Tensors in Image Processing and Computer Vision (eBook)

Preface 5
Contents 7
A Review of Tensors and Tensor Signal Processing 10
1 Introduction 11
2 Tensor Definition 11
3 Example of Tensors in Mathematics 12
4 Example of Tensors in Physics and Engineering 17
5 Tensors Usage in Medical Imaging 20
6 Tensors Usage in Computer Vision 29
7 Conclusion 38
References 39
Part I Tensors and Tensor Field Processing 42
Segmentation of Tensor Fields: Recent Advances and Perspectives 43
1 Introduction 43
2 Tensors in Image Processing 44
3 Scalar Diffusion and Anisotropy Measures 47
4 Tensor Dissimilarity Measures 50
5 Tensor Field Segmentation Techniques 55
6 Summary 59
References 60
A Variational Approach to the Registration of Tensor- Valued Images 67
1 Introduction 67
2 The Variational Model 69
3 Experiments 75
4 Summary and Outlook 82
References 84
Quality Assessment of Tensor Images 86
1 Introduction 86
2 Background on Quality Assessment in Image Processing 88
3 Background on Tensor Images 91
4 Application of Quality Measures to Tensor Images 93
5 Experiments 98
6 Discussion 106
7 Conclusions 108
References 109
Algorithms for Nonnegative Tensor Factorization 111
1 Introduction 112
2 Formulating Nonnegative Tensor Factorizations 113
3 Solving the Optimization Problems Using Multiplicative Update Rules 118
4 Supervised Nonnegative Tensor Factorization 123
5 Applications of Nonnegative Tensor Factorizationss 125
6 Conclusions and Discussion 128
References 129
PDE-based Morphology for Matrix Fields: Numerical Solution Schemes 131
1 Introduction 132
2 Morphology for Grey Scale Images 133
3 Ordering Based Morphology for Matrix Fields 140
4 PDE-based morphology for Matrix Fields 144
5 Experimental Comparison of the Numerical Schemes 146
6 Conclusion 150
References 150
Part II Tensors in Image Processing 157
Spherical Tensor Calculus for Local Adaptive Filtering 158
1 Introduction 158
2 Spherical Tensor Analysis 161
3 Tensorial Harmonic Expansion 166
4 Local Adaptive Filtering with Tensorial Harmonics 171
5 Spherical Tensor Derivatives 174
6 local adaptive filtering with STDs 177
7 Conclusion 180
Appendix 181
References 182
On Geometric Transformations of Local Structure Tensors 184
1 Introduction 184
2 Background 185
3 Tensor Transformations 186
4 Local Structure Tensor Fields 188
5 Local Structure Analysis 191
6 Experiments 194
7 Results 195
8 Summary 197
References 197
Part III Tensors in Computer Vision 199
Multi-View Matching Tensors from Lines for General Camera Models 200
1 Introduction 200
2 Plücker Coordinates 202
3 Lines in Space 203
4 Multi-Focal Tensors from Lines 204
5 Constrained Camera Models 207
6 Experimental Results 212
7 Conclusions 213
Acknowledgements 214
References 214
Binocular Full-Body Pose Recognition and Orientation Inference Using Multilinear Analysis 218
1 Introduction 218
2 Overview of the Proposed Approach 220
3 Theoretical Background 221
4 Multilinear Analysis of Dance Pose Images 225
5 SVM-based Pose Recognition Using Pose Vectors 228
6 Body Orientation Estimation Using Orientation Vector through Manifold Learning and Nonlinear Minimization 230
7 Experimental Results 232
8 Conclusions and FutureWork 237
References 238
Applications of Multiview Tensors in Higher Dimensions 240
1 Introduction 241
2 Notation and Background Material 242
3 Higher Dimensional Spaces As Frameworks For Some Dynamic And Segmented Scenes 243
4 Multiview Tensors in Higher Dimension 244
5 Algorithms 246
6 Critical Configurations and Their Loci 248
7 Examples of Critical Loci for Projective Reconstruction 254
8 Instability Results 256
References 262
Constraints for the Trifocal Tensor 264
1 Introduction 264
2 The Trifocal Tensor 265
3 Constraints via Group Theory 267
4 Constraints via Geometry 269
5 A Minimal Set of Constraints 270
References 271
Part IV Diffusion Tensor Imaging and Medical Applications 273
Review of Techniques for Registration of Diffusion Tensor Imaging 274
1 Introduction 274
2 Registration of DT-MRI 276
3 Review of DT-MRI registration Algorithms 279
4 Similarity Measures 284
5 Diffusion Tensor Reorientation 287
6 Performance Evaluation 290
7 Discussion and Conclusions 292
Appendix. Scalar Measures of the Diffusion 294
References 296
Practical and Intuitive Basis for Tensor Field Processing with Invariant Gradients and Rotation Tangents 299
1 Introduction 299
2 Mathematical Background 300
3 Conceptual Overview 302
4 Invariant Gradients 303
5 Rotation Tangents 305
6 Example Application: Edge Detection 306
7 Other Applications 311
8 Discussion 311
References 312
From Second to Higher Order Tensors in Diffusion- MRI 315
1 Introduction 315
2 Principles of Diffusion MRI 316
3 Higher Order Tensors in D-MRI 324
4 Positivity Constraints 328
5 Conclusion 332
References 333
DT-MRI Connectivity and/or Tractography?: Two New Algorithms 335
1 Introduction 336
2 Connectivity or Tractography? 338
3 Lattice-of-Springs (LoS) 339
4 Split and Merge Tractography (SMT) 345
5 Concluding Remarks 349
References 351
Strain Rate Tensor Estimation in Cine Cardiac MRI Based on Elastic Image Registration 354
1 Introduction 354
2 Strain Tensor Definition 356
3 MRI Modalities to Estimate Strain and Strain Rate Tensor 358
4 Registration Methods 360
5 Phantom Design 362
6 Consistency Checking 364
7 Visualization 368
8 Real Data Results 370
9 Conclusion 375
References 376
Strain Tensor Elastography: 2D and 3D Visualizations 379
1 Introduction 379
2 Background 380
3 Strain Tensor Elastography (STE) 385
4 Experiments 389
5 2D Strain Tensor Elastography 390
6 3D Strain Tensor Elastography 392
7 Conclusions 397
References 399
Part V Storage, Visualization and Interfaces 402
Similar Tensor Arrays – A Framework for Storage of Tensor Array Data 403
1 Introduction 404
2 RelatedWork 405
3 Geometric Arrays 406
4 Scalar Arrays 407
5 Tensor Arrays 409
6 Examples of Use 413
7 Discussion 417
Appendix – Tensor Mathematics 417
References 423
User Interfaces to Interact with Tensor Fields 425
1 Introduction 425
2 Diffusion Tensor MRI 426
3 Visualization of Diffusion Tensor Fields 429
4 Command Line Interfaces 433
5 Graphical User Interfaces for Tensor Processing 435
6 Multimodal Interfaces for Tensor Processing 442
7 Discussion 444
8 Conclusions 446
References 447
T-flash: Tensor Visualization in Medical Studio 451
1 Introduction 451
2 The Tflash Tensor Glyph 452
3 Implementation as a MedicalStudio Component 455
4 Result 456
5 Conclusion 457
Appendix: Matlab code for Tflash 457
References 462
Index 463