Visualization and Processing of Tensor Fields

1. Front Matter

   Pages I-XV

   PDF

2. Introduction

   1. Front Matter

      Pages 1-1

      PDF

   2. An Introduction to Tensors

      • Hans Hagen, Christoph Garth

      Pages 3-13

3. Feature Detection with Tensors

   1. Front Matter

      Pages 15-15

      PDF

   2. Adaptive Structure Tensors and their Applications

      • Thomas Brox, Rein van den Boomgaard, François Lauze, Joost van de Weijer, Joachim Weickert, Pavel Mrázek et al.

      Pages 17-47

   3. On the Concept of a Local Greyvalue Distribution and the Adaptive Estimation of a Structure Tensor

      • Hans-Hellmut Nagel

      Pages 49-61

   4. Low-level Feature Detection Using the Boundary Tensor

      • Ullrich Köthe

      Pages 63-79

4. Diffusion Tensor Imaging

   1. Front Matter

      Pages 81-81

      PDF

   2. An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond

      • Daniel C. Alexander

      Pages 83-106

   3. Random Noise in Diffusion Tensor Imaging, its Destructive Impact and Some Corrections

      • Klaus R. Hahn, Sergej Prigarin, Susanne Heim, Khader Hasan

      Pages 107-119

   4. An Introduction to Visualization of Diffusion Tensor Imaging and Its Applications

      • A. Vilanova, S. Zhang, G. Kindlmann, D. Laidlaw

      Pages 121-153

   5. Anatomy-Based Visualizations of Diffusion Tensor Images of Brain White Matter

      • James C. Gee, Hui Zhang, Abraham Dubb, Brian B. Avants, Paul A. Yushkevich, Jeffrey T. Duda

      Pages 155-163

   6. Variational Regularization of Multiple Diffusion Tensor Fields

      • Ofer Pasternak, Nir Sochen, Yaniv Assaf

      Pages 165-176

   7. Higher Rank Tensors in Diffusion MRI

      • Evren Özarslan, Baba C. Vemuri, Thomas H. Mareci

      Pages 177-187

5. Visualization of Tensor Fields

   1. Front Matter

      Pages 189-189

      PDF

   2. Strategies for Direct Visualization of Second-Rank Tensor Fields

      • Werner Benger, Hans-Christian Hege

      Pages 191-214

   3. Tensor Invariants and their Gradients

      • Gordon Kindlmann

      Pages 215-224

   4. Visualizing the Topology of Symmetric, Second-Order, Time-Varying Two-Dimensional Tensor Fields

      • Xavier Tricoche, Xiaoqiang Zheng, Alex Pang

      Pages 225-240

   5. Degenerate 3D Tensors

      • Xiaoqiang Zheng, Xavier Tricoche, Alex Pang

      Pages 241-256

   6. Locating Closed Hyperstreamlines in Second Order Tensor Fields

      • Thomas Wischgoll, Joerg Meyer

      Pages 257-267

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.

• 3D
• differential equation
• differential geometry
• diffusion tensor imaging
• image processing
• measure
• scientific visualization
• sets
• tensor fields
• topology
• visualization

"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)

• Faculty of Mathematics and Computer Science, Saarland University, Saarbrücken, Germany

  Joachim Weickert

• Computer Graphics and Visualization Group, Technical University of Kaiserslautern, Kaiserslautern, Germany

  Hans Hagen

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.

Policies and ethics