Anisotropy Across Fields and Scales

✍ Scribed by Evren Özarslan, Thomas Schultz, Eugene Zhang, Andrea Fuster (eds.)

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 284
• Series: Mathematics and Visualization
• Category: Library

✦ Table of Contents

Preface
Contents
Foundations
Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions
1 Introduction
1.1 Outline
2 Preliminaries
2.1 Tensor Notation and Representations
2.2 Invariants, Traces and Decompositions
3 Rabcd as a Quadratic Form on mathbbR3
3.1 Representation of the Canonically Derived Parts of Rabcd
3.2 The Behaviour of Mij Under a Rotation of the Coordinate System in Va
4 The Equivalence Problem for Rabcd
4.1 Different Ways to Characterize the Equivalence of Rabcd and widetildeRabcd
5 Discussion
References
Degenerate Curve Bifurcations in 3D Linear Symmetric Tensor Fields
1 Introduction
2 Previous Work
3 Background on Tensors and Tensor Fields
3.1 Tensors
3.2 Tensor Field Topology
3.3 3D Linear Tensor Fields
4 Bifurcations
4.1 Degenerate Curve Removal and Generation
4.2 Degenerate Curve Reconnection
4.3 Transition Point Pair Cancellation and Generation
4.4 Transition Point Relocation
5 Conclusion
References
Continuous Histograms for Anisotropy of 2D Symmetric Piece-Wise Linear Tensor Fields
1 Introduction
2 Context and Related Work
2.1 Continuous Histograms
2.2 Notes on Tensor Field Interpolation
2.3 Contour Trees, a Topological Summary of Scalar Functions
3 Problem Statement and Solution Overview
4 Background and Notations
4.1 Second Order Symmetric Tensors and Anisotropy
4.2 Barycentric Coordinates and Piece-Wise Linear Interpolation
4.3 Bivariate Quadratic Functions and Their Critical Points
5 Anisotropy for 2D Piece-Wise Linear Tensor Fields
5.1 Field Normalization Using Coordinate Transformations
6 Subdivision in Monotonous Sub-triangles
7 Computation of the Histogram for ν
7.1 Implementation
8 Results
8.1 Synthetic Data
8.2 Simulation Data
8.3 Measurement Data
9 Conclusions
References
Image Processing and Visualization
Tensor Approximation for Multidimensional and Multivariate Data
1 Introduction
1.1 Higher-Order Data Decompositions
1.2 TA Applications in Graphics and Visualization
1.3 Motivation and Contributions
2 Singular Value Decomposition
3 Tensor Approximation Notation and Definitions
3.1 General Notation
3.2 Computing with Tensors
3.3 Rank of a Tensor
4 Tensor Decompositions
4.1 Tucker Model
5 Tensor Rank Reduction
5.1 Rank-R and Rank-(R1, R2, …, RN) Approximations
5.2 Truncated Tensor Decomposition
6 Tucker Decomposition Algorithms
7 Tensor Reconstruction
7.1 Element-Wise Reconstruction
7.2 Optimized Tucker Reconstruction
8 Useful TA Properties for Scientific Visualization
8.1 Spatial Selectivity and Subsampling
8.2 Approximation and Rank Reduction
9 Application to Multivariate Data
9.1 Dataset
9.2 Vector Field Magnitude and Angle
9.3 Vorticity and Divergence
10 Conclusions
References
Fourth-Order Anisotropic Diffusion for Inpainting and Image Compression
1 Introduction
2 Background and Related Work
2.1 Diffusion-Based Inpainting
2.2 From Linear to Anisotropic Nonlinear Diffusion
2.3 From Second to Fourth Order Diffusion
2.4 Alternative Approaches to Image Compression
3 Method
3.1 Anisotropic Edge-Enhancing Fourth Order PDE
3.2 A Unifying Framework for Fourth-Order Diffusion
3.3 Discretization and Stability
4 Experimental Results
4.1 Reconstruction From a Sparse Set of Pixels
4.2 Scratch Removal
4.3 Effect of Diffusivity Function and Contrast Parameter
5 Conclusions
References
Uncertainty in the DTI Visualization Pipeline
1 Introduction
2 Background
2.1 Diffusion Tensor
2.2 Fiber Tracking
3 Sources of Uncertainty
3.1 Image Acquisition
3.2 Diffusion Tensor Calculation
3.3 Fiber Tracking
3.4 Visualization
4 Uncertainty Modeling
4.1 Analytical Methods
4.2 Stochastic Methods
5 Uncertainty Visualization
5.1 Local Uncertainty Visualization
5.2 Global Uncertainty Visualization
6 Conclusion
References
Challenges for Tractogram Filtering
1 Introduction
2 Approaches for Tractogram Filtering
2.1 Explainability of the Diffusion Signal
2.2 Inclusion and Exclusion ROIs
2.3 Streamline Geometry or Shape
2.4 Streamline Similarity and Clustering
2.5 Multiapproaches
3 Challenges and Perspective
4 Conclusion
References
Modeling Anisotropy
Single Encoding Diffusion MRI: A Probe to Brain Anisotropy
1 Accessing Brain Anisotropy Using Diffusion MRI
1.1 Introduction
1.2 Anisotropy as Reflected by Water Motion
1.3 Structural Brain Anisotropy
1.4 Measuring Anisotropy Using Diffusion MRI
2 Diffusion MRI: Introduction to a Non-Invasive Imaging Technique
2.1 Diffusion MRI Acquisition Sequence
2.2 Mathematical Foundations
2.3 Acquisition Strategies
2.4 Difficulties
3 Quantifying Anisotropy via Signal Representation
3.1 Cumulant Expansion
3.2 Other Representations
3.3 Limitations
4 Biophysical Modeling to Measure Anisotropy
4.1 Multi-compartmental Model
4.2 Neurites as Sticks
4.3 Standard Model of Diffusion in Neural Tissue
4.4 Standard Model Parameter Estimation Using Constraints
4.5 Lemonade
5 Summary and Above
References
Conceptual Parallels Between Stochastic Geometry and Diffusion-Weighted MRI
1 Introduction
2 Specific Volumes and the Short-Time Limit
3 Stationarity and the Long-Time Limit
4 Directional Measures and the Strong-Gradient Limit
5 Perspectives
References
Magnetic Resonance Assessment of Effective Confinement Anisotropy with Orientationally-Averaged Single and Double Diffusion Encoding
1 Introduction
2 Double Diffusion Encoding at the Compartment Level
3 Double Diffusion Encoding: Powder Average
3.1 Axisymmetric Confinement
3.2 Insights from Two Dimensions
3.3 One-Dimensional Diffusion Under High Gradient: g-2 Scaling
4 Single Diffusion Encoding
4.1 Axisymmetry and the Power-Laws for Confined diffusion
5 Discussion
6 Conclusion
References
Riemann-DTI Geodesic Tractography Revisited
1 Introduction
2 Theory
3 Experiments
4 Conclusion and Discussion
References
Measuring Anisotropy
Magnetic Resonance Imaging of T2- and Diffusion Anisotropy Using a Tiltable Receive Coil
1 Introduction
1.1 Background
1.2 Scope of This Work
2 Methods
2.1 Data Acquisition
2.2 MRI Signal Processing
2.3 Estimation
3 Results
4 Discussion
4.1 Incorporating Tiltable Coil in –diffusion correlation experiments
4.2 Origin of –Contrast and -Anisotropy in WM
4.3 Considerations in Data Processing
5 Conclusion
References
Anisotropy in the Human Placenta in Pregnancies Complicated by Fetal Growth Restriction
1 Introduction
1.1 Placental Microstructure
1.2 Placental MRI
2 Methods
2.1 Recruitment
3 Results
4 Discussion and Conclusion
References
Index