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Format: 1 online resource (284 pages)

Edition: 1st ed.

ISBN: 9783030562151

Series Statement: Mathematics and Visualization Series

Note: 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

Additional Edition: Print version Özarslan, Evren Anisotropy Across Fields and Scales Cham : Springer International Publishing AG,c2021 ISBN 9783030562144

Language: English

Keywords: Electronic books

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