Anisotropy Across Fields and Scales.

• Superior document: Mathematics and Visualization Series
• Author: Özarslan, Evren.
• TeilnehmendeR: Schultz, Thomas.; Zhang, Eugene.; Fuster, Andrea.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2021.; {copy}2021.
• Year of Publication: 2021
• Edition: 1st ed.
• Language: English
• Series: Mathematics and Visualization Series
• Physical Description: 1 online resource (284 pages)

Table of Contents

• Intro
• 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.