Visualization and Processing of Tensor Fields

Matrix-valued data sets – so-called second order tensor fields – have gained significant importance in scientific visualization and image processing due to recent developments such as diffusion tensor imaging. This book is the first edited volume that presents the state of the art in the visualization and processing of tensor fields. It contains some longer chapters dedicated to surveys and tutorials of specific topics, as well as a great deal of original work by leading experts that has not been published before. It serves as an overview for the inquiring scientist, as a basic foundation for developers and practitioners, and as as a textbook for specialized classes and seminars for graduate and doctoral students.

An Introduction to Tensors.- Feature Detection with Tensors.- Adaptive Structure Tensors and their Applications.- On the Concept of a Local Greyvalue Distribution and the Adaptive Estimation of a Structure Tensor.- Low-level Feature Detection Using the Boundary Tensor.- Diffusion Tensor Imaging.- An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond.- Random Noise in Diffusion Tensor Imaging, its Destructive Impact and Some Corrections.- An Introduction to Visualization of Diffusion Tensor Imaging and Its Applications.- Anatomy-Based Visualizations of Diffusion Tensor Images of Brain White Matter.- Variational Regularization of Multiple Diffusion Tensor Fields.- Higher Rank Tensors in Diffusion MRI.- Visualization of Tensor Fields.- Strategies for Direct Visualization of Second-Rank Tensor Fields.- Tensor Invariants and their Gradients.- Visualizing the Topology of Symmetric, Second-Order, Time-Varying Two-Dimensional Tensor Fields.- Degenerate 3D Tensors.- Locating Closed Hyperstreamlines in Second Order Tensor Fields.- Tensor Field Visualization Using a Metric Interpretation.- Tensor Field Transformations.- Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization.- Continuous Tensor Field Approximation of Diffusion Tensor MRI data.- Tensor Field Interpolation with PDEs.- Diffusion-Tensor Image Registration.- Image Processing Methods for Tensor Fields.- Tensor Median Filtering and M-Smoothing.- Mathematical Morphology on Tensor Data Using the Loewner Ordering.- A Local Structure Measure for Anisotropic Regularization of Tensor Fields.- Tensor Field Regularization using Normalized Convolution and Markov Random Fields in a Bayesian Framework.- PDEs for Tensor Image Processing.

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.