Eduardo Bayro-Corrochano. Gerik Scheuermann. Editors. Geometric. Algebra. Computing. in Engineering. and Computer Science.

Transcription

1 Eduardo Bayro-Corrochano Gerik Scheuermann Editors Geometric Algebra Computing in Engineering and Computer Science & Springer

2 Contents Part I Geometric Algebra New Tools for Computational Geometry and Rejuvenation of Screw Theory 3 David Hestenes 1 Introduction 3 2 Universal Geometric Algebra 4 3 Group Theory with Geometric Algebra 6 4 Euclidean Geometry with Conformal GA 8 5 Invariant Euclidean Geometry 10 6 Projective Geometry 13 7 Covariant Euclidean Geometry with Conformal Splits 14 8 Rigid Displacements 18 9 Framing a Rigid Body Rigid Body Kinematics Rigid Body Dynamics Screw Theory Conformal Split and Matrix Representation Linked Rigid Bodies & Robotics 31 References 33 Tutorial: Structure-Preserving Representation of Euclidean Motions Through Conformal Geometric Algebra 35 Leo Dorst 1 Introduction 36 2 Conformal Geometric Algebra Trick 1: Representing Euclidean Points in Minkowski Space Trick 2: Orthogonal Transformations as Multiple Reflections in a Sandwiching Representation Trick 3: Constructing Elements by Anti-Symmetry Trick 4: Dual Specification of Elements Permits Intersection 43

3 X Contents 3 Bonus: The Elements of Euclidean Geometry as Blades 45 4 Bonus: Euclidean Motions Through Sandwiching 46 5 Bonus: Structure Preservation and the Transfer Principle 47 6 Trick 5: Exponential Representation of Versors 49 7 Trick 6: Sparse Implementation at Compiler Level 51 References 52 Engineering Graphics in Geometric Algebra 53 Alyn Rockwood and Dietmar Hildenbrand 1 Introduction Benefits of Geometric Algebra for Computational Engineering 2.1 Unification of Mathematical Systems Uniform Handling of Different Geometric Primitives. 2.3 Simplified Rigid Body Motion Curl, Vorticity and Rotation More Efficient Implementations 55 3 Some Applications 55 4 The Geometric Primitives in More Detail Planes as a Limit of Spheres Distances Based on the Inner Product Approximation of Points with the Help of Planes or Spheres 62 5 Computational Efficiency of Geometric Algebra using Gaalop Conclusion 67 References Parameterization of 3D Conformal Transformations in Conformal Geometric Algebra 71 Hongbo Li 1 Terminology and Notations 71 2 Exponential Map and Exterior Exponential Map 74 3 Twisted Vahlen Matrices and Quaternionic Vahlen Matrices 77 4 Cayley Transform 85 5 Conclusion 90 References 90 Part II Clifford Fourier Transform Two-Dimensional Clifford Windowed Fourier Transform 93 Mawardi Bahri, Eckhard M.S. Hitzer, and Sriwulan Adji 1 Introduction 93 2 Real Clifford Algebra $ Clifford Fourier Transform (CFT) D Clifford Windowed Fourier Transform Definition of the CWFT Properties of the CWFT Examples of the CWFT 103

4 Contents *i 5 Comparison of CFT and CWFT Conclusion 105 References 106 The Cylindrical Fourier Transform 107 Fred Brackx, Nele De Schepper, and Frank Sommen 1 Introduction The Clifford Analysis Toolkit The Clifford-Fourier Transform The Cylindrical Fourier Transform Ill 4.1 Definition Ill 4.2 Properties Spectrum of the,2-basis Consisting of Generalized Clifford-Hermite Functions Application Potential of the Cylindrical Fourier Transform Conclusion 118 References 119 Analyzing Real Vector Fields with Clifford Convolution and Clifford-Fourier Transform 121 Wieland Reich and Gerik Scheuermann 1 Fluid Flow Analysis Geometric Algebra Clifford Convolution Clifford-Fourier Transform Relation to Other Fourier Transforms H-Holomorphic Functions and Fourier Series Biquaternion Fourier Transform Two-Sided Quaternion Fourier Transform Conclusion 132 References, Clifford-Fourier Transform for Color Image Processing 135 Thomas Batard, Michel Berthier, and Christophe Saint-Jean 1 Introduction Fourier Transform and Group Actions Clifford-Fourier Transform in L2(R2, (R", Q)) TheCasesn = 3,4:GroupMorphismsfromK2toSpin(4) The Cases n = 3,4: The Clifford-Fourier Transform Application to Color Image Filtering Clifford-Fourier Transform of Color Images Color Image Filtering Related Works The Hypercomplex Fourier Transform of Sangwine et al. 5.2 The Quaternionic Fourier Transform of Biilow Conclusion 155

5 xii Contents Appendix 155 A.l Lie Groups Representations and Fourier Transforms A.2 Clifford Algebras 158 A.3 The Spinor Group Spin(n) 160 References 161 Hilbert Transforms in Clifford Analysis 163 Fred Brackx, Bram De Knock, and Hennie De Schepper 1 Introduction: The Hilbert Transform on the Real Line Hilbert Transforms in Euclidean Space Definition and Properties Analytic Signals Monogenic Extensions of Analytic Signals Example Example Generalized Hilbert Transforms in Euclidean Space First generalization Second Generalization The Anisotropic Hilbert Transform Definition of the Anisotropic Hilbert Transform Properties of the Anisotropic Hilbert Transform Example Conclusion 185 References 186 Part III Image Processing, Wavelets and Neurocomputing Geometric Neural Computing for 2D Contour and 3D Surface Reconstruction 191 Jorge Rivera-Rovelo, Eduardo Bayro-Corrochano, and Ruediger Dillmann 1 Introduction Geometric Algebra The OPNS and IPNS Conformal Geometric Algebra Rigid Body Motion Determining the Shape of an Object Automatic Samples Selection Using GGVF Learning the Shape Using Versors Experiments Conclusion 208 References 209 Geometric Associative Memories and Their Applications to Pattern Classification 211 Benjamin Cruz, Ricardo Barron, and Humberto Sossa 1 Introduction Classic Associative Memory Models 212

6 Contents xiii 2 Basics of Conformal Geometric Algebra Geometric Algebra Classification Models Geometric Associative Memories Creating Spheres Pattern Learning and Classification Conditions for Perfect Classification Conditions for Robust Classification Numerical Examples Real Examples Conclusions and Future Work 228 References Classification and Clustering of Spatial Patterns with Geometric Algebra 231 Minh Tuan Pham, Kanta Tachibana, Eckhard M.S. Hitzer, Tomohiro Yoshikawa, and Takeshi Furuhashi 1 Introduction Method Feature Extraction for Geometric Data Distribution Learning and Its Mixture for Classification 2.3 GA Kernel and Alignment and Semi-Supervised Learning for Clustering Experimental Results and Discussion Classification of Handwritten Digits Kernel Alignment and Web Questionnaire Analysis Results Conclusions 244 References 246 QWT: Retrospective and New Applications 249 Yi Xu, Xiaokang Yang, Li Song, Leonardo Traversoni, and Wei Lu 1 Introduction Evolution of Qwt and Principles of Quaternion Wavelet Construction Evolution of QWT Principles of Quaternion Wavelet Construction The Mechanism of Adaptive Scale Representation in QWT The Potential Use of QWT in Image Registration The Potential Use of QWT in Image Fusion The Potential Use of QWT in Color Image Recognition Conclusion 272 References 272

7 xiv Contents Part IV Computer Vision Image Sensor Model Using Geometric Algebra: From Calibration to Motion Estimation 277 Thibaud Debaecker, Ryad Benosman, and Sio H. Ieng 1 Introduction Introduction to Conformal Geometric Algebra Geometric Algebras Conformal Geometric Algebra (CGA) General Model of a Cone-Pixels Camera Geometric Settings The General Model of a Central Cone-Pixel Camera Intersection of Cones General Cone-Pixel Camera Calibration Experimental Protocol Calibration Experimental Results Motion Estimation Problem Formulation Cone Intersection Score Functions Simulation Experiments for Motion Estimation Using Cone Intersection Criterion Conclusion and Future Works 296 References 296 Model-Based Visual Self-localization Using Gaussian Spheres 299 David Gonzalez-Aguirre, Tamim Asfour, Eduardo Bayro-Corrochano, and Ruediger Dillmann 1 Motivation Outline of Visual Self-localization Visual Acquisition of Landmarks Data Association for Model Matching Pose-Estimation Optimization Uncertainty Image-to-Space Uncertainty Space-to-Ego Uncertainty Geometry and Uncertainty Model Gaussian Spheres Radial Space Restriction Lines Restriction Hyperplanes Duality and Uniqueness Conclusion 323 References 324

8 Contents xv Part V Conformal Mapping and Fluid Analysis Geometric Characterization of M-Conformal Mappings 327 K. Giirlebeck and J. Morais 1 Introduction Preliminaries Monogenic-Conforrnal Mappings and Their Relation to Monogenic Functions Influence of the Linear Part of a Monogenic Function Observations and Perspectives 340 References 342 Fluid Flow Problems with Quaternionic Analysis An Alternative Conception 345 K. Giirlebeck and W. SproBig 1 Introduction Operator Calculus Operator Triple Plemelj-Type Projections Examples of L-Holomorphy Quaternionic Analysis Bergman-Hodge Decomposition Quaternionic Operator Calculus Discrete Quaternionic Analysis Fluid Flow Problems A Brief History of Fluid Dynamics Stationary Linear Stokes Problem Nonlinear Stokes Problem Stationary Navier-Stokes Problem Navier-Stokes Equations with Heat Conduction Continuous and Discrete Teodorescu Transforms Discrete Version of Navier-Stokes Equations Stationary Magneto-Hydromechanics Time-Dependent Fluid Flow Problems Characterization of Fluids Rothe's Method of Semi-Discretization Time-Dependent Stokes Problem Oseen's Equation A Special Discretization Method (D + i'a)-holomorphic Functions Approximation and Stability Approximation Property Stability Representation Formulae 374

9 xvi Contents.. 7 More Relevant Problems in Fluid Dynamics Magnetic Benard's Problem Boussinesq's Formulation of Poisson-Stokes' Problem Shallow Water Equations Forecasting Equations Numerical Examples Conclusions 380 References 380 Part VI Crystallography, Holography and Complexity Interactive 3D Space Group Visualization with CLUCalc and Crystallographic Subperiodic Groups in Geometric Algebra 385 Eclchard M.S. Hitzer, Christian Perwass, and Daisuke Ichikawa 1 Introduction Point Groups and Space Groups in Clifford Geometric Algebra Cartan-Dieudonne and Geometric Algebra Two-Dimensional Point Groups Three-Dimensional Point Groups Space Groups Interactive Software Implementation The Space Group Visualizer GUI Space Group and Symmetry Selection Mouse Pointer Interactivity Visualization Options in Detail Integration with the Online International Tables of Crystallography Subperiodic Groups Represented in Clifford Geometric. Algebra 4.1 Frieze Groups Rod Groups Layer Groups Conclusion 397 References 399 Geometric Algebra Model of Distributed Representations 401 Agnieszka Patyk 1 Introduction Geometric Algebra Model Recognition Right-Hand-Side Questions Appropriate-Hand-Side Reversed Questions Other Measures of Similarity Matrix Representation The Hamming Measure of Similarity The Euclidean Measure of Similarity Performance of Hamming and Euclidean Measures

10 Contents xvii 5 The Average Number of Potential Answers Comparison with Previously Developed Models Conclusion 429 References 429 Computational Complexity Reductions Using Clifford Algebras 431 Rene Schott and G. Stacey Staples 1 Introduction Preliminaries Graph Preliminaries Complexity Reduction for Graph Problems: Nilpotent Adjacency Matrix Approach Matrix-Free Approach to Representing Graphs Conclusion 451 References 452 Part VII Efficient Computing with Clifford (Geometric) Algebra. Efficient Algorithms for Factorization and Join of Blades 457 Daniel Fontijne and Leo Dorst 1 Introduction Blade Factorization New Algorithm for Blade Factorization Algorithms for Computing the Join of Blades Fast Join Algorithm Computational Example Grade Stability of Fast Join Algorithm Improved Fast Join Algorithm Numerical Stability of the Fast Join Algorithms Implementation Code Generation Implementation of the Fast Factorization Algorithm Implementation of the Fast Join Algorithm Implementation of the Delta Product Benchmarks Discussion Fast Factorization Algorithm FasJoin Algorithms Simultaneous Computation of Meet and Join Costs More Conclusion 475 References 476 Gaalop High Performance Parallel Computing Based on Conformal Geometric Algebra 477 Dietmar Hildenbrand, Joachim Pitt, and Andreas Koch 1 Introduction 477

11 xviii Contents 2 Related Work Software Implementations Hardware Implementations Our Proof-of-Concept Approach Conformal Geometric Algebra Concepts Symbolic Optimization Use of Inherent Fine-Grained Parallel Structure The Architecture of Gaalop Inverse Kinematics of the Leg of a Humanoid Robot Solving the Inverse Kinematics Algorithm Symbolic Optimization of the Kinematic Chain Current Status and Future Perspectives Conclusion 494 References 494 Some Applications of Grobner Bases in Robotics and Engineering Rafat Ablamowicz 1 Introduction Grobner Basis Theory in Polynomial Rings Examples of Using Grobner Bases Fermat Curves and B6zier Cubics Fermat Curves Bezier Cubics Conclusions 516 References 516 Index 519