Visualization and Processing of Tensor Fields

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.

• 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.

"This timely Springer monograph collects a total of 25 survey and research articles on the analysis, processing, and visualization of tensor fields, an area with increasing impact and visibility in contemporary imaging and medical image analysis. Compared with many existing volumes of similar styles or missions, this collection has been carefully edited to yield a smooth and continuous flow of topics and techniques. The categorization into five self-contained parts has been enormously helpful for a generic reader to attain an organic view of the entire dynamic field. This volume will be an invaluable research tool or information source for all readers who are interested in tensor processing and visualization, including, e.g., image processing experts, graphic and visualization engineers, medical imaging scientists, and radiologists, as well as graduate students who are pursuing a degree in the related research subjects." (Jianhong Shen, Mathematical Reviews)