Handbook of Geometric Computing: Applications to Pattern Recognition, Computer Vision, Neural Computing, and Robotics

Preface......Page 3
Contents......Page 11
--- Neuroscience......Page 14
Introduction......Page 15
The Basic Geometry of Neural Representation: Maps and Pathways......Page 16
Example: Maps and Pathways in Coding Object Boundaries......Page 19
Summary......Page 37
References......Page 38
Introduction......Page 42
Motivations From the Visual System......Page 44
Overview of the Model......Page 48
The Structure of Light as a Signal......Page 50
Symmetry and Signal Representations......Page 52
The Space of Features......Page 58
Information Dynamics......Page 66
Application and Computation......Page 69
References......Page 73
--- Neural Networks......Page 77
Introduction......Page 78
Space of Neural Networks......Page 79
Learning in Neuromanifolds......Page 84
Problem of Singular Structures......Page 92
Discussion and Conclusions......Page 103
References......Page 104
Some Elementary Concepts From Lattice Algebra......Page 106
Morphological Associative Memories......Page 109
Kernels and Morphological Independence......Page 116
Dendritic Computation Based on Lattice Algebra......Page 119
Computational Capability of Perceptrons Based on the Dendritic Model......Page 121
Training Algorithms for SLMPs with Dendritic Structures......Page 124
SLMPs with Fuzzy-Valued Outputs......Page 130
Conclusions......Page 131
References......Page 133
Introduction......Page 137
Linear Algebra......Page 139
Kernel Methods......Page 143
Dimensionality Reduction: PCA, (R)CCA, PLS......Page 148
Classification: Fisher Discriminant Analysis (FDA)......Page 164
Spectral Methods for Clustering......Page 168
Conclusions......Page 171
References......Page 174
--- Image Processing......Page 176
Geometrical Aspects of Image Processing......Page 177
Theory......Page 182
Image Processing......Page 189
References......Page 205
Diffusion Flows and Geometric Filters......Page 208
Extending the Beltrami Framework......Page 219
Numerical Schemes......Page 224
Kernels......Page 228
Conclusion......Page 232
References......Page 233
Introduction......Page 236
Chaos-Based Encryption Schemes......Page 243
Chaos-Based Image Encryption......Page 247
Chaos-Based Block Ciphers for Image Encryption......Page 248
Chaos-Based Stream Cipher for Image Encryption......Page 258
Conclusion: An Engineer’s Perspective......Page 267
References......Page 268
--- Computer Vision......Page 271
Introduction......Page 272
Scanner Geometry......Page 274
Problem Formulation......Page 275
Structure and Motion Problems Without Missing Data......Page 277
Structure and Motion Problems With Missing Data......Page 285
Ambiguous Cases of Structure and Motion Problems......Page 300
Conclusions......Page 305
References......Page 306
Introduction......Page 308
Projective Geometry......Page 309
Camera Modelling......Page 315
Multiple-View Geometry......Page 324
Structure and Motion from Image Sequences......Page 334
Autocalibration......Page 344
Conclusions......Page 347
Acknowledgements......Page 348
References......Page 349
Dynamic Pn to Pn Alignment......Page 351
Introduction......Page 352
oftheProjectivePlane......Page 356
Homography Tensors of......Page 363
Homography Tensors for......Page 370
The Structure of......Page 372
Experiments and Applications......Page 375
Summary......Page 378
Representation Theory Digest......Page 379
References......Page 383
Introduction......Page 384
A Compositional Viewpoint......Page 385
Existing Approaches......Page 386
Ambiguity in 3D Motion Estimation......Page 387
Types of Independently Moving Objects......Page 388
Phase Correlation......Page 390
Motion-Based Clustering......Page 393
Ordinal Depth from Occlusion Filling Using Three Frames......Page 394
Experiments......Page 397
Conclusion......Page 398
References......Page 401
--- Perception & Action......Page 403
Introduction......Page 404
Geometric Algebra: An Outline......Page 405
Conformal Geometry......Page 412
The 3-D Affine Plane......Page 427
-D Rigid-Motion Estimation......Page 432
Body-Eye Calibration......Page 434
Strategies for Object Manipulation......Page 439
Omnidirectional Vision Using Conformal Geometric Algebra......Page 447
Conclusion......Page 456
References......Page 457
--- Uncertainty in Geometric Computations......Page 458
What Is Geometric Inference?......Page 459
Asymptotic Analysis for Geometric Inference......Page 464
Standard Statistical Analysis vs. Geometric Inference......Page 469
Linear Geometric Fitting......Page 475
Nuisance Parameters and Semiparametric Model......Page 484
References......Page 486
Introduction......Page 490
Uncertainty......Page 494
Geometric Relations......Page 511
Conditioning and Normalization......Page 527
Conclusions......Page 528
Appendix: Uncertainty of Tranformed 3D Lines......Page 529
References......Page 530
Introduction......Page 532
Related Work on Perceptual Organization......Page 533
The Original Second-Order Tensor Voting Framework......Page 538
First-Order Voting......Page 545
Applications......Page 551
Conclusion......Page 561
References......Page 563
--- Computer Graphics & Visualization......Page 566
Introduction......Page 567
Categorization......Page 568
Types of Deformation......Page 573
Specific Approaches......Page 575
Surface Warping......Page 579
Volumetric Warping......Page 584
References......Page 589
Introduction......Page 599
Types of Implicit Functions......Page 602
Implicit Models......Page 603
Re-tooling Implicit Models......Page 610
Regularizing Nonzero Space......Page 612
Putting it together......Page 616
Summary and Future Development......Page 620
References......Page 622
--- Geometry & Robotics......Page 623
Plücker Coordinates......Page 624
Dual Plücker Coordinates......Page 629
Basis Change......Page 630
Grassmann–Cayley Algebra; The Join Operation......Page 631
Application to Instantaneous Kinematics......Page 634
The Meet Operation......Page 638
Application to Statics......Page 640
Application to Series–Parallel Mechanisms......Page 642
An Example of a Robot Arm......Page 643
Cayley Factorization......Page 644
Rigidity of Frameworks......Page 645
References......Page 650
Introduction......Page 652
The Clifford algebra......Page 653
Serial Robots......Page 660
Parallel Robots......Page 667
Conclusions......Page 672
References......Page 673
Introduction......Page 674
The Geometry of Rigid Body Motion......Page 676
An SVD-Based Method for Interpolation on......Page 681
Optimal Motion Generation for Groups of Robots......Page 688
Conclusion......Page 698
References......Page 699
--- Reaching & Motion Planning......Page 702
Overview......Page 703
Introduction......Page 704
Spatial Displacements......Page 705
Articulated Serial Chains......Page 707
Linear Product Decomposition......Page 709
The Plane......Page 711
The Sphere......Page 712
The Circular Cylinder......Page 713
The Circular Hyperboloid......Page 716
The Elliptic Cylinder......Page 718
The Circular Torus......Page 721
The General Torus......Page 723
Generalization of the Problem......Page 725
Conclusion......Page 726
References......Page 727
Planning Collision-free Paths using Probabilistic Roadmaps......Page 730
The Evolution of Sampling-Based Methods......Page 731
An Overview of the Approach......Page 732
Generating Sample Configurations......Page 735
Sample Rejection......Page 739
Sample Transformation......Page 748
Connecting the Vertices......Page 753
Enhancement......Page 755
References......Page 757
Index......Page 761