Visualization and Processing of Tensor Fields: Proceedings of the Dagstuhl Workshop

482 Pages·2005·20.905 MB·English

by  Joachim Weickert, Hans Hagen

VisualizationandProcessingofTensorFields
Joachim Weickert Hans Hagen (Editors)
Visualization and
Processing of Tensor
Fields
ABA C
ProfessorJoachimWeickert ProfessorHansHagen
FacultyofMathematicsand ComputerGraphicsandVisualizationGroup
ComputerScience TechnicalUniversityofKaiserslautern
SaarlandUniversity,Building27 POBox3049
66041Saarbrücken 67653Kaiserslautern
Germany Germany
E-mail:[email protected] E-mail:[email protected]
LibraryofCongressControlNumber:2005931221
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, 86-08
ISBN-10 3-540-25032-8SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-25032-6SpringerBerlinHeidelbergNewYork
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Preface
Matrix-valued data sets – so-called tensor fields – have gained significant im-
portance 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 mea-
surementsof stressandstrain,inertia, permeability anddiffusion.Inthefield
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 descrip-
tors 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 struc-
ture 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,mostsignalandimageprocessingmethodshavebeen
developedforscalar-andvector-valueddatasets,andonlyrecentlyresearchers
havetriedtoinvestigatehowtheycanbeextendedtotensorfields.Inthiscase
one has to take into account a number 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 work-
shopthattookplaceatSchlossDagstuhl,GermanyfromApril18to23,2004.
In that week 30 invited scientists, representing many of the world-wide lead-
ing 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.
Oneofthesefruitswasthewishofallparticipantstocompiletheirknowl-
edge in a single edited volume. The present book – which is the first of its
kind in this field – is the result of these efforts. Its goal is to present the
VI Preface
state-of-the-artinthevisualizationandprocessingoftensorfields,bothasan
overview for the inquiring scientist, and as a basic foundation for developers
and practitioners. The book contains some longer chapters dedicated to sur-
veys 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.
Anothergoalofthisbookistoprovidethebasicmaterialforteachingstate-of-
the-arttechniquesintensorfieldvisualizationandprocessing.Itcantherefore
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.
ThebookstartswithanintroductorychapterbyHagenandGarth.Itgives
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 tensor-
valuedfeaturesthataresuitableforcornerdetection,textureanalysisoroptic
flow estimation. Structure tensors are the most prominent representatives of
these concepts. Chapter 2 by Brox et al. is a survey chapter on adaptive
structuretensors,whileChap.3byNagelanalyzesclosedformsolutionsfora
structuretensorconceptinimagesequenceanalysis.Analternativeframework
for tensor-valued feature detection is presented in Chap. 4 by Ko¨¨the 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 tech-
nique measures the diffusion properties of water molecules and has gained
significant popularity in medical imaging of the brain. Chapter 5 by Alexan-
dergivesageneraloverviewoftheprinciplesofbiomedicaldiffusionMRIand
algorithms for reconstructing the diffusion tensor fields, while the subsequent
chapter by Hahn et al. describesthe empirical origin of noiseand analyzesits
influenceontheDT-MRIvariables.Afterhavingunderstoodtheformationof
DT-MRimagestheiradequatevisualizationremainsachallengingtask.Thisis
thetopicofthesurveychapterbyVilanovaetal.thatalsosketchestheclinical
impact of DT-MR imaging. A more specific medical application is treated in
Chap.8byGeeetal.whostudyanatomical labelingofcerebralwhitematter
in DT-MR images. For conventional DT-MR imaging, the identification and
Preface VII
analysisoffibrecrossingsconstitutesaseveredifficulty.InChap.9,Pasternak
et al. introduce a variational image processing framework for resolving these
ambiguities.AnalternativestrategyisinvestigatedinChap.10byO¨zarslanet
al. They reconstruct higher-order tensors from the MR measurements. These
allow to encode a richer orientational heterogeneity.
In the third part, general visualization strategies for tensor fields are ex-
plored. This part starts with a review chapter by Benger and 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. 13–16 investigate different strategies for
understanding the complex nature of tensor fields by extracting suitable dif-
ferential 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
investigatethedetectionofalternativetopologicalfeatures,namelyclosedhy-
perstreamlines.ThethirdpartisconcludedwithachapterbyHotzetal.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 pos-
itive 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
Welkintroducearotationallyinvariantframeworkfortensorfieldapproxima-
tion, interpolation and inpainting. It is based on partial differential equations
(PDEs). Finally, Chap. 20 by Gee and Alexander treats the problem of diffu-
sion 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 process-
ingmethodsthatarespecificallydevelopedtodealwithtensorfields.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 can-
notbeappliedandalternativegeneralizationsbecomenecessary.InChap.23,
Sua´rez-Santana et al. review adaptive local filters for tensor field regulariza-
tion and interpolation that are steered by a structure tensor, while Chap. 24
byWestinetal.isconcernedwithtwootherregularizationtechniquesforten-
sor 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.
VIII Preface
Acknowledgements
It is a pleasure to thank all authors for their excellent contributions. We are
very grateful to Reinhard Wilhelm and the Scientific Directorate for accept-
ing our proposal for a Dagstuhl Perspective Workshop on Visualization and
Processing of Tensor Fields, and the local team at Schloss Dagstuhl for pro-
viding 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.
Saarbru¨¨cken and Kaiserslautern, Joachim Weickert
June 2005 Hans Hagen
Contents
Introduction
1 An Introduction to Tensors
H. Hagen and C. Garth........................................... 3
1.1 Some Linear Algebra......................................... 3
1.2 Fundamentals of Differential Geometry......................... 7
1.3 Tensor Fields – A Mathematical Concept....................... 10
References ...................................................... 13
Part I Feature Detection with Tensors
2 Adaptive Structure Tensors and their Applications
T. Brox, R. van den Boomgaard, F. Lauze, J. van de Weijer,
J. Weickert, P. Kornprobst........................................ 17
2.1 Introduction ................................................ 17
2.2 Data-adaptive Structure Tensors .............................. 20
2.3 Optic Flow Estimation ....................................... 28
2.4 Texture Analysis ............................................ 37
2.5 Corner Detection............................................ 39
2.6 Summary................................................... 43
References ...................................................... 44
3 On the Concept of a Local Greyvalue Distribution
and the Adaptive Estimation of a Structure Tensor
H.-H. Nagel ..................................................... 49
3.1 Introduction ................................................ 49
3.2 Greyvalue Structure Tensor of a Gaussian Bell .................. 50
3.3 Weighted Average of the Hessian .............................. 53
3.4 Determination of Parameters of a Gaussian Bell................. 55
3.5 Discussion.................................................. 56
References ...................................................... 60
X Contents
4 Low-level Feature Detection Using the Boundary Tensor
U. K¨o¨the........................................................ 63
4.1 Introduction ................................................ 63
4.2 The Boundary Tensor........................................ 66
4.3 Analysis of the Boundary Tensor as a Quadratic Filter ........... 68
4.4 Efficient Computation of the Boundary Tensor .................. 71
4.5 Applications ................................................ 73
4.6 Conclusions................................................. 76
References ...................................................... 78
Part II Diffusion Tensor Imaging
5 An Introduction to Computational Diffusion MRI:
the Diffusion Tensor and Beyond
D. C. Alexander ................................................. 83
5.1 Introduction ................................................ 83
5.2 Diffusion-Weighted MRI...................................... 85
5.3 Diffusion MRI Reconstruction Algorithms ...................... 88
5.4 Applications ................................................ 98
5.5 Discussion..................................................100
References ......................................................103
6 Random Noise in Diffusion Tensor Imaging,
its Destructive Impact and Some Corrections
K. R. Hahn, S. Prigarin, S. Heim, K. Hasan ........................107
6.1 Introduction ................................................107
6.2 Noise Impact ...............................................108
6.3 Corrections of Noise Effects...................................113
6.4 Conclusion .................................................117
References ......................................................117
7 An Introduction to Visualization of Diffusion Tensor
Imaging and Its Applications
A. Vilanova, S. Zhang, G. Kindlmann, D. Laidlaw ...................121
7.1 Introduction ................................................121
7.2 Diffusion Tensor Imaging ....................................123
7.3 DTI Visualization ...........................................125
7.4 Applications ................................................139
7.5 Open Problems .............................................145
7.6 Summary and Conclusions....................................147
References ......................................................148
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......................................................155
8.1 Introduction ................................................155
8.2 Methods ...................................................157
8.3 Discussion..................................................159
References ......................................................162
9 Variational Regularization
of Multiple Diffusion Tensor Fields
O. Pasternak, N. Sochen, Y. Assaf.................................165
9.1 Introduction ................................................165
9.2 Variational Approach for DTI Denoising........................166
9.3 Multiple Tensor Variational Framework for Fitting
and Regularizing Diffusion Weighted Images ....................169
9.4 Simulations.................................................172
9.5 Concluding Remarks.........................................174
References ......................................................175
10 Higher Rank Tensors in Diffusion MRI
E. O¨zarslan, B. C. Vemuri, T. H. Mareci ...........................177
10.1 Introduction ................................................177
10.2 Quantification of Anisotropy from Higher Rank Tensors ..........180
10.3 Fiber Orientations Implied by Higher Rank Tensors..............184
References ......................................................187
Part III Visualization of Tensor Fields
11 Strategies for Direct Visualization
of Second-Rank Tensor Fields
W. Benger and H.-C. Hege........................................191
11.1 Introduction ................................................191
11.2 Visualization via Integral Manifolds............................195
11.3 Vertex-Based Visualization Methods ...........................202
11.4 Summary...................................................210
References ......................................................213
12 Tensor Invariants and their Gradients
G. Kindlmann...................................................215
12.1 Background and Notation ....................................215
12.2 From Principal Invariants to Eigenvalues .......................216
12.3 Eigenvalue Wheel ...........................................217
12.4 Anatomical Significance of Eigenvalue Statistics .................219
12.5 Edge Detection with Invariant Gradients .......................220