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Format: 1 Online-Ressource.

ISBN: 978-3-030-56215-1

Series Statement: Mathematics and visualization

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-030-56214-4

Language: English

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DOI: 10.1007/978-3-030-56215-1

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UID:

Format: 1 online resource (X, 280 p. 109 illus., 91 illus. in color.)

Edition: 1st ed. 2021.

ISBN: 3-030-56215-8

Series Statement: Mathematics and Visualization,

Content: This open access book focuses on processing, modeling, and visualization of anisotropy information...

Note: Tensor approximation for multidimensional and multivariate data -- Tensor field topology without symmetrization using Hermitian tensors -- Continuous histograms for Anisotropy of 2D symmetric piece-wise linear tensor fields -- Riemann-DTI geodesic tractography revisited -- Fourth-order anisotropic diffusion for inpainting and image compression -- Advanced deep learning for processing orientation-dependent diffusion magnetic resonance imaging data - A review -- On the variance measure of diffusion tensors in two dimensions -- Magnetic resonance assessment of effective confinement anisotropy with powder-averaged single and double diffusion encoding -- Merge trees, neutral surfaces, and tensor field topology -- Asymmetric tensor analysis -- Anisotropy issues in shape-based object analysis -- Tractogram filtering -- Multispectral image processing in astronomy -- Anisotropy in the human placenta in pregnancies complicated by fetal growth restriction -- The case for spatially homogeneous models in high gradient strength diffusion-weighted MRI: A position paper on the potential applicability of stochastic geometry. , English

Additional Edition: ISBN 3-030-56214-X

Language: English

DOI: 10.1007/978-3-030-56215-1

URL: kostenfrei

UID:

Format: 1 Online-Ressource (280 p.)

ISBN: 9783030562151

Series Statement: Mathematics and Visualization

Content: This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018

Note: English

Language: English

URL: kostenfrei

UID:

Format: X, 280 p. 109 illus., 91 illus. in color. , online resource.

Edition: 1st ed. 2021.

ISBN: 9783030562151

Series Statement: Mathematics and Visualization,

Content: This open access book focuses on processing, modeling, and visualization of anisotropy information...

Note: Tensor approximation for multidimensional and multivariate data -- Tensor field topology without symmetrization using Hermitian tensors -- Continuous histograms for Anisotropy of 2D symmetric piece-wise linear tensor fields -- Riemann-DTI geodesic tractography revisited -- Fourth-order anisotropic diffusion for inpainting and image compression -- Advanced deep learning for processing orientation-dependent diffusion magnetic resonance imaging data - A review -- On the variance measure of diffusion tensors in two dimensions -- Magnetic resonance assessment of effective confinement anisotropy with powder-averaged single and double diffusion encoding -- Merge trees, neutral surfaces, and tensor field topology -- Asymmetric tensor analysis -- Anisotropy issues in shape-based object analysis -- Tractogram filtering -- Multispectral image processing in astronomy -- Anisotropy in the human placenta in pregnancies complicated by fetal growth restriction -- The case for spatially homogeneous models in high gradient strength diffusion-weighted MRI: A position paper on the potential applicability of stochastic geometry.

In: Springer Nature eBook

Additional Edition: Printed edition: ISBN 9783030562144

Additional Edition: Printed edition: ISBN 9783030562168

Additional Edition: Printed edition: ISBN 9783030562175

Language: English

DOI: 10.1007/978-3-030-56215-1

URL: https://doi.org/10.1007/978-3-030-56215-1

UID:

Format: 1 online resource (X, 280 p. 109 illus., 91 illus. in color.)

Edition: 1st ed. 2021.

ISBN: 3-030-56215-8

Series Statement: Mathematics and Visualization,

Content: This open access book focuses on processing, modeling, and visualization of anisotropy information...

Note: Tensor approximation for multidimensional and multivariate data -- Tensor field topology without symmetrization using Hermitian tensors -- Continuous histograms for Anisotropy of 2D symmetric piece-wise linear tensor fields -- Riemann-DTI geodesic tractography revisited -- Fourth-order anisotropic diffusion for inpainting and image compression -- Advanced deep learning for processing orientation-dependent diffusion magnetic resonance imaging data - A review -- On the variance measure of diffusion tensors in two dimensions -- Magnetic resonance assessment of effective confinement anisotropy with powder-averaged single and double diffusion encoding -- Merge trees, neutral surfaces, and tensor field topology -- Asymmetric tensor analysis -- Anisotropy issues in shape-based object analysis -- Tractogram filtering -- Multispectral image processing in astronomy -- Anisotropy in the human placenta in pregnancies complicated by fetal growth restriction -- The case for spatially homogeneous models in high gradient strength diffusion-weighted MRI: A position paper on the potential applicability of stochastic geometry. , English

Additional Edition: ISBN 3-030-56214-X

Language: English

DOI: 10.1007/978-3-030-56215-1

UID:

Format: 1 online resource (X, 280 p. 109 illus., 91 illus. in color.)

Edition: 1st ed. 2021.

ISBN: 3-030-56215-8

Series Statement: Mathematics and Visualization,

Content: This open access book focuses on processing, modeling, and visualization of anisotropy information...

Note: Tensor approximation for multidimensional and multivariate data -- Tensor field topology without symmetrization using Hermitian tensors -- Continuous histograms for Anisotropy of 2D symmetric piece-wise linear tensor fields -- Riemann-DTI geodesic tractography revisited -- Fourth-order anisotropic diffusion for inpainting and image compression -- Advanced deep learning for processing orientation-dependent diffusion magnetic resonance imaging data - A review -- On the variance measure of diffusion tensors in two dimensions -- Magnetic resonance assessment of effective confinement anisotropy with powder-averaged single and double diffusion encoding -- Merge trees, neutral surfaces, and tensor field topology -- Asymmetric tensor analysis -- Anisotropy issues in shape-based object analysis -- Tractogram filtering -- Multispectral image processing in astronomy -- Anisotropy in the human placenta in pregnancies complicated by fetal growth restriction -- The case for spatially homogeneous models in high gradient strength diffusion-weighted MRI: A position paper on the potential applicability of stochastic geometry. , English

Additional Edition: ISBN 3-030-56214-X

Language: English

DOI: 10.1007/978-3-030-56215-1

UID:

Format: 1 online resource (284 pages)

ISBN: 9783030562151

Series Statement: Mathematics and Visualization Ser.

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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