Visualization and Processing of Tensor Fields

Transcription

1 Visualization and Processing of Tensor Fields

2 Joachim Weickert Hans Hagen (Editors) Visualization and Processing of Tensor Fields ABC

3 Professor Joachim Weickert Faculty of Mathematics and Computer Science Saarland University, Building Saarbrücken Germany Professor Hans Hagen Computer Graphics and Visualization Group Technical University of Kaiserslautern PO Box Kaiserslautern Germany Library of Congress Control Number: Mathematics Subject Classification Numbers (2000) 15A48, 15A72, 35J60, 35K55, 53-02, 53-06, 54-06, 62-07, 65D05, 65D17, 65D18, 68T45, 68U10, 65K10, 83-08, ISBN-10 ISBN Springer Berlin Heidelberg New York Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm orin any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German CopyrightLaw. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag BerlinHeidelberg 2006 Printed inthe Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LALT A EX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: /TechBooks

4 Preface Matrix-valued data sets so-called tensor fields have gained significant importance in the fields of scientific visualization and image processing. The tensor concept is a common physical description of anisotropic behaviour, especially in solid mechanics and civil engineering. It arises e.g. in the measurements of stress and strain, inertia, permeability and diffusion. In the field of medical imaging, diffusion tensor magnetic resonance imaging (DT-MRI) has become widespread in order to gain valuable insights into connectivity properties of the brain. Tensors have also shown their use as feature descriptors in image analysis, segmentation and grouping. These recent developments have created the need for appropriate tools for visualizing and processing tensor fields. Due to the multivariate structure of the data and their multidimensional variation in space, tensor field visualization belongs to the most challenging topics in the area of scientific visualization. Moreover, most signal and image processing methods have been developed for scalar- and vector-valued data sets, and only recently researchers have tried to investigate how they can be extended to tensor fields. In this case one has to take into account anumber of additional constraints such as an appropriate coupling of the different channels, and preservation of properties such as positive semidefiniteness during the filtering process. Unfortunately the results in the field of tensor-valued visualization and image processing are scattered in the literature, and often researchers in one application area are not aware of recent progress in another area. In order to address this problem, the editors of this book organized a perspective workshop that took place at Schloss Dagstuhl, Germany from April 18 to 23, In that week 30 invited scientists, representing many of the world-wide leading experts in tensor field visualization and processing, had the opportunity to exchange ideas in a highly inspiring atmosphere. Since many of them met for the first time, this exchange proved to be particularly fruitful. One of these fruits was the wish of all participants to compile their knowledge inasingle edited volume. The present book which is the first of its kindinthisfield is the result of these efforts. Its goal is to present the

5 VI Preface state-of-the-art in the visualization and processing of tensor fields, both as an overview for the inquiring scientist, and as a basic foundation for developers and practitioners. The book contains some longer chapters dedicated to surveys and tutorials of specific topics, as well as a great deal of original work not previously published. In all cases the emphasis has been on presenting the details necessary for others to reproduce the techniques and algorithms. Another goal of this book is to provide the basic material for teaching state-ofthe-art techniques in tensor field visualization and processing. It can therefore also serve as a textbook for specialized classes and seminars for graduate and doctoral students. An extended bibliography is included at the end of each chapter pointing out where to obtain further information. Organization of the Book This volume consists of 25 chapters. Each of them has been reviewed by two experts and carefully revised according to their suggestions. The chapters are arranged in five thematic areas. Color plates can be found in the Appendix. The book starts with an introductory chapter by Hagen and Garth. It gives the mathematical background from linear algebra and differential geometry that is necessary for understanding the concept of tensor fields. Part I of the book is devoted to feature detection using tensors. Here one typically starts with scalar- or vector-valued images and creates tensorvalued features that are suitable for corner detection, texture analysis or optic flow estimation. Structure tensors are the most prominent representatives of these concepts. Chapter 2 by Brox et al. is a survey chapter on adaptive structure tensors, while Chap. 3 by Nagel analyzes closed form solutions for a structure tensor concept in image sequence analysis. An alternative framework for tensor-valued feature detection is presented in Chap. 4 by Köthe who shows that the so-called boundary tensor may overcome some problems of more traditional approaches. Part II deals with the currently most important technique for creating tensor-valued images, namely Diffusion Tensor Magnetic Resonance Imaging (DT-MRI), often simply called Diffusion Tensor Imaging (DTI). This technique measures the diffusion properties of water molecules and has gained significant popularity in medical imaging of the brain. Chapter 5 by Alexander gives a general overview of the principles of biomedical diffusion MRI and algorithms for reconstructing the diffusion tensor fields, while the subsequent chapter by Hahn et al. describes the empirical origin of noise and analyzes its influence on the DT-MRI variables. After having understood the formation of DT-MR images their adequate visualization remains a challenging task. This is the topic of the survey chapter by Vilanova et al. that also sketches the clinical impact of DT-MR imaging. A more specific medical application is treated in Chap. 8 by Gee et al. who study anatomical labeling of cerebral white matter in DT-MR images. For conventional DT-MR imaging, the identification and

6 Preface VII analysis of fibre crossings constitutes a severe difficulty. In Chap. 9, Pasternak et al. introduce a variational image processing framework for resolving these ambiguities. An alternative strategy is investigated in Chap. 10 by Özarslan et al. They reconstruct higher-order tensors from the MR measurements. These allow to encodearicher orientational heterogeneity. In the third part, general visualization strategies for tensor fields are explored. This part starts with a review chapter by Bengerand Hege who also consider applications in relativity theory. Kindlmann s chapter is concerned with visualizing discontinuities in tensor fields by computing the gradients of invariants. The subsequent Chaps investigate different strategies for understanding the complex nature of tensor fields by extracting suitable differential geometric information. While Chap. 13 by Tricoche et al. deals with the topology and simplification of static and time-variant 2-D tensor fields, Chap. 14 by Zheng et al. is concerned with a novel and numerically stable analysis of degerated tensors in 3-D fields. In Chap. 15, Wischgoll and Meyer investigate the detection of alternative topological features, namely closed hyperstreamlines. Thethird part is concluded with achapter by Hotzetal. who introduce specific visualization concepts for stress and strain tensor fields by interpreting them as distortions of a flat metric. Part IV of the book is concerned with transformations of tensor fields, in particular interpolation and registration strategies. In Chap. 17, Moakher and Batchelor perform a differential geometric analysis of the space of positive definite tensors in order to derive appropriate interpolation methods. The next chapter by Pajevic et al. deals with non-uniform rational B-splines (NURBS) as a flexible interpolation tool, while in Chap. 19 Weickert and Welk introduce a rotationally invariant framework for tensor field approximation, interpolation and inpainting. It is based on partial differential equations (PDEs). Finally, Chap. 20 by Gee and Alexander treats the problem of diffusion tensor registration. Compared to scalar-valued registration approaches, incorporating the orientation information provides additional challenges. The fifth part is a collection of contributions on signal and image processing methods that are specifically developed to deal with tensor fields. Chapter 21 by Welk et al. as well as Chap. 22 by Burgeth et al. show that seemingly simple ideas like median filtering and morphological image processing can create substantial difficulties when one wants to generalize them to tensor fields. Since tensor lack a full ordering, many straightforward concepts cannot be applied and alternative generalizations become necessary. In Chap. 23, Suarez-Santana et al. review adaptive local filters for tensor field regularization and interpolation that are steered by a structure tensor, while Chap. 24 by Westin et al. is concerned with two other regularization techniques for tensor fields: normalized convolution and Markov random fields. These ideas a complemented by Chap. 25 where Weickert et al. survey the most important PDE approaches for discontinuity-preserving smoothing and segmentation of tensor fields.

7 VIII Preface Acknowledgements It isapleasure to thank all authors for their excellent contributions. We are very grateful to Reinhard Wilhelm and the Scientific Directorate for accepting our proposal for a Dagstuhl Perspective Workshop on Visualization and Processing of Tensor Fields, and the local team at Schloss Dagstuhl for providing a very pleasant atmosphere that allowed all participants to enjoy this workshop with real scientific benefits. Moreover, we thank the editors of the Springer book series Mathematics and Visualization as well as Martin Peters and Ute McCrory (Springer, Heidelberg) for their support to publish this edited volume in their series. Last but not least, we are greatly indebted to Martin Welk (Saarland University) who was of substantial help in the final compilation of all chapters to a coherent, single volume. Saarbrücken and Kaiserslautern, June 2005 Joachim Weickert Hans Hagen

8 Contents Introduction 1AnIntroduction to Tensors H. Hagen and C. Garth SomeLinearAlgebra Fundamentals of Differential Geometry Tensor Fields A Mathematical Concept References Part I Feature Detection with Tensors 2Adaptive Structure Tensors and their Applications T. Brox, R. van den Boomgaard, F. Lauze, J. van de Weijer, J. Weickert, P. Kornprobst Introduction Data-adaptiveStructureTensors Optic Flow Estimation TextureAnalysis CornerDetection Summary References On the Concept of a Local Greyvalue Distribution and the Adaptive Estimation of a Structure Tensor H.-H. Nagel Introduction Greyvalue Structure Tensor of a GaussianBell Weighted Average of the Hessian Determination of Parameters of a GaussianBell Discussion References... 60

9 X Contents 4 Low-level Feature Detection Using the Boundary Tensor U. K the Introduction The BoundaryTensor Analysis of the Boundary Tensor as a Quadratic Filter Efficient Computation of the Boundary Tensor Applications Conclusions References Part II Diffusion Tensor Imaging 5 An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond D. C. Alexander Introduction Diffusion-WeightedMRI Diffusion MRI Reconstruction Algorithms Applications Discussion References Random Noise in Diffusion Tensor Imaging, its Destructive Impact and Some Corrections K. R. Hahn, S. Prigarin, S. Heim, K. Hasan Introduction Noise Impact Corrections of Noise Effects Conclusion References An Introduction to Visualization of Diffusion Tensor Imaging and Its Applications A. Vilanova, S. Zhang, G. Kindlmann, D. Laidlaw Introduction Diffusion Tensor Imaging DTI Visualization Applications OpenProblems Summary and Conclusions References...148

10 Contents XI 8 Anatomy-Based Visualizations of Diffusion Tensor Images of Brain White Matter J. C. Gee, H. Zhang, A. Dubb, B. B. Avants, P. A. Yushkevich, J. T.Duda Introduction Methods Discussion References Variational Regularization of Multiple Diffusion Tensor Fields O. Pasternak, N. Sochen, Y. Assaf Introduction VariationalApproachforDTIDenoising Multiple Tensor Variational Framework for Fitting and Regularizing Diffusion Weighted Images Simulations Concluding Remarks References Higher Rank Tensors in Diffusion MRI E. Özarslan, B. C. Vemuri, T. H. Mareci Introduction Quantification of Anisotropy from Higher RankTensors Fiber Orientations Implied by HigherRankTensors References Part III Visualization of Tensor Fields 11 Strategies for Direct Visualization of Second-Rank Tensor Fields W. Benger and H.-C. Hege Introduction Visualization via Integral Manifolds Vertex-Based Visualization Methods Summary References Tensor Invariants and their Gradients G. Kindlmann Background and Notation From Principal Invariants to Eigenvalues Eigenvalue Wheel Anatomical Significance of Eigenvalue Statistics Edge Detection with Invariant Gradients...220

11 XII Contents 12.6 ApplicationtoDiffusionTensorImages Discussion References Visualizing the Topology of 2D Tensor Fields X. Tricoche, X. Zheng, A. Pang Fundamental Notions of Two-Dimensional Tensor Field Topology Basic Topology Visualization TopologySimplification Topology Tracking Conclusion References Degenerate 3D Tensors X. Zheng, X. Tricoche, A. Pang Introduction Dimensionality of Degenerate Features ImplicitFunctionApproach GeometricApproach Topological Feature Lines Results OpenProblems Conclusion References Locating Closed Hyperstreamlines in Second Order Tensor Fields T. Wischgoll and J. Meyer Introduction Mathematical Background Detection of Closed Hyperstreamlines Results Conclusion References Tensor Field Visualization Using a Metric Interpretation I. Hotz, L. Feng, H.Hagen, B. Hamann, K. Joy Introduction RelatedWork Metric Definition ResultsandConclusions References...280

12 Contents XIII Part IV Tensor Field Transformations 17 Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization M. Moakher and P. G. Batchelor Introduction GeometryoftheSpaceofSPDMatrices AnisotropyIndices Means Interpolation References Continuous Tensor Field Approximation of DT-MRI data S. Pajevic, A. Aldroubi, P. J. Basser Introduction Continuous Approximation and Representation of DiscreteTensorData B-Spline Approximation Non-Uniform Rational B-Splines (NURBS) B-spline vs NURBS Comparison on Curvature Estimation Discussion and Conclusion References Tensor Field Interpolation with PDEs J. Weickert and M. Welk Introduction ScalarInterpolation TensorInterpolation Summary References Diffusion-Tensor Image Registration J. C. Gee and D. C. Alexander Introduction Background Warping DT-MRIs Review of Current DT-MRI Registration Literature Discussion References...340

13 XIV Contents Part V Image Processing Methods for Tensor Fields 21 Tensor-Valued Median Filtering and M-Smoothing M. Welk, C. Feddern, B. Burgeth, J. Weickert Introduction Scalar-Valued Median Filters Tensor-Valued Median Filters Mid-Range Filters and M-Smoothers Algorithmic Aspects Experiments Conclusion References Mathematical Morphology on Tensor Data Using the Loewner Ordering B. Burgeth, M. Welk, C. Feddern, J. Weickert Introduction Brief Review of Scalar Morphology Extremal Matrices in the Loewner Ordering Experimental Results Conclusions References A Local Structure Measure for Anisotropic Regularization of Tensor Fields E. Suarez-Santana, M. A. Rodriguez-Florido, C. Castaño-Moraga, C.-F. Westin, J. Ruiz-Alzola Introduction The StructureTensor Anisotropic Tensor Field Filtering Applications References Tensor Field Regularization using Normalized Convolution and Markov Random Fields in a Bayesian Framework C.-F. Westin, M. Martin-Fernandez, C. Alberola-Lopez, J. Ruiz-Alzola, H. Knutsson Introduction Normalized Convolution Bayesian Regularization using Multivariate Gaussian Markov RandomFields Conclusion References...397

14 Contents XV 25 PDEs for Tensor Image Processing J. Weickert, C. Feddern, M. Welk, B. Burgeth, T. Brox Introduction Structure Analysis of Tensor-Valued Data Diffusion Filtering Regularisation Methods Mean Curvature Motion Self-Snakes Geodesic Active Contour Models SummaryandConclusions References Appendix Color Plates Index...473