Visualization and Processing of Tensor Fields (eBook)

Advances and Perspectives

David H. Laidlaw, Joachim Weickert (Herausgeber)

eBook Download: PDF

2009 | 2009
XVII, 376 Seiten
Springer Berlin (Verlag)
978-3-540-88378-4 (ISBN)

Infos

This book provides researchers an inspirational look at how to process and visualize complicated 2D and 3D images known as tensor fields. With numerous color figures, it details both the underlying mathematics and the applications of tensor fields.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 30.3.2009
• Reihe/Serie: Mathematics and Visualization
• Reihe/Serie: Mathematics and Visualization
• Zusatzinfo: XVII, 376 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Grafik / Design
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Calculus • Diffusion • diffusion tensor imaging • Imaging • Rendering • tensor field • Visualization
• ISBN-10: 3-540-88378-9 / 3540883789
• ISBN-13: 978-3-540-88378-4 / 9783540883784