Visualization And Processing of Tensor Fields,9783540250326

Visualization And Processing of Tensor Fields

by ;
ISBN13: 9783540250326
ISBN10: 3540250328
Format: Hardcover
Pub. Date: 2005-12-30
Publisher(s): Springer Verlag
List Price: $181.88

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Summary

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.

Author Biography

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.

Table of Contents

Introduction
1 An Introduction to Tensors
H. Hagen and C. Garth
3(80)
1.1 Some Linear Algebra
3(4)
1.2 Fundamentals of Differential Geometry
7(3)
1.3 Tensor Fields — A Mathematical Concept
10(3)
References
13(4)
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(32)
2.1 Introduction
17(3)
2.2 Data-adaptive Structure Tensors
20(8)
2.3 Optic Flow Estimation
28(9)
2.4 Texture Analysis
37(2)
2.5 Corner Detection
39(4)
2.6 Summary
43(1)
References
44(5)
3 On the Concept of a Local Greyvalue Distribution and the Adaptive Estimation of a Structure Tensor
H.-H. Nagel
49(14)
3.1 Introduction
49(1)
3.2 Greyvalue Structure Tensor of a Gaussian Bell
50(3)
3.3 Weighted Average of the Hessian
53(2)
3.4 Determination of Parameters of a Gaussian Bell
55(1)
3.5 Discussion
56(4)
References
60(3)
4 Low-level Feature Detection Using the Boundary Tensor
U. Köthe
63(20)
4.1 Introduction
63(3)
4.2 The Boundary Tensor
66(2)
4.3 Analysis of the Boundary Tensor as a Quadratic Filter
68(3)
4.4 Efficient Computation of the Boundary Tensor
71(2)
4.5 Applications
73(3)
4.6 Conclusions
76(2)
References
78(5)
Part II Diffusion Tensor Imaging
5 An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond
D.C. Alexander
83(24)
5.1 Introduction
83(2)
5.2 Diffusion-Weighted MRI
85(3)
5.3 Diffusion MRI Reconstruction Algorithms
88(10)
5.4 Applications
98(2)
5.5 Discussion
100(3)
References
103(4)
6 Random Noise in Diffusion Tensor Imaging, its Destructive Impact and Some Corrections
K.R. Hahn, S. Prigarin, S. Heim, K. Hasan
107(14)
6.1 Introduction
107(1)
6.2 Noise Impact
108(5)
6.3 Corrections of Noise Effects
113(4)
6.4 Conclusion
117(1)
References
117(4)
7 An Introduction to Visualization of Diffusion Tensor Imaging and Its Applications
A. Vilanova, S. Zhang, G. Kindlmann, D. Laidlaw
121(34)
7.1 Introduction
121(2)
7.2 Diffusion Tensor Imaging
123(2)
7.3 DTI Visualization
125(14)
7.4 Applications
139(6)
7.5 Open Problems
145(2)
7.6 Summary and Conclusions
147(1)
References
148(7)
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(10)
8.1 Introduction
155(2)
8.2 Methods
157(2)
8.3 Discussion
159(3)
References
162(3)
9 Variational Regularization of Multiple Diffusion Tensor Fields
O. Pasternak, N. Sochen, Y. Assaf
165(12)
9.1 Introduction
165(1)
9.2 Variational Approach for DTI Denoising
166(3)
9.3 Multiple Tensor Variational Framework for Fitting and Regularizing Diffusion Weighted Images
169(3)
9.4 Simulations
172(2)
9.5 Concluding Remarks
174(1)
References
175(2)
10 Higher Rank Tensors in Diffusion MRI
E. Özarslan, B.C. Vemuri, T.H. Mareci
177(14)
10.1 Introduction
177(3)
10.2 Quantification of Anisotropy from Higher Rank Tensors
180(4)
10.3 Fiber Orientations Implied by Higher Rank Tensors
184(3)
References
187(4)
Part III Visualization of Tensor Fields
11 Strategies for Direct Visualization of Second-Rank Tensor Fields
W. Benger and H.-C. Hege
191(24)
11.1 Introduction
191(4)
11.2 Visualization via Integral Manifolds
195(7)
11.3 Vertex-Based Visualization Methods
202(8)
11.4 Summary
210(3)
References
213(2)
12 Tensor Invariants and their Gradients
G. Kindlmann
215(10)
12.1 Background and Notation
215(1)
12.2 From Principal Invariants to Eigenvalues
216(1)
12.3 Eigenvalue Wheel
217(2)
12.4 Anatomical Significance of Eigenvalue Statistics
219(1)
12.5 Edge Detection with Invariant Gradients
220(1)
12.6 Application to Diffusion Tensor Images
221(2)
12.7 Discussion
223(1)
References
223(2)
13 Visualizing the Topology of 2D Tensor Fields
X. Tricoehe, X. Zheng, A. Pang
225(16)
13.1 Fundamental Notions of Two-Dimensional Tensor Field Topology
225(8)
13.2 Basic Topology Visualization
233(2)
13.3 Topology Simplification
235(2)
13.4 Topology Tracking
237(2)
13.5 Conclusion
239(1)
References
240(1)
14 Degenerate 3D Tensors
X. Zheng, X. Tricoche, A. Pang
241(16)
14.1 Introduction
241(2)
14.2 Dimensionality of Degenerate Features
243(1)
14.3 Implicit Function Approach
244(4)
14.4 Geometric Approach
248(3)
14.5 Topological Feature Lines
251(1)
14.6 Results
251(3)
14.7 Open Problems
254(1)
14.8 Conclusion
255(1)
References
256(1)
15 Locating Closed Hyperstreamlines in Second Order Tensor Fields
T. Wischgoll and J. Meyer
257(12)
15.1 Introduction
257(2)
15.2 Mathematical Background
259(1)
15.3 Detection of Closed Hyperstreamlines
260(3)
15.4 Results
263(2)
15.5 Conclusion
265(1)
References
266(3)
16 Tensor Field Visualization Using a Metric Interpretation
I. Hotz, L. Feng, H. Hagen, B. Hamann. K. Joy
269(16)
16.1 Introduction
269(1)
16.2 Related Work
270(1)
16.3 Metric Definition
271(6)
16.4 Results and Conclusions
277(3)
References
280(5)
Part IV Tensor Field Transformations
17 Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization
M. Moakher and P.G. Batchelor
285(14)
17.1 Introduction
285(1)
17.2 Geometry of the Space of SPD Matrices
286(3)
17.3 Anisotropy Indices
289(2)
17.4 Means
291(4)
17.5 Interpolation
295(2)
References
297(2)
18 Continuous Tensor Field Approximation of DT-MRI data
S. Pajevic, A. Aldroubi, P.J. Basser
299(16)
18.1 Introduction
299(2)
18.2 Continuous Approximation and Representation of Discrete Tensor Data
301(1)
18.3 B-Spline Approximation
302(3)
18.4 Non-Uniform Rational B-Splines (NURBS)
305(4)
18.5 B-spline vs NURBS Comparison on Curvature Estimation
309(1)
18.6 Discussion and Conclusion
310(2)
References
312(3)
19 Tensor Field Interpolation with PDEs
J. Weickert and M. Welk
315(12)
19.1 Introduction
315(1)
19.2 Scalar Interpolation
316(4)
19.3 Tensor Interpolation
320(3)
19.4 Summary
323(1)
References
324(3)
20 Diffusion-Tensor Image Registration
J.C. Gee and D.C. Alexander
327(18)
20.1 Introduction
327(1)
20.2 Background
328(5)
20.3 Warping DT-MRIs
333(3)
20.4 Review of Current DT-MRI Registration Literature
336(2)
20.5 Discussion
338(2)
References
340(5)
Part V Image Processing Methods for Tensor Fields
21 Tensor-Valued Median Filtering and M-Smoothing
M. Welk, C. Feddern, B. Burgeth, J. Weickert
345(12)
21.1 Introduction
345(1)
21.2 Scalar-Valued Median Filters
346(1)
21.3 Tensor-Valued Median Filters
347(2)
21.4 Mid-Range Filters and M-Smoothers
349(2)
21.5 Algorithmic Aspects
351(1)
21.6 Experiments
352(3)
21.7 Conclusion
355(1)
References
355(2)
22 Mathematical Morphology on Tensor Data Using the Loewner Ordering
B. Burgeth,, M. Welk, C. Feddern, J. Weickert
357(12)
22.1 Introduction
357(2)
22.2 Brief Review of Scalar Morphology
359(1)
22.3 Extremal Matrices in the Loewner Ordering
360(2)
22.4 Experimental Results
362(4)
22.5 Conclusions
366(1)
References
367(2)
23 A Local Structure Measure for Anisotropic Regularization of Tensor Fields
E. Suárez-Santana, M.A. Rodriguez-Florido, C. Castaño-Moraga, C.-F. Westin, J. Ruiz-Alzola
369(12)
23.1 Introduction
369(1)
23.2 The Structure Tensor
370(3)
23.3 Anisotropic Tensor Field Filtering
373(2)
23.4 Applications
375(5)
References
380(1)
24 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-Alzoln,, H. Knutsson
381(18)
24.1 Introduction
381(1)
24.2 Normalized Convolution
382(4)
24.3 Bayesian Regularization using Multivariate Gaussian Markov Random Fields
386(11)
24.4 Conclusion
397(1)
References
397(2)
25 PDEs for Tensor Image Processing
J. Weickert, C. Feddern, M. Welk, B. Burgeth, T. Brox
399(16)
25.1 Introduction
399(1)
25.2 Structure Analysis of Tensor-Valued Data
400(2)
25.3 Diffusion Filtering
402(4)
25.4 Regularisation Methods
406(1)
25.5 Mean Curvature Motion
407(1)
25.6 Self-Snakes
408(1)
25.7 Geodesic Active Contour Models
409(3)
25.8 Summary and Conclusions
412(1)
References
412(3)
Appendix Color Plates 415(58)
Index 473

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