Visualization and Processing of Tensor Fields (eBook)

Joachim Weickert is Full Professor of Mathematics and Computer Science at    Saarland University (Saarbr/"ucken, Germany) where he heads the Mathematical    Image Analysis Group. He performs research in image processing, computer    vision and scientific computing, focusing on techniques based on partial    differential equations and variational methods.     Hans Hagen is heading the research group for Computer Graphics and   Computer Geometry at the University of Kaiserslautern, Germany, and is   Scientific Director of the research lab Intelligent Visualization and   Simulation at the German Research Center for Artificial Intelligence   (DFKI). His research domains are geometric modeling and scientific    visualization.

Preface 5
Contents 9
Introduction 16
1 An Introduction to Tensors 17
Feature Detection with Tensors 28
2 Adaptive Structure Tensors and their Applications 29
3 On the Concept of a Local Greyvalue Distribution and the Adaptive Estimation of a Structure Tensor 60
4 Low-level Feature Detection Using the Boundary Tensor 73
Diffusion Tensor Imaging 90
5 An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond 91
6 Random Noise in Diffusion Tensor Imaging, its Destructive Impact and Some Corrections 115
7 An Introduction to Visualization of Diffusion Tensor Imaging and Its Applications 128
8 Anatomy-Based Visualizations of Diffusion Tensor Images of Brain White Matter 161
9 Variational Regularization of Multiple Diffusion Tensor Fields 170
10 Higher Rank Tensors in Diffusion MRI 182
Visualization of Tensor Fields 193
11 Strategies for Direct Visualization of Second- Rank Tensor Fields 194
12 Tensor Invariants and their Gradients 218
13 Visualizing the Topology of Symmetric, Second- Order, Time- Varying Two- Dimensional Tensor Fields 228
14 Degenerate 3D Tensors 244
15 Locating Closed Hyperstreamlines in Second Order Tensor Fields 260
16 Tensor Field Visualization Using a Metric Interpretation 271
Tensor Field Transformations 284
17 Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization 285
18 Continuous Tensor Field Approximation of Diffusion Tensor MRI data 299
19 Tensor Field Interpolation with PDEs 315
20 Diffusion-Tensor Image Registration 326
Image Processing Methods for Tensor Fields 342
21 Tensor Median Filtering and M-Smoothing 343
22 Mathematical Morphology on Tensor Data Using the Loewner Ordering 355
23 A Local Structure Measure for Anisotropic Regularization of Tensor Fields 367
24 Tensor Field Regularization using Normalized Convolution and Markov Random Fields in a Bayesian Framework 379
25 PDEs for Tensor Image Processing 397
A Color Plates 413
Index 470