Geometric Computing - for Wavelet Transforms, Robot Vision, Learning, Control and Action (eBook)

Geometric Computing - for Wavelet Transforms, Robot Vision, Learning, Control and Action (eBook)

This book offers a gentle introduction to Clifford geometric algebra, an advanced mathematical framework, for applications in perception action systems. Part I, is written in an accessible way allowing readers to easily grasp the mathematical system of Clifford algebra. Part II presents related topics. While Part 3 features practical applications for Computer Vision, Robotics, Image Processing and Neural Computing.Topics and Features include: theory and application of the quaternion Fourier and wavelet transforms, thorough discussion on geometric computing under uncertainty, an entire chapter devoted to the useful conformal geometric algebra, presents examples and hints for the use of public domain computer programs for geometric algebra.The modern framework for geometric computing highlighted will be of great use for communities working on image processing, computer vision, artificial intelligence, neural networks, neuroscience, robotics, control engineering, human and robot interfaces, haptics and humanoids.

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Inhaltsangabe

Foreword ... 6
Preface ... 7
Part I Fundamentals of Geometric Algebra ... 27
1 Introduction to Geometric Algebra ... 28
1.1 History of Geometric Algebra ... 28
1.2 What Is Geometric Algebra? ... 30
1.2.1 Basic Definitions ... 30
1.2.2 Nonorthonormal Frames and Reciprocal Frames ... 32
1.2.3 Some Useful Formulas ... 33
1.2.4 Multivector Products ... 33
1.2.5 Further Properties of the Geometric Product ... 36
1.2.6 Dual Blades and Duality in the Geometric Product ... 44
1.2.7 Multivector Operations ... 45
1.3 Linear Algebra ... 47
1.4 Simplexes ... 48
1.5 Geometric Calculus ... 50
1.5.1 Multivector-Valued Functions and the Inner Product ... 50
1.5.2 The Multivector Integral ... 51
1.5.3 The Vector Derivative ... 52
1.5.4 Grad, Div, and Curl ... 52
1.5.5 Multivector Fields ... 53
1.5.6 Convolution and Correlation of Scalar Fields ... 54
1.5.7 Clifford Convolution and Correlation ... 54
1.5.8 Linear Algebra Derivations ... 55
1.5.9 Reciprocal Frames with Curvilinear Coordinates ... 56
1.5.10 Geometric Calculus in 2D ... 56
1.5.11 Electromagnetism: The Maxwell Equations ... 57
1.5.12 Spinors, Shrödinger Pauli, and Dirac Equations ... 60
1.5.13 Spinor Operators ... 62
1.6 Exercises ... 64
2 Geometric Algebra for Modeling in Robot Physics ... 69
2.1 The Roots of Geometry and Algebra ... 69
2.2 Geometric Algebra: A Unified Mathematical Language ... 71
2.3 What Does Geometric Algebra Offer for GeometricComputing? ... 72
2.3.1 Coordinate-Free Mathematical System ... 72
2.3.2 Models for Euclidean and Pseudo-Euclidean Geometry ... 73
2.3.3 Subspaces as Computing Elements ... 74
2.3.4 Representation of Orthogonal Transformations ... 75
2.3.5 Objects and Operators ... 75
2.3.6 Extension of Linear Transformations ... 76
2.3.7 Signals and Wavelets in the Geometric Algebra Framework ... 77
2.3.8 Kinematics and Dynamics ... 77
2.4 Solving Problems in Perception and Action Systems ... 78
Part II Euclidean, Pseudo-Euclidean, Lie and Incidence Algebras,and Conformal Geometries ... 84
3 2D, 3D, and 4D Geometric Algebras ... 85
3.1 Complex, Double, and Dual Numbers ... 85
3.2 2D Geometric Algebras of the Plane ... 86
3.3 3D Geometric Algebra for the Euclidean 3D Space ... 88
3.3.1 The Algebra of Rotors ... 89
3.3.2 Orthogonal Rotors ... 92
3.3.3 Recovering a Rotor ... 93
3.4 Quaternion Algebra ... 94
3.5 Lie Algebras and Bivector Algebras ... 96
3.5.1 Lie Group of Rotors ... 97
3.5.2 Bivector Lie Algebra ... 98
3.5.3 Complex Structures and Unitary Groups ... 99
3.5.4 Hermitian Inner Product and Unitary Groups ... 100
3.6 4D Geometric Algebra for 3D Kinematics ... 102
3.6.1 Motor Algebra ... 102
3.6.2 Motors, Rotors, and Translators in G+3,0,1 ... 104
3.6.3 Properties of Motors ... 107
3.7 4D Geometric Algebra for Projective 3D Space ... 109
3.8 Conclusion ... 110
3.9 Exercises ... 110
4 Kinematics of the 2D and 3D Spaces ... 115
4.1 Introduction ... 115
4.2 Representation of Points, Lines, and Planes Using 3D Geometric Algebra ... 115
4.3 Representation of Points, Lines, and PlanesUsing Motor Algebra ... 117
4.4 Representation of Points, Lines, and Planes Using 4D Geometric Algebra ... 118
4.5 Motion of Points, Lines, and Planes in 3D Geometric Algebra ... 119
4.6 Motion of Points, Lines, and Planes Using Motor Algebra ... 120
4.7 Motion of Points, Lines, and Planes Using 4D Geometric Algebra ... 122
4.8 Spatial Velocity of Points, Lines, and Planes ... 123
4.8.1 Rigid-Body Spatial Velocity Using Matrices ... 123
4.8.2 Angular Velocity Using Rotors ... 127
4.8.3 Rigid-Body Spatial Velocity Using Motor Algebra ... 130
4.8.4 Point, Line, and Plane Spatial Velocities Using Motor Algebra ... 131
4.9 Incidence Relations Between Points, Lines, and Planes ... 132
4.9.1 Flags of Points, Lines, and Planes ... 133
4.10 Conclusion ... 134
4.11 Exercises ... 134
5 Lie Algebras and the Algebra of Incidence Using the Null Cone and Affine Plane ... 138
5.1 Introduction ... 138
5.2 Geometric Algebra of Reciprocal Null Cones ... 139
5.2.1 Reciprocal Null Cones ... 139
5.2.2 The Universal Geometric Algebra Gn,n ... 140
5.2.3 The Lie Algebra of Null Spaces ... 141
5.2.4 The Standard Bases of Gn,n ... 143
5.2.5 Representations and Operations Using Bivector Matrices ... 144
5.2.6 Bivector Representation of Linear Operators ... 145
5.3 Horosphere and n-Dimensional Affine Plane ... 146
5.4 The General Linear Group ... 148
5.4.1 The General Linear Algebra gl(N) of the General Linear Lie Group GL(N) ... 150
5.4.2 The Orthogonal Groups ... 151
5.5 Computing Rigid Motion in the Affine Plane ... 154
5.6 The Lie Algebra of the Affine Plane ... 155
5.7 The Algebra of Incidence ... 159
5.7.1 Incidence Relations in the Affine n-Plane ... 161
5.7.2 Directed Distances ... 162
5.7.3 Incidence Relations in the Affine 3-Plane ... 163
5.7.4 Geometric Constraints as Flags ... 165
5.8 Conclusion ... 165
5.9 Exercises ... 166
6 Conformal Geometric Algebra ... 169
6.1 Introduction ... 169
6.1.1 Conformal Split ... 170
6.1.2 Conformal Splits for Points and Simplexes ... 171
6.1.3 Euclidean and Conformal Spaces ... 172
6.1.4 Stereographic Projection ... 175
6.1.5 Inner- and Outer-Product Null Spaces ... 177
6.1.6 Spheres and Planes ... 178
6.1.7 Geometric Identities, Meet and Join Operations, Duals, and Flats ... 180
6.1.8 Meet, Pair of Points, and Plunge ... 186
6.1.9 Simplexes and Spheres ... 188
6.2 The 3D Affine Plane ... 189
6.2.1 Lines and Planes ... 190
6.2.2 Directed Distance ... 191
6.3 The Lie Algebra ... 192
6.4 Conformal Transformations ... 192
6.4.1 Inversion ... 193
6.4.2 Reflection ... 195
6.4.3 Translation ... 196
6.4.4 Transversion ... 196
6.4.5 Rotation ... 197
6.4.6 Rigid Motion Using Flags ... 197
6.4.7 Dilation ... 199
6.4.8 Involution ... 199
6.4.9 Conformal Transformation ... 200
6.5 Ruled Surfaces ... 200
6.5.1 Cone and Conics ... 200
6.5.2 Cycloidal Curves ... 201
6.5.3 Helicoid ... 202
6.5.4 Sphere and Cone ... 202
6.5.5 Hyperboloid, Ellipsoids, and Conoid ... 203
6.6 Exercises ... 203
7 Programming Issues ... 209
7.1 Main Issues for an Efficient Implementation ... 209
7.1.1 Specific Aspects for the Implementation ... 210
7.2 Implementation Practicalities ... 211
7.2.1 Specification of the Geometric Algebra, Gp,q ... 211
7.2.2 The General Multivector Class ... 211
7.2.3 Optimization of Multivector Functions ... 212
7.2.4 Factorization ... 213
7.2.5 Speeding Up Geometric Algebra Expressions ... 214
7.2.6 Multivector Software Packets ... 215
Part III Geometric Computing for Image Processing, Computer Vision, and Neurocomputing ... 218
8 Clifford–Fourier and Wavelet Transforms ... 219
8.1 Introduction ... 219
8.2 Image Analysis in the Frequency Domain ... 219
8.2.1 The One-Dimensional Fourier Transform ... 220
8.2.2 The Two-Dimensional Fourier Transform ... 221
8.2.3 Quaternionic Fourier Transform ... 221
8.2.4 2D Analytic Signals ... 223
8.2.5 Properties of the QFT ... 226
8.2.6 Discrete QFT ... 229
8.3 Image Analysis Using the Phase Concept ... 231
8.3.1 2D Gabor Filters ... 231
8.3.2 The Phase Concept ... 232
8.4 Clifford–Fourier Transforms ... 232
8.4.1 Tri-Dimensional Clifford–Fourier Transform ... 235
8.4.2 Space and Time Geometric AlgebraFourier Transform ... 236
8.4.3 n-Dimensional Clifford–Fourier Transform ... 237
8.5 From Real to Clifford Wavelet Transforms for Multiresolution Analysis ... 237
8.5.1 Real Wavelet Transform ... 238
8.5.2 Discrete Wavelets ... 238
8.5.3 Wavelet Pyramid ... 241
8.5.4 Complex Wavelet Transform ... 241
8.5.5 Quaternion Wavelet Transform ... 243
8.5.6 Quaternionic Wavelet Pyramid ... 247
8.5.7 The Tridimensional Clifford Wavelet Transform ... 250
8.5.8 The Continuous Conformal Geometric Algebra Wavelet Transform ... 252
8.5.9 The n-Dimensional Clifford Wavelet Transform ... 253
8.6 Conclusion ... 254
9 Geometric Algebra of Computer Vision ... 255
9.1 Introduction ... 255
9.2 The Geometric Algebras of 3D and 4D Spaces ... 255
9.2.1 3D Space and the 2D Image Plane ... 256
9.2.2 The Geometric Algebra of 3D Euclidean Space ... 258
9.2.3 A 4D Geometric Algebra for Projective Space ... 258
9.2.4 Projective Transformations ... 259
9.2.5 The Projective Split ... 260
9.3 The Algebra of Incidence ... 262
9.3.1 The Bracket ... 263
9.3.2 The Duality Principle and Meet and Join Operations ... 264
9.4 Algebra in Projective Space ... 265
9.4.1 Intersection of a Line and a Plane ... 266
9.4.2 Intersection of Two Planes ... 267
9.4.3 Intersection of Two Lines ... 268
9.4.4 Implementation of the Algebra ... 268
9.5 Projective Invariants ... 269
9.5.1 The 1D Cross-Ratio ... 269
9.5.2 2D Generalization of the Cross-Ratio ... 271
9.5.3 3D Generalization of the Cross-Ratio ... 272
9.6 Visual Geometry of n-Uncalibrated Cameras ... 273
9.6.1 Geometry of One View ... 273
9.6.2 Geometry of Two Views ... 277
9.6.3 Geometry of Three Views ... 279
9.6.4 Geometry of n-Views ... 281
9.7 Omnidirectional Vision ... 282
9.7.1 Omnidirectional Vision and Geometric Algebra ... 283
9.7.2 Point Projection ... 284
9.7.3 Inverse Point Projection ... 285
9.8 Invariants in the Conformal Space ... 286
9.8.1 Invariants and Omnidirectional Vision ... 287
9.8.2 Projective and Permutation p2-Invariants ... 289
9.9 Conclusion ... 291
9.10 Exercises ... 291
10 Geometric Neuralcomputing ... 294
10.1 Introduction ... 294
10.2 Real-Valued Neural Networks ... 295
10.3 Complex MLP and Quaternionic MLP ... 296
10.4 Geometric Algebra Neural Networks ... 297
10.4.1 The Activation Function ... 297
10.4.2 The Geometric Neuron ... 298
10.4.3 Feedforward Geometric Neural Networks ... 300
10.4.4 Generalized Geometric Neural Networks ... 301
10.4.5 The Learning Rule ... 302
10.4.6 Multidimensional Back-Propagation Training Rule ... 302
10.4.7 Simplification of the Learning Rule Using the Density Theorem ... 303
10.4.8 Learning Using the Appropriate Geometric Algebras ... 304
10.5 Support Vector Machines in Geometric Algebra ... 305
10.6 Linear Clifford Support Vector Machinesfor Classification ... 305
10.7 Nonlinear Clifford Support Vector Machines For Classification ... 309
10.8 Clifford SVM for Regression ... 311
10.9 Conclusion ... 313
Part IV Geometric Computing of Robot Kinematics and Dynamics ... 314
11 Kinematics ... 315
11.1 Introduction ... 315
11.2 Elementary Transformations of Robot Manipulators ... 315
11.2.1 The Denavit–Hartenberg Parameterization ... 316
11.2.2 Representations of Prismatic and Revolute Transformations ... 317
11.2.3 Grasping by Using Constraint Equations ... 320
11.3 Direct Kinematics of Robot Manipulators ... 322
11.3.1 MAPLE Program for Motor Algebra Computations ... 323
11.4 Inverse Kinematics of Robot ManipulatorsUsing Motor Algebra ... 324
11.4.1 The Rendezvous Method ... 325
11.4.2 Computing 1, 2, and d3 Using a Point ... 325
11.4.3 Computing 4 and 5 Using a Line ... 328
11.4.4 Computing 6 Using a Plane Representation ... 330
11.5 Inverse Kinematics Using the 3D Affine Plane ... 331
11.6 Inverse Kinematic Using Conformal Geometric Algebra ... 334
11.7 Conclusion ... 338
12 Dynamics ... 340
12.1 Introduction ... 340
12.2 Differential Kinematics ... 340
12.3 Dynamics ... 343
12.3.1 Kinetic Energy ... 343
12.3.2 Potential Energy ... 350
12.3.3 Lagrange's Equations ... 350
12.4 Complexity Analysis ... 358
12.4.1 Computing M ... 358
12.4.2 Computing G ... 358
12.5 Conclusion ... 359
Part V Applications I: Image Processing, Computer Vision,and Neurocomputing ... 360
13 Applications of Lie Filters, and Quaternion Fourier and Wavelet Transforms ... 361
13.1 Lie Filters in the Affine Plane ... 361
13.1.1 The Design of an Image Filter ... 362
13.1.2 Recognition of Hand Gestures ... 363
13.2 Representation of Speech as 2D Signals ... 364
13.3 Preprocessing of Speech 2D Representations Using the QFT and Quaternionic Gabor Filter ... 366
13.3.1 Method 1 ... 366
13.3.2 Method 2 ... 368
13.4 Recognition of French Phonemes Using Neurocomputing ... 369
13.5 Application of QWT ... 371
13.5.1 Estimation of the Quaternionic Phase ... 372
13.5.2 Confidence Interval ... 373
13.5.3 Discussion on Similarity Distance and the Phase Concept ... 374
13.5.4 Optical Flow Estimation ... 375
13.6 Conclusion ... 379
14 Invariants Theory in Computer Vision and Omnidirectional Vision ... 380
14.1 Introduction ... 380
14.2 Conics and Pascal's Theorem ... 381
14.3 Computing Intrinsic Camera Parameters ... 384
14.4 Projective Invariants ... 385
14.4.1 The 1D Cross-Ratio ... 386
14.4.2 2D Generalization of the Cross-Ratio ... 387
14.4.3 3D Generalization of the Cross-Ratio ... 389
14.4.4 Generation of 3D Projective Invariants ... 390
14.5 3D Projective Invariants from Multiple Views ... 394
14.5.1 Projective Invariants Using Two Views ... 394
14.5.2 Projective Invariant of Points Using Three Uncalibrated Cameras ... 396
14.5.3 Comparison of the Projective Invariants ... 398
14.6 Visually Guided Grasping ... 400
14.6.1 Parallel Orienting ... 400
14.6.2 Centering ... 402
14.6.3 Grasping ... 402
14.6.4 Holding the Object ... 403
14.7 Camera Self-Localization ... 403
14.8 Projective Depth ... 404
14.9 Shape and Motion ... 406
14.9.1 The Join-Image ... 407
14.9.2 The SVD Method ... 408
14.9.3 Completion of the 3D Shape Using Invariants ... 409
14.10 Omnidirectional Vision Landmark Identification Using Projective Invariants ... 411
14.10.1 Learning Phase ... 411
14.10.2 Recognition Phase ... 412
14.10.3 Omnidirectional Vision and Invariants for Robot Navigation ... 413
14.10.4 Learning Phase ... 414
14.10.5 Recognition Phase ... 414
14.10.6 Quantitative Results ... 415
14.11 Conclusions ... 416
15 Registration of 3D Points Using GA and Tensor Voting ... 418
15.1 Problem Formulation ... 418
15.1.1 The Geometric Constraint ... 419
15.2 Tensor Voting ... 422
15.2.1 Tensor Representation in 3D ... 422
15.2.2 Voting Fields in 3D ... 423
15.2.3 Detection of 3D Surfaces ... 427
15.2.4 Estimation of 3D Correspondences ... 428
15.3 Experimental Analysis ... 430
15.3.1 Correspondences Between 3D Pointsby Rigid Motion ... 430
15.3.2 Multiple Overlapping Motions and Nonrigid Motion ... 432
15.3.3 Extension to Nonrigid Motion ... 433
15.4 Conclusions ... 435
16 Applications in Neuralcomputing ... 437
16.1 Experiments Using Geometric Feedforward Neural Networks ... 437
16.1.1 Learning a High Nonlinear Mapping ... 437
16.1.2 Encoder–Decoder Problem ... 438
16.1.3 Prediction ... 440
16.2 Experiments Using Clifford Support Vector Machines ... 441
16.2.1 3D Spiral: Nonlinear Classification Problem ... 442
16.2.2 Object Recognition ... 444
16.2.3 Multi-Case Interpolation ... 452
16.3 Conclusion ... 454
17 Neural Computing for 2D Contour and 3D Surface Reconstruction ... 455
17.1 Determining the Shape of an Object ... 455
17.1.1 Automatic Sample Selection Using GGVF ... 456
17.1.2 Learning the Shape Using Versors ... 458
17.2 Experiments ... 460
17.3 Conclusion ... 467
Part VI Applications II: Robotics and Medical Robotics ... 468
18 Rigid Motion Estimation Using Line Observations ... 469
18.1 Introduction ... 469
18.2 Batch Estimation Using SVD Techniques ... 469
18.2.1 Solving AX = XB Using Motor Algebra ... 471
18.2.2 Estimation of the Hand–Eye Motor Using SVD ... 474
18.3 Experimental Results ... 476
18.4 Discussion ... 480
18.5 Recursive Estimation Using Kalman Filter Techniques ... 480
18.5.1 The Kalman Filter ... 480
18.5.2 The Extended Kalman Filter ... 482
18.5.3 The Rotor-Extended Kalman Filter ... 484
18.6 The Motor-Extended Kalman Filter ... 487
18.6.1 Representation of the Line Motion Model in Linear Algebra ... 488
18.6.2 Linearization of the Measurement Model ... 490
18.6.3 Enforcing a Geometric Constraint ... 491
18.6.4 Operation of the MEKF Algorithm ... 493
18.6.5 Estimation of the Relative Positioning of a Robot End-Effector ... 496
18.7 Conclusion ... 500
19 Tracker Endoscope Calibration and Body-Sensors' Calibration ... 501
19.1 Camera Device Calibration ... 501
19.1.1 Rigid Body Motion in CGA ... 501
19.1.2 Hand–Eye Calibration in CGA ... 503
19.1.3 Tracker Endoscope Calibration ... 504
19.2 Body-Sensor Calibration ... 507
19.2.1 Body–Eye Calibration ... 508
19.2.2 Algorithm Simplification ... 511
19.3 Conclusions ... 513
20 Tracking, Grasping, and Object Manipulation ... 514
20.1 Tracking ... 514
20.1.1 Exact Linearization via Feedback ... 515
20.1.2 Visual Jacobian ... 517
20.1.3 Exact Linearization via Feedback ... 518
20.1.4 Experimental Results ... 519
20.2 Barrett Hand Direct Kinematics ... 521
20.3 Pose Estimation ... 523
20.3.1 Segmentation ... 524
20.3.2 Object Projection ... 525
20.4 Grasping Objects ... 527
20.4.1 First Style of Grasping ... 528
20.4.2 Second Style of Grasping ... 530
20.4.3 Third Style of Grasping ... 530
20.5 Target Pose ... 531
20.5.1 Object Pose ... 533
20.6 Visually Guided Grasping ... 533
20.6.1 Results ... 534
20.7 Fuzzy Logic and Conformal Geometric Algebra for Grasping ... 534
20.7.1 Mandami Fuzzy System ... 535
20.7.2 Direct Kinematics of the Barrett Hand ... 536
20.7.3 Fuzzy Grasping of Objects ... 537
20.8 Conclusion ... 540
21 3D Maps, Navigation, and Relocalization ... 541
21.1 Map Building ... 541
21.1.1 Matching Laser Readings ... 541
21.1.2 Map Building ... 544
21.1.3 Line Map ... 544
21.1.4 3D Map Building ... 546
21.2 Navigation ... 548
21.2.1 Localization ... 548
21.2.2 Adding Objects to the 3D Map ... 548
21.2.3 Path Following ... 549
21.3 3D Map Building Using Laser and Stereo Vision ... 553
21.3.1 Laser Rangefinder ... 556
21.3.2 Stereo Camera System with Pan-Tilt Unit ... 558
21.4 Relocation Using Lines and the Hough Transform ... 559
21.5 Experiments ... 562
21.6 Conclusions ... 563
22 Modeling and Registration of Medical Data ... 564
22.1 Background ... 564
22.1.1 Union of Spheres ... 564
22.1.2 The Marching Cubes Algorithm ... 565
22.2 Segmentation ... 566
22.3 Marching Spheres ... 570
22.3.1 Experimental Results for Modeling ... 571
22.4 Registration of Two Models ... 574
22.4.1 Sphere Matching ... 574
22.4.2 Experimental Results for Registration ... 577
22.5 Conclusions ... 579
Part VII Appendix ... 580
23 Clifford Algebras and Related Algebras ... 581
23.1 Clifford Algebras ... 581
23.1.1 Basic Properties ... 581
23.1.2 Definitions and Existence ... 582
23.1.3 Real and Complex Clifford Algebras ... 583
23.1.4 Involutions ... 585
23.1.5 Structure and Classification of Clifford Algebras ... 585
23.1.6 Clifford Groups, Pin and Spin Groups, and Spinors ... 587
23.2 Related Algebras ... 590
23.2.1 Gibbs' Vector Algebra ... 590
23.2.2 Exterior Algebras ... 592
23.2.3 Grassmann–Cayley Algebras ... 596
24 Notation ... 601
25 Useful Formulas for Geometric Algebra ... 602
References ... 607
Index ... 616

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Inhaltsangabe

Introduction to Geometric Algebra.- Geometric Algebra for Modeling in Robot Physics.- 2D, 3D and 4D Geometric Algebras.- Kinematics of the 2D and 3D Spaces.- Lie Algebras and Algebra of Incidence Using The Null Cone and Affine Plane.- Conformal Geometric Algebra.- Programming Issues.- Clifford Fourier and Wavelet Transforms.- Geometric Algebra of Computer Vision.- Geometric Neuralcomputing.- Kinematics.- Dynamics.- Applications of Lie Filters, Quaternion Fourier and Wavelet Transforms.- Invariants Theory in Computer Vision and Omnidirectional Vision.- Registration of 3D Points Using GA and Tensor Voting.- Applications in Neuralcomputing.- Neural Computing for 2D Contour and 3D Surface Reconstruction.- Rigid Motion Estimation Using Line Observations.- Tracker Endoscope Calibration and Body-Sensors Calibration.- Tracking, Grasping and Object Manipulation.- 3D Maps, Navigation and Relocalization.- Modeling and Registration of Medical Data.