Visualization and Processing of Tensor Fields (eBook)

Preface 6
Contents 7
List of Contributors 10
Part I Models for Diffusion MRI 15
Modelling, Fitting and Sampling in Diffusion MRI 16
1 Introduction 16
2 Background 17
3 Methods 19
4 Experiments and Results 26
5 Discussion 29
References 29
Tensors, Polynomials and Models for Directional Data 34
1 Introduction 34
2 Tensors, Symmetric Tensors and Multivariate Polynomials 34
3 Signals and Directional Signals 41
4 Summary and Conclusions 48
Acknowledgments 48
References 48
A Mixture of Wisharts (MOW) Model for Multifiber Reconstruction 51
1 Introduction 51
2 A Mixture of Wisharts Statistical Model 53
3 Stable, Sparse, and Positive Deconvolution 57
4 Experimental Results 60
5 Conclusions 65
Acknowledgments 65
References 65
The Algebra of Fourth-Order Tensors with Application to Diffusion MRI 69
1 Introduction 69
2 Second-Order Tensors 71
3 Fourth-Order Tensors 74
4 Symmetries of Fourth-Order Tensors 75
5 Fourth-Order Tensors in 3D Space 77
6 Material Symmetries 79
7 Orientation Distributions and Orientation Tensors 84
8 Fourth-Order Diffusion Tensors 89
References 91
Part II Higher-Level Analysis of Diffusion Images 93
Structure-Specific Statistical Mapping of White Matter Tracts* 94
1 Introduction 94
2 Background 96
3 Methods and Materials 97
4 Experimental Evaluation 106
5 Discussion and Conclusions 115
Acknowledgements 119
References 119
Analysis of Distance/Similarity Measures for Diffusion Tensor Imaging 124
1 Introduction 124
2 Notation 125
3 Properties 126
4 Measures 128
5 Methods 133
6 Experiments 136
7 Conclusions 144
Acknowledgments 145
References 145
Part III Tensor Field Visualization 148
Tensor Glyph Warping: Visualizing Metric Tensor Fields using Riemannian Exponential Maps 149
1 Introduction 149
2 Glyphs and Glyph Warping 150
3 Related Work 151
4 Tensors and Index Notation 155
5 The Metric and Metric Spheres 157
8 Solving the Geodesic Equation 161
9 Geodesic Spheres and Warped Coordinate Systems 162
10 The Logarithmic Map 162
11 Experiments 164
12 Conclusion 167
Acknowledgments 168
References 168
Interactive Volume Rendering of Diffusion Tensor Data 171
1 Introduction 171
2 Volume Rendering 174
3 Application and Results 178
4 Conclusions and Future Work 183
Acknowledgements 183
References 184
Dense Glyph Sampling for Visualization 187
1 Introduction 187
2 Related Work 188
3 Assumptions and Goals 189
4 Algorithm 190
5 Structural Behavior 193
6 Results 196
7 Conclusion 200
Acknowledgments 201
References 201
Part IV Tensor Field Analysis in the Physical Sciences 204
The Role of Tensor Fields for Satellite Gravity Gradiometry 205
1 Satellite Gravity Gradiometry 205
2 Nonuniqueness Results 209
3 Spherical Approach for SGG 210
4 Decomposition of Spherical Tensor Fields 211
5 Existence and Uniqueness of the SGG Problem 216
6 Tensorial Zonal Kernels 218
References 219
Tensor Visualization and Defect Detection for Nematic Liquid Crystals using Shape Characteristics 221
1 Introduction 221
2 Background 222
3 Tensor Shape Characteristics for NLC Alignment 225
4 Shape-Based Tensor Glyphs for NLC Visualization 231
5 Detecting Topological Defects with Shape Characteristics 236
6 Combining the Two Methods 241
7 Summary and Conclusions 243
Acknowledgments 244
References 244
A Tensor Approach to Elastography Analysis and Visualization 247
1 Introduction 247
2 Background: The Theory of Elasticity 249
3 Methods 253
4 Experimental Set-up and Results 255
5 Discussion 259
6 Conclusion and Ongoing Research 266
Acknowledgments 266
References 266
Part V Tensor Image Structure Models 269
A Higher-Order Structure Tensor 270
1 Introduction 270
2 A Higher-Order Structure Tensor 271
3 Glyphs for Higher-Order Tensors 272
4 A Mathematical Toolbox 275
5 Integrated Edge and Junction Detection 281
6 Conclusions and Future Work 283
Acknowledgements 284
References 284
Monogenic Curvature Tensor as Image Model* 287
1 Introduction 287
2 Related Work 289
3 Monogenic Curvature Tensor and its Evaluation 297
4 Conclusions 304
References 304
Part VI Filtering with Tensors 308
A General Structure Tensor Concept and Coherence-Enhancing Diffusion Filtering for Matrix Fields 309
1 Introduction 309
2 Synopsis of Coherence Enhancing Diffusion 312
3 Basic Differential Calculus for Matrix Fields 313
4 The Full Structure Tensor SL for Matrix Fields 316
5 Coherence-Enhancing Diffusion Filtering for Matrix Fields 320
6 Experiments 321
7 Summary and Future Work 325
Acknowledgments 325
References 325
Coordinates-Based Diffusion Over the Space of Symmetric Positive-Definite Matrices 328
1 Introduction 328
2 Images and Fiber Bundles 330
3 Calculus of Variations in Fiber Bundles 332
4 Diffusion on the Space of SPD Matrices 334
5 Summary 341
References 341
Variational Methods for Denoising Matrix Fields 344
1 Introduction 344
2 Variational Methods for Scalar-Valued Images 345
3 Variational Methods for Matrix-Valued Images 349
4 Numerical Results 353
A Proofs 357
Acknowledgements 361
References 361
An Operator Algebraic Inverse Scale Space Method for Symmetric Matrix Valued Images 364
1 Introduction 364
2 Inverse Methods for Matrix Valued Images 367
3 Numerical Implementation 370
4 Numerical Experiments 370
5 Conclusions 375
Acknowledgements 376
References 377