line–Line Intersections
		5.3.4 Plane–Circle Intersections
		5.3.5 Plane–Line Intersections
		5.3.6 Plane Intersection
		5.3.7 Sphere–Circle Intersection
		5.3.8 Sphere–Line Intersections
		5.3.9 Sphere–Plane Intersections
		5.3.10 Sphere–Sphere Intersection
	Reference
6 The Geometric Algebras G6,0,2+, G6,3, G9,3+, G6,0,6+
	6.1 Introduction
	6.2 The Double Motor Algebra G6,0,2+
		6.2.1 The Shuffle Product
		6.2.2 Equations of Motion
	6.3 The Geometric Algebra G6,3
		6.3.1 Additive Split of G6,3
		6.3.2 Geometric Entities of G6,3
		6.3.3 Intersection of Surfaces
		6.3.4 Transformations of G6,3
	6.4 The Geometric Subalgebras G9,3+ and G6,0,6+
	6.5 Exercises
	References
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
		7.2.7 Specialized Hardware to Speed Up Geometric Algebra Algorithms
	References
Part II Information Theory, Machine Learning and Quantum Computing
8 Information Theory
	8.1 Classical Information Theory
		8.1.1 Shannon's Entropy
		8.1.2 Joint Entropy
		8.1.3 Conditional Entropy
		8.1.4 Mutual Information
	8.2 Quantum Information Theory
		8.2.1 Bloch Sphere
		8.2.2 Quantum State (Wave Function)
		8.2.3 Quantum Measurement
		8.2.4 Quantum Entanglement
		8.2.5 Quantum Gates and Operations
		8.2.6 Quantum Superposition
		8.2.7 Quantum No-Cloning Theorem
		8.2.8 Quantum Teleportation
	8.3 Examples of Quantum Computing
	8.4 Quantum Computing
		8.4.1 The Hardware of Quantum Computers
		8.4.2 Quantum Computer Software Architecture
	References
9 Integral Transforms
	9.1 Introduction
		9.1.1 Integral Transform
		9.1.2 Fourier Transform
		9.1.3 The Kernel and the Generalization of the Fourier Transform
		9.1.4 The Inverse Fourier Transform
	9.2 Quaternion and Clifford Fourier Transforms
		9.2.1 The One-Dimensional Fourier Transform
		9.2.2 The Two-Dimensional Fourier Transform
		9.2.3 Fourier Phase, Instantaneous Phase, and Local Phase
		9.2.4 Quaternionic Fourier Transform
		9.2.5 2D Analytic Signals
		9.2.6 Properties of the QFT
		9.2.7 Discrete QFT
	9.3 Quaternion Split Fast Fourier Transform
	9.4 Gabor Filters and Atomic Functions
		9.4.1 2D Gabor Filters
		9.4.2 Atomic Functions
		9.4.3 The dup(x)
		9.4.4 Quaternion Atomic Function Qup(x)
	9.5 Quaternionic Analytic Signal, Monogenic Signal, Hilbert …
		9.5.1 Local Phase Information
		9.5.2 Quaternionic Analytic Signal
		9.5.3 Monogenic Signal
		9.5.4 Hilbert Transform Using AF
		9.5.5 Riesz Transform Using AF
	9.6 Clifford Fourier Transforms
		9.6.1 Tri-dimensional Clifford Fourier Transform
		9.6.2 Space and Time Geometric Algebra Fourier Transform
		9.6.3 n-Dimensional Clifford Fourier Transform
	9.7 From Real to Clifford Wavelet Transforms for Multiresolution Analysis
		9.7.1 Real Wavelet Transform
		9.7.2 Discrete Wavelets
		9.7.3 Wavelet Pyramid
		9.7.4 Complex Wavelet Transform
		9.7.5 Quaternion Wavelet Transform
		9.7.6 Quaternionic Wavelet Pyramid
		9.7.7 The Tri-dimensional Clifford Wavelet Transform
		9.7.8 The Continuous Conformal Geometric Algebra Wavelet Transform
		9.7.9 The n-Dimensional Clifford Wavelet Transform
	9.8 Quaternion Quantum Fourier Transform
		9.8.1 Quantum Fourier Transform
		9.8.2 Quaternion Quantum Fourier Transform
	9.9 Radon Transform of Functionals
	9.10 3D Radon Transform
		9.10.1 Getting 3D Radon Transform from Cone Beam Data
	9.11 The Spherical Radon Transform
	9.12 Conclusion
	References
10 Hough Transform and Geometric Algebra
	10.1 The Classic 2D Hough Transform
	10.2 The Hough Transform in 3D
	10.3 The Hough Transform and Geometric Conformal Algebra
	10.4 Representation of 3D Objects in Hough Domain
	10.5 Perception of the Environment
	10.6 Variation of Only One Angle in Objects
	10.7 Conclusion
	References
11 Color Image Processing Using Geometric Algebra
	11.1 Introduction
		11.1.1 Properties of the Multivectors
		11.1.2 Quaternion Algebra
		11.1.3 Split Quaternion Algebra
		11.1.4 Rotors in G3
		11.1.5 Motors in G3,1
		11.1.6 Motors in G3,0,1+
		11.1.7 Split Rotors and Split Motors in Conformal Geometric Algebra A Motor Belongs to mathcalG+3,0,1
	11.2 The Representational Viewpoint in Color Theory
		11.2.1 Color Models
		11.2.2 Grassman's Laws
		11.2.3 Representation of Grassmann Structures Using Quaternion Algebra mathbbH for the Color Model RGB
		11.2.4 Representation of Grassmann Structures Using Quaternion Split Algebra mathbbHS for the Color Model HSV
	References
12 Geometric Neural Computing
	12.1 Introduction
	12.2 Real-Valued Neural Networks
	12.3 Complex MLP and Quaternionic MLP
	12.4 Quaternion Neural Networks
	12.5 Matrix Representation of Split Quaternions
		12.5.1 The Extended Kalman Filter Procedure
		12.5.2 Learning Algorithm
	12.6 Geometric Algebra Neural Networks
		12.6.1 The Activation Function
		12.6.2 The Geometric Neuron
		12.6.3 Feedforward Geometric Neural Networks
		12.6.4 Generalized Geometric Neural Networks
		12.6.5 The Learning Rule
		12.6.6 Multidimensional Back-Propagation Training Rule
		12.6.7 Simplification of the Learning Rule Using the Density Theorem
		12.6.8 Learning Using the Appropriate Geometric Algebras
	12.7 Geometric Radial Basis Function Networks
	12.8 Support Vector Machines in Geometric Algebra
	12.9 Linear Clifford Support Vector Machines for Classification
	12.10 Nonlinear Clifford Support Vector Machines for Classification
	12.11 Clifford SVM for Regression
	12.12 Conclusion
	References
13 Neurocontrol
	13.1 Quaternionic Spike Neural Networks
		13.1.1 Threshold and Firing Models
		13.1.2 Perfect Integrate and Fire Model
		13.1.3 Learning Method
		13.1.4 Error-Backpropagation
		13.1.5 Quaternion Spike Neural Networks
		13.1.6 Comparison SNN Against QSNN
		13.1.7 Kinematic Control of a Manipulator of 6 DOF Using QSNN
	13.2 Quaternion Wavelet Neural Network
		13.2.1 Rotation with Quaternions
		13.2.2 Jacobian with Quaternions
		13.2.3 Quaternion Wavelet Neural Network
		13.2.4 Design QWNN
		13.2.5 Adaptive PID Controller
	13.3 Conclusion
	References
14 Deep Learning Using Geometric Algebra
	14.1 Deep Learning
	14.2 Quaternionic Convolutional Neural Network
		14.2.1 Fully Connected Layer
		14.2.2 Correlation
		14.2.3 Convolutional Layer
		14.2.4 Quaternion Convolution in Terms of Kernels
		14.2.5 Quaternion Weight Initialization
	14.3 Geometric (Clifford) Algebra CNN
		14.3.1 Clifford Fully Connected and Convolutional Layers
		14.3.2 Clifford Cross-correlation and Convolution
		14.3.3 Clifford Weight Initialization
	14.4 Conclusion
	References
15 Geometric Quantum Computing
	15.1 Quantum Computing
		15.1.1 Multiparticle Quantum Theory in Geometric Algebra
		15.1.2 Quantum Bits in Geometric Algebra
		15.1.3 A Spinor–Quaternion Map
		15.1.4 Quantum Bit Operators Action and Observables in Geometric Algebra
		15.1.5 Measurement of Probabilities in Geometric Algebra
		15.1.6 The 2-Qubit Space–Time Algebra
		15.1.7 Gates in Geometric Algebra
		15.1.8 Two-Qubit Quantum Computing
		15.1.9 Quaternion–Quantum Neural Computing in Geometric Algebra
	15.2 Clifford Group a Set of Quantum Computing Operations
	15.3 Quaternionic–Quantum Neural Network
		15.3.1 Quaternionic Qubit
		15.3.2 Architecture of the Quaternion–Quantum Neural Network
		15.3.3 One-Hot Encoding for the QQNN
		15.3.4 Feed-Forward Phase
		15.3.5 Update Phases
		15.3.6 Learning Rate Schedule
		15.3.7 Activation Operators
	References
Part III Applications of Integral Transforms, and Geometric Methods in Computer Vision
16 Applications of Quaternion Fourier and Wavelet Transforms, and Radon Transform
	16.1 Representation of Speech as 2D Signals
	16.2 Preprocessing of Speech 2D Representations Using QFT …
		16.2.1 Method 1
		16.2.2 Method 2
	16.3 Recognition of French Phonemes Using Neurocomputing
	16.4 Application of QWT
		16.4.1 Estimation of the Quaternionic Phase
		16.4.2 Confidence Interval
		16.4.3 Discussion on the Similarity Distance and the Phase Concept
		16.4.4 Optical Flow Estimation
	16.5 Riesz Transform and Multi-resolution Processing
		16.5.1 Gabor, Log-Gabor, and Atomic Function Filters Using Riesz Transform in Multi-resolution Pyramid
		16.5.2 Multi-resolution Analysis Using the Quaternion Wavelet Atomic Function
		16.5.3 Radon Transform for Circle Detection by Color Images Using the Quaternion Atomic Functions
		16.5.4 Feature Extraction Using Symmetry
	16.6 Conclusion
	References
17 Applications of Color Image Processing Using Geometric Algebra
	17.1 Color Image Processing Using the Quaternion Split Fast Fourier Transform
		17.1.1 Quaternion Split Fast Fourier Transform
		17.1.2 Symplectic Form of the Quaternion
		17.1.3 The Fourier Transform for Symplectic Form of the Quaternion
		17.1.4 Constructing the Symplectic Form of an Image
		17.1.5 Implementation of the Fourier Transform for Symplectic Form of the Quaternion
	17.2 Interpolation Using Split Motors of the Conformal Geometric Algebra G4,1
	17.3 Split Quaternion Neural Network Using the HSV for Model …
		17.3.1 Dataset for Training
		17.3.2 Cost Function for Training
		17.3.3 A Particular Image for Testing
	17.4 Color Image Enhancement Using the Quaternion Split Neural Network
	References
18 Applications of Incidence Algebra and Hough Transform
	18.1 Application of Incidence Algebra
		18.1.1 Inverse Kinematics Using Geometric Primitives and Geometric Constraints
		18.1.2 Interpolation of Geometric Entities
		18.1.3 Procedures for Kidney Surgery
		18.1.4 Interpolation of Surgery Motion
	18.2 Application of Hough Transform in Conformal Geometric Algebra
		18.2.1 Randomized Hough Transform
		18.2.2 CGA Representation of Lines, Circles, Planes, and Spheres
		18.2.3 Conformal Geometric Hough Transform
		18.2.4 Detection of Lines, Circles, Planes, and Spheres Using the Conformal Geometric Hough Transform
	18.3 Relocation Using Lines and the 2D Hough Transform
	18.4 Recognizing Objects for Robot Localization Using the 3D the Hough Transform
	18.5 Experiments Using 3D the Hough Transform
		18.5.1 3D Shape Perception
	18.6 Conclusion
	References
Part IV Applications of Neurocomputing, CNN Deep-Learning and Quantum Computing
19 Applications in Neuralcomputing
	19.1 Experiments Using Geometric Feedforward Neural Networks
		19.1.1 Learning a High Nonlinear Mapping
		19.1.2 Encoder–Decoder Problem
		19.1.3 Prediction
	19.2 Recognition of 2D and 3D Low-Level Structures
		19.2.1 Recognition of 2D Low-Level Structures
		19.2.2 Recognition of 3D Low-Level Structures
		19.2.3 Solving the Hand–Eye Calibration Problem Using the GRFB Network
	19.3 Experiments Using Clifford Support Vector Machines
		19.3.1 3D Spiral: Nonlinear Classification Problem
		19.3.2 Object Recognition
		19.3.3 Multi-case Interpolation
	19.4 Conclusion
	References
20 Robot Neurocontrol
	20.1 Quaternion Spiking Neural Networks
		20.1.1 Neuronal Model
		20.1.2 Neuronal Architecture
	20.2 Training Algorithm
		20.2.1 Rotation with Quaternions
		20.2.2 Quaternion Spike Neural Network
	20.3 Control of a Simulated Nonlinear System
	20.4 Control of a Real Nonlinear System
		20.4.1 Forward Kinematics
		20.4.2 The Jacobian
		20.4.3 Inverse Kinematics
		20.4.4 Control
	20.5 Neural Network Signal Processing
		20.5.1 Myo Bracelet
		20.5.2 Physiology of Robotic Prosthesis
		20.5.3 Preprocessing and Training
		20.5.4 Evaluation and Control
	20.6 Control of a Hand Prosthesis
		20.6.1 Complete Description of the Identification of Signals
		20.6.2 Final Check
	20.7 Experimental Results Using the Quaternion Wavelet Neural Network
	20.8 Conclusions
	References
21 Applications of Quantum Computing and Geometric Algebra Convolutional Neural Networks
	21.1 CoCoQNN
	21.2 Experimental Results
		21.2.1 CIFAR10: Reconstruction and Upsampling
		21.2.2 Set5: Reconstruction and Upsampling
		21.2.3 Retinal OCT Denoising
	21.3 Quanvolution
	21.4 Architecture of the Quaternion and Geometric Clifford …
		21.4.1 Experimental Results
		21.4.2 Colorectal Histology MNIST Dataset
		21.4.3 Covid-Qu-Ex Dataset
	21.5 Conclusions
	References
22 Applications of Geometric Quantum Computing
	22.1 Quaternion Qubit Neural Network
		22.1.1 Quaternion Qubit Neural Network
		22.1.2 Qubit Neuron Model
		22.1.3 Quaternion Qubit Neural Network
		22.1.4 Learning Rule for the Quaternion Qubit Neural Network
	22.2 Classification Experiments of the Quaternion Quantum Neural Network
		22.2.1 QQNN Training and Testing Algorithms
		22.2.2 Classification Using Quaternion Quantum Neural Network
	22.3 Quaternion Quantum Fourier Transform
	22.4 Application of the Quaternion Quantum Fast Fourier Transform
		22.4.1 Quaternion Quantum Image
		22.4.2 Quantum Quaternion Fast Fourier Transform
	22.5 Quantum Image Processing
		22.5.1 Quantum Edge Detection
		22.5.2 Quantum Adaptive Median Filtering
	22.6 Conclusion
	References
Appendix  Appendix
A.1  Table of Notation
A.2  Useful Formulas for Geometric Algebra
A.3  Matrix Representation of Split Quaternions
A.4  Derivatives of SQNN for EKF Algorithm
A.5  Appendix for the Quaternion Quantum Neural Network
A.5.1 Component-Wise Sigmoid with Binary Cross-Entropy Loss
A.5.2 Component-Wise Softmax with Categorical Cross-Entropy Loss
	References
Index