Anisotropy Across Fields and Scales.
Saved in:
Superior document: | Mathematics and Visualization Series |
---|---|
: | |
TeilnehmendeR: | |
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
|
Online Access: | |
Physical Description: | 1 online resource (284 pages) |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
id |
5006478284 |
---|---|
ctrlnum |
(MiAaPQ)5006478284 (Au-PeEL)EBL6478284 (OCoLC)1240211601 |
collection |
bib_alma |
record_format |
marc |
spelling |
Özarslan, Evren. Anisotropy Across Fields and Scales. 1st ed. Cham : Springer International Publishing AG, 2021. {copy}2021. 1 online resource (284 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Mathematics and Visualization Series 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. Description based on publisher supplied metadata and other sources. Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries. Electronic books. Schultz, Thomas. Zhang, Eugene. Fuster, Andrea. Print version: Özarslan, Evren Anisotropy Across Fields and Scales Cham : Springer International Publishing AG,c2021 9783030562144 ProQuest (Firm) https://ebookcentral.proquest.com/lib/oeawat/detail.action?docID=6478284 Click to View |
language |
English |
format |
eBook |
author |
Özarslan, Evren. |
spellingShingle |
Özarslan, Evren. Anisotropy Across Fields and Scales. Mathematics and Visualization Series 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. |
author_facet |
Özarslan, Evren. Schultz, Thomas. Zhang, Eugene. Fuster, Andrea. |
author_variant |
e o eo |
author2 |
Schultz, Thomas. Zhang, Eugene. Fuster, Andrea. |
author2_variant |
t s ts e z ez a f af |
author2_role |
TeilnehmendeR TeilnehmendeR TeilnehmendeR |
author_sort |
Özarslan, Evren. |
title |
Anisotropy Across Fields and Scales. |
title_full |
Anisotropy Across Fields and Scales. |
title_fullStr |
Anisotropy Across Fields and Scales. |
title_full_unstemmed |
Anisotropy Across Fields and Scales. |
title_auth |
Anisotropy Across Fields and Scales. |
title_new |
Anisotropy Across Fields and Scales. |
title_sort |
anisotropy across fields and scales. |
series |
Mathematics and Visualization Series |
series2 |
Mathematics and Visualization Series |
publisher |
Springer International Publishing AG, |
publishDate |
2021 |
physical |
1 online resource (284 pages) |
edition |
1st ed. |
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. |
isbn |
9783030562151 9783030562144 |
callnumber-first |
Q - Science |
callnumber-subject |
QA - Mathematics |
callnumber-label |
QA76 |
callnumber-sort |
QA 276.9 I52 |
genre |
Electronic books. |
genre_facet |
Electronic books. |
url |
https://ebookcentral.proquest.com/lib/oeawat/detail.action?docID=6478284 |
illustrated |
Not Illustrated |
oclc_num |
1240211601 |
work_keys_str_mv |
AT ozarslanevren anisotropyacrossfieldsandscales AT schultzthomas anisotropyacrossfieldsandscales AT zhangeugene anisotropyacrossfieldsandscales AT fusterandrea anisotropyacrossfieldsandscales |
status_str |
n |
ids_txt_mv |
(MiAaPQ)5006478284 (Au-PeEL)EBL6478284 (OCoLC)1240211601 |
carrierType_str_mv |
cr |
hierarchy_parent_title |
Mathematics and Visualization Series |
is_hierarchy_title |
Anisotropy Across Fields and Scales. |
container_title |
Mathematics and Visualization Series |
author2_original_writing_str_mv |
noLinkedField noLinkedField noLinkedField |
_version_ |
1792331058866290688 |
fullrecord |
<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>08711nam a22004693i 4500</leader><controlfield tag="001">5006478284</controlfield><controlfield tag="003">MiAaPQ</controlfield><controlfield tag="005">20240229073839.0</controlfield><controlfield tag="006">m o d | </controlfield><controlfield tag="007">cr cnu||||||||</controlfield><controlfield tag="008">240229s2021 xx o ||||0 eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783030562151</subfield><subfield code="q">(electronic bk.)</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="z">9783030562144</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(MiAaPQ)5006478284</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(Au-PeEL)EBL6478284</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)1240211601</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">MiAaPQ</subfield><subfield code="b">eng</subfield><subfield code="e">rda</subfield><subfield code="e">pn</subfield><subfield code="c">MiAaPQ</subfield><subfield code="d">MiAaPQ</subfield></datafield><datafield tag="050" ind1=" " ind2="4"><subfield code="a">QA76.9.I52</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Özarslan, Evren.</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Anisotropy Across Fields and Scales.</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">1st ed.</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Cham :</subfield><subfield code="b">Springer International Publishing AG,</subfield><subfield code="c">2021.</subfield></datafield><datafield tag="264" ind1=" " ind2="4"><subfield code="c">{copy}2021.</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">1 online resource (284 pages)</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">computer</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">online resource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Mathematics and Visualization Series</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">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.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">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.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">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.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">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.</subfield></datafield><datafield tag="588" ind1=" " ind2=" "><subfield code="a">Description based on publisher supplied metadata and other sources.</subfield></datafield><datafield tag="590" ind1=" " ind2=" "><subfield code="a">Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries. </subfield></datafield><datafield tag="655" ind1=" " ind2="4"><subfield code="a">Electronic books.</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Schultz, Thomas.</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Zhang, Eugene.</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Fuster, Andrea.</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Print version:</subfield><subfield code="a">Özarslan, Evren</subfield><subfield code="t">Anisotropy Across Fields and Scales</subfield><subfield code="d">Cham : Springer International Publishing AG,c2021</subfield><subfield code="z">9783030562144</subfield></datafield><datafield tag="797" ind1="2" ind2=" "><subfield code="a">ProQuest (Firm)</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Mathematics and Visualization Series</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://ebookcentral.proquest.com/lib/oeawat/detail.action?docID=6478284</subfield><subfield code="z">Click to View</subfield></datafield></record></collection> |