Geometric Computing: for Wavelet Transforms, Robot Vision, Learning, Control and Action

Cover......Page 1
Geometric Computing\r......Page 2
ISBN 1848829280......Page 4
Foreword......Page 6
Preface......Page 7
Part I Fundamentals of Geometric Algebra......Page 27
1.1 History of Geometric Algebra......Page 28
1.2.1 Basic Definitions......Page 30
1.2.2 Nonorthonormal Frames and Reciprocal Frames......Page 32
1.2.4 Multivector Products......Page 33
1.2.5 Further Properties of the Geometric Product......Page 36
1.2.6 Dual Blades and Duality in the Geometric Product......Page 44
1.2.7 Multivector Operations......Page 45
1.3 Linear Algebra......Page 47
1.4 Simplexes......Page 48
1.5.1 Multivector-Valued Functions and the Inner Product......Page 50
1.5.2 The Multivector Integral......Page 51
1.5.4 Grad, Div, and Curl......Page 52
1.5.5 Multivector Fields......Page 53
1.5.7 Clifford Convolution and Correlation......Page 54
1.5.8 Linear Algebra Derivations......Page 55
1.5.10 Geometric Calculus in 2D......Page 56
1.5.11 Electromagnetism: The Maxwell Equations......Page 57
1.5.12 Spinors, Shrödinger Pauli, and Dirac Equations......Page 60
1.5.13 Spinor Operators......Page 62
1.6 Exercises......Page 64
2.1 The Roots of Geometry and Algebra......Page 69
2.2 Geometric Algebra: A Unified Mathematical Language......Page 71
2.3.1 Coordinate-Free Mathematical System......Page 72
2.3.2 Models for Euclidean and Pseudo-Euclidean Geometry......Page 73
2.3.3 Subspaces as Computing Elements......Page 74
2.3.5 Objects and Operators......Page 75
2.3.6 Extension of Linear Transformations......Page 76
2.3.8 Kinematics and Dynamics......Page 77
2.4 Solving Problems in Perception and Action Systems......Page 78
Part II Euclidean, Pseudo-Euclidean, Lie and Incidence Algebras,and Conformal Geometries......Page 84
3.1 Complex, Double, and Dual Numbers......Page 85
3.2 2D Geometric Algebras of the Plane......Page 86
3.3 3D Geometric Algebra for the Euclidean 3D Space......Page 88
3.3.1 The Algebra of Rotors......Page 89
3.3.2 Orthogonal Rotors......Page 92
3.3.3 Recovering a Rotor......Page 93
3.4 Quaternion Algebra......Page 94
3.5 Lie Algebras and Bivector Algebras......Page 96
3.5.1 Lie Group of Rotors......Page 97
3.5.2 Bivector Lie Algebra......Page 98
3.5.3 Complex Structures and Unitary Groups......Page 99
3.5.4 Hermitian Inner Product and Unitary Groups......Page 100
3.6.1 Motor Algebra......Page 102
3.6.2 Motors, Rotors, and Translators in G+3,0,1......Page 104
3.6.3 Properties of Motors......Page 107
3.7 4D Geometric Algebra for Projective 3D Space......Page 109
3.9 Exercises......Page 110
4.2 Representation of Points, Lines, and Planes Using 3D Geometric Algebra......Page 115
4.3 Representation of Points, Lines, and PlanesUsing Motor Algebra......Page 117
4.4 Representation of Points, Lines, and Planes Using 4D Geometric Algebra......Page 118
4.5 Motion of Points, Lines, and Planes in 3D Geometric Algebra......Page 119
4.6 Motion of Points, Lines, and Planes Using Motor Algebra......Page 120
4.7 Motion of Points, Lines, and Planes Using 4D Geometric Algebra......Page 122
4.8.1 Rigid-Body Spatial Velocity Using Matrices......Page 123
4.8.2 Angular Velocity Using Rotors......Page 127
4.8.3 Rigid-Body Spatial Velocity Using Motor Algebra......Page 130
4.8.4 Point, Line, and Plane Spatial Velocities Using Motor Algebra......Page 131
4.9 Incidence Relations Between Points, Lines, and Planes......Page 132
4.9.1 Flags of Points, Lines, and Planes......Page 133
4.11 Exercises......Page 134
5.1 Introduction......Page 138
5.2.1 Reciprocal Null Cones......Page 139
5.2.2 The Universal Geometric Algebra Gn,n......Page 140
5.2.3 The Lie Algebra of Null Spaces......Page 141
5.2.4 The Standard Bases of Gn,n......Page 143
5.2.5 Representations and Operations Using Bivector Matrices......Page 144
5.2.6 Bivector Representation of Linear Operators......Page 145
5.3 Horosphere and n-Dimensional Affine Plane......Page 146
5.4 The General Linear Group......Page 148
5.4.1 The General Linear Algebra gl(N) of the General Linear Lie Group GL(N)......Page 150
5.4.2 The Orthogonal Groups......Page 151
5.5 Computing Rigid Motion in the Affine Plane......Page 154
5.6 The Lie Algebra of the Affine Plane......Page 155
5.7 The Algebra of Incidence......Page 159
5.7.1 Incidence Relations in the Affine n-Plane......Page 161
5.7.2 Directed Distances......Page 162
5.7.3 Incidence Relations in the Affine 3-Plane......Page 163
5.8 Conclusion......Page 165
5.9 Exercises......Page 166
6.1 Introduction......Page 169
6.1.1 Conformal Split......Page 170
6.1.2 Conformal Splits for Points and Simplexes......Page 171
6.1.3 Euclidean and Conformal Spaces......Page 172
6.1.4 Stereographic Projection......Page 175
6.1.5 Inner- and Outer-Product Null Spaces......Page 177
6.1.6 Spheres and Planes......Page 178
6.1.7 Geometric Identities, Meet and Join Operations, Duals, and Flats......Page 180
6.1.8 Meet, Pair of Points, and Plunge......Page 186
6.1.9 Simplexes and Spheres......Page 188
6.2 The 3D Affine Plane......Page 189
6.2.1 Lines and Planes......Page 190
6.2.2 Directed Distance......Page 191
6.4 Conformal Transformations......Page 192
6.4.1 Inversion......Page 193
6.4.2 Reflection......Page 195
6.4.4 Transversion......Page 196
6.4.6 Rigid Motion Using Flags......Page 197
6.4.8 Involution......Page 199
6.5.1 Cone and Conics......Page 200
6.5.2 Cycloidal Curves......Page 201
6.5.4 Sphere and Cone......Page 202
6.6 Exercises......Page 203
7.1 Main Issues for an Efficient Implementation......Page 209
7.1.1 Specific Aspects for the Implementation......Page 210
7.2.2 The General Multivector Class......Page 211
7.2.3 Optimization of Multivector Functions......Page 212
7.2.4 Factorization......Page 213
7.2.5 Speeding Up Geometric Algebra Expressions......Page 214
7.2.6 Multivector Software Packets......Page 215
Part III Geometric Computing for Image Processing, Computer Vision, and Neurocomputing......Page 218
8.2 Image Analysis in the Frequency Domain......Page 219
8.2.1 The One-Dimensional Fourier Transform......Page 220
8.2.3 Quaternionic Fourier Transform......Page 221
8.2.4 2D Analytic Signals......Page 223
8.2.5 Properties of the QFT......Page 226
8.2.6 Discrete QFT......Page 229
8.3.1 2D Gabor Filters......Page 231
8.4 Clifford–Fourier Transforms......Page 232
8.4.1 Tri-Dimensional Clifford–Fourier Transform......Page 235
8.4.2 Space and Time Geometric AlgebraFourier Transform......Page 236
8.5 From Real to Clifford Wavelet Transforms for Multiresolution Analysis......Page 237
8.5.2 Discrete Wavelets......Page 238
8.5.4 Complex Wavelet Transform......Page 241
8.5.5 Quaternion Wavelet Transform......Page 243
8.5.6 Quaternionic Wavelet Pyramid......Page 247
8.5.7 The Tridimensional Clifford Wavelet Transform......Page 250
8.5.8 The Continuous Conformal Geometric Algebra Wavelet Transform......Page 252
8.5.9 The n-Dimensional Clifford Wavelet Transform......Page 253
8.6 Conclusion......Page 254
9.2 The Geometric Algebras of 3D and 4D Spaces......Page 255
9.2.1 3D Space and the 2D Image Plane......Page 256
9.2.3 A 4D Geometric Algebra for Projective Space......Page 258
9.2.4 Projective Transformations......Page 259
9.2.5 The Projective Split......Page 260
9.3 The Algebra of Incidence......Page 262
9.3.1 The Bracket......Page 263
9.3.2 The Duality Principle and Meet and Join Operations......Page 264
9.4 Algebra in Projective Space......Page 265
9.4.1 Intersection of a Line and a Plane......Page 266
9.4.2 Intersection of Two Planes......Page 267
9.4.4 Implementation of the Algebra......Page 268
9.5.1 The 1D Cross-Ratio......Page 269
9.5.2 2D Generalization of the Cross-Ratio......Page 271
9.5.3 3D Generalization of the Cross-Ratio......Page 272
9.6.1 Geometry of One View......Page 273
9.6.2 Geometry of Two Views......Page 277
9.6.3 Geometry of Three Views......Page 279
9.6.4 Geometry of n-Views......Page 281
9.7 Omnidirectional Vision......Page 282
9.7.1 Omnidirectional Vision and Geometric Algebra......Page 283
9.7.2 Point Projection......Page 284
9.7.3 Inverse Point Projection......Page 285
9.8 Invariants in the Conformal Space......Page 286
9.8.1 Invariants and Omnidirectional Vision......Page 287
9.8.2 Projective and Permutation p2-Invariants......Page 289
9.10 Exercises......Page 291
10.1 Introduction......Page 294
10.2 Real-Valued Neural Networks......Page 295
10.3 Complex MLP and Quaternionic MLP......Page 296
10.4.1 The Activation Function......Page 297
10.4.2 The Geometric Neuron......Page 298
10.4.3 Feedforward Geometric Neural Networks......Page 300
10.4.4 Generalized Geometric Neural Networks......Page 301
10.4.6 Multidimensional Back-Propagation Training Rule......Page 302
10.4.7 Simplification of the Learning Rule Using the Density Theorem......Page 303
10.4.8 Learning Using the Appropriate Geometric Algebras......Page 304
10.6 Linear Clifford Support Vector Machinesfor Classification......Page 305
10.7 Nonlinear Clifford Support Vector Machines For Classification......Page 309
10.8 Clifford SVM for Regression......Page 311
10.9 Conclusion......Page 313
Part IV Geometric Computing of Robot Kinematics and Dynamics......Page 314
11.2 Elementary Transformations of Robot Manipulators......Page 315
11.2.1 The Denavit–Hartenberg Parameterization......Page 316
11.2.2 Representations of Prismatic and Revolute Transformations......Page 317
11.2.3 Grasping by Using Constraint Equations......Page 320
11.3 Direct Kinematics of Robot Manipulators......Page 322
11.3.1 MAPLE Program for Motor Algebra Computations......Page 323
11.4 Inverse Kinematics of Robot ManipulatorsUsing Motor Algebra......Page 324
11.4.2 Computing 1, 2, and d3 Using a Point......Page 325
11.4.3 Computing 4 and 5 Using a Line......Page 328
11.4.4 Computing 6 Using a Plane Representation......Page 330
11.5 Inverse Kinematics Using the 3D Affine Plane......Page 331
11.6 Inverse Kinematic Using Conformal Geometric Algebra......Page 334
11.7 Conclusion......Page 338
12.2 Differential Kinematics......Page 340
12.3.1 Kinetic Energy......Page 343
12.3.3  Lagrange\'s Equations......Page 350
12.4.2 Computing G......Page 358
12.5 Conclusion......Page 359
Part V Applications I: Image Processing, Computer Vision,and Neurocomputing......Page 360
13.1 Lie Filters in the Affine Plane......Page 361
13.1.1 The Design of an Image Filter......Page 362
13.1.2 Recognition of Hand Gestures......Page 363
13.2 Representation of Speech as 2D Signals......Page 364
13.3.1 Method 1......Page 366
13.3.2 Method 2......Page 368
13.4 Recognition of French Phonemes Using Neurocomputing......Page 369
13.5 Application of QWT......Page 371
13.5.1 Estimation of the Quaternionic Phase......Page 372
13.5.2 Confidence Interval......Page 373
13.5.3 Discussion on Similarity Distance and the Phase Concept......Page 374
13.5.4 Optical Flow Estimation......Page 375
13.6 Conclusion......Page 379
14.1 Introduction......Page 380
14.2 Conics and Pascal\'s Theorem......Page 381
14.3 Computing Intrinsic Camera Parameters......Page 384
14.4 Projective Invariants......Page 385
14.4.1 The 1D Cross-Ratio......Page 386
14.4.2 2D Generalization of the Cross-Ratio......Page 387
14.4.3 3D Generalization of the Cross-Ratio......Page 389
14.4.4 Generation of 3D Projective Invariants......Page 390
14.5.1 Projective Invariants Using Two Views......Page 394
14.5.2 Projective Invariant of Points Using Three Uncalibrated Cameras......Page 396
14.5.3 Comparison of the Projective Invariants......Page 398
14.6.1 Parallel Orienting......Page 400
14.6.3 Grasping......Page 402
14.7 Camera Self-Localization......Page 403
14.8 Projective Depth......Page 404
14.9 Shape and Motion......Page 406
14.9.1 The Join-Image......Page 407
14.9.2 The SVD Method......Page 408
14.9.3 Completion of the 3D Shape Using Invariants......Page 409
14.10.1 Learning Phase......Page 411
14.10.2 Recognition Phase......Page 412
14.10.3 Omnidirectional Vision and Invariants for Robot Navigation......Page 413
14.10.5 Recognition Phase......Page 414
14.10.6 Quantitative Results......Page 415
14.11 Conclusions......Page 416
15.1 Problem Formulation......Page 418
15.1.1 The Geometric Constraint......Page 419
15.2.1 Tensor Representation in 3D......Page 422
15.2.2 Voting Fields in 3D......Page 423
15.2.3 Detection of 3D Surfaces......Page 427
15.2.4 Estimation of 3D Correspondences......Page 428
15.3.1 Correspondences Between 3D Pointsby Rigid Motion......Page 430
15.3.2 Multiple Overlapping Motions and Nonrigid Motion......Page 432
15.3.3 Extension to Nonrigid Motion......Page 433
15.4 Conclusions......Page 435
16.1.1 Learning a High Nonlinear Mapping......Page 437
16.1.2 Encoder–Decoder Problem......Page 438
16.1.3 Prediction......Page 440
16.2 Experiments Using Clifford Support Vector Machines......Page 441
16.2.1 3D Spiral: Nonlinear Classification Problem......Page 442
16.2.2 Object Recognition......Page 444
16.2.3 Multi-Case Interpolation......Page 452
16.3 Conclusion......Page 454
17.1 Determining the Shape of an Object......Page 455
17.1.1 Automatic Sample Selection Using GGVF......Page 456
17.1.2 Learning the Shape Using Versors......Page 458
17.2 Experiments......Page 460
17.3 Conclusion......Page 467
Part VI Applications II: Robotics and Medical Robotics......Page 468
18.2 Batch Estimation Using SVD Techniques......Page 469
18.2.1 Solving AX = XB Using Motor Algebra......Page 471
18.2.2 Estimation of the Hand–Eye Motor Using SVD......Page 474
18.3 Experimental Results......Page 476
18.5.1 The Kalman Filter......Page 480
18.5.2 The Extended Kalman Filter......Page 482
18.5.3 The Rotor-Extended Kalman Filter......Page 484
18.6 The Motor-Extended Kalman Filter......Page 487
18.6.1 Representation of the Line Motion Model in Linear Algebra......Page 488
18.6.2 Linearization of the Measurement Model......Page 490
18.6.3 Enforcing a Geometric Constraint......Page 491
18.6.4 Operation of the MEKF Algorithm......Page 493
18.6.5 Estimation of the Relative Positioning of a Robot End-Effector......Page 496
18.7 Conclusion......Page 500
19.1.1 Rigid Body Motion in CGA......Page 501
19.1.2 Hand–Eye Calibration in CGA......Page 503
19.1.3 Tracker Endoscope Calibration......Page 504
19.2 Body-Sensor Calibration......Page 507
19.2.1 Body–Eye Calibration......Page 508
19.2.2 Algorithm Simplification......Page 511
19.3 Conclusions......Page 513
20.1 Tracking......Page 514
20.1.1 Exact Linearization via Feedback......Page 515
20.1.2 Visual Jacobian......Page 517
20.1.3 Exact Linearization via Feedback......Page 518
20.1.4 Experimental Results......Page 519
20.2 Barrett Hand Direct Kinematics......Page 521
20.3 Pose Estimation......Page 523
20.3.1 Segmentation......Page 524
20.3.2 Object Projection......Page 525
20.4 Grasping Objects......Page 527
20.4.1 First Style of Grasping......Page 528
20.4.3 Third Style of Grasping......Page 530
20.5 Target Pose......Page 531
20.6 Visually Guided Grasping......Page 533
20.7 Fuzzy Logic and Conformal Geometric Algebra for Grasping......Page 534
20.7.1 Mandami Fuzzy System......Page 535
20.7.2 Direct Kinematics of the Barrett Hand......Page 536
20.7.3 Fuzzy Grasping of Objects......Page 537
20.8 Conclusion......Page 540
21.1.1 Matching Laser Readings......Page 541
21.1.3 Line Map......Page 544
21.1.4 3D Map Building......Page 546
21.2.2 Adding Objects to the 3D Map......Page 548
21.2.3 Path Following......Page 549
21.3 3D Map Building Using Laser and Stereo Vision......Page 553
21.3.1 Laser Rangefinder......Page 556
21.3.2 Stereo Camera System with Pan-Tilt Unit......Page 558
21.4 Relocation Using Lines and the Hough Transform......Page 559
21.5 Experiments......Page 562
21.6 Conclusions......Page 563
22.1.1 Union of Spheres......Page 564
22.1.2 The Marching Cubes Algorithm......Page 565
22.2 Segmentation......Page 566
22.3 Marching Spheres......Page 570
22.3.1 Experimental Results for Modeling......Page 571
22.4.1 Sphere Matching......Page 574
22.4.2 Experimental Results for Registration......Page 577
22.5 Conclusions......Page 579
Part VII Appendix......Page 580
23.1.1 Basic Properties......Page 581
23.1.2 Definitions and Existence......Page 582
23.1.3 Real and Complex Clifford Algebras......Page 583
23.1.5 Structure and Classification of Clifford Algebras......Page 585
23.1.6 Clifford Groups, Pin and Spin Groups, and Spinors......Page 587
23.2.1 Gibbs\' Vector Algebra......Page 590
23.2.2 Exterior Algebras......Page 592
23.2.3 Grassmann–Cayley Algebras......Page 596
24 Notation......Page 601
25 Useful Formulas for Geometric Algebra......Page 602
References......Page 607
Index......Page 616