Handbook of Geometric Computing. Applications in Pattern Recognition, Computer Vision, Neuralcomputing and Robotics

Description

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