Modeling, Analysis, and Visualization of Anisotropy.

"Preface" -- "Contents" -- "Part I Features and Visualization" -- "Robustness for 2D Symmetric Tensor Field Topology" -- "1 Introduction" -- "2 Related Work" -- "3 Preliminaries on Robustness for Vector Fields" -- "4 Tensor F...

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Bibliographic Details
Main Author: Schultz, Thomas (Author)
Other Involved Persons: Özarslan, Evren (Contributor) ; Hotz, Ingrid (Contributor)
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, z.Hd. Alexander Grossmann 2017
ISBN:9783319613581
3319613588
9783319613574
331961357X
Series:Mathematics and Visualization
Physical Description:1 online resource (406 pages)
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Other Editions:Show all 3 Editions
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Description:
  • "Preface" -- "Contents" -- "Part I Features and Visualization" -- "Robustness for 2D Symmetric Tensor Field Topology" -- "1 Introduction" -- "2 Related Work" -- "3 Preliminaries on Robustness for Vector Fields" -- "4 Tensor Fields and Bidirectional Anisotropy Vector Fields" -- "4.1 Background in Tensor Field Topology" -- "4.2 Space of Bidirectional Anisotropy Vectors" -- "4.3 The Anisotropy Vector Field" -- "4.4 Notes on the Topology of the Anisotropy Vector Field " -- "5 Robustness for Tensor Fields" -- "5.1 Indexes of Degenerate Points and Degree Theory" -- "5.2 r-Perturbation of Anisotropy Vector Field" -- "5.3 Robustness of Degenerate Points" -- "6 Discussion" -- "Appendix A: Notations" -- "Appendix B: Triangle Inequality for the Distance Measure Between Bidirectional Anisotropy Vectors (Eq.(6))" -- "Appendix C: Derivations for Eqs.(12) and (13)" -- "References" -- "Applying 2D Tensor Field Topology to Solid Mechanics Simulations" -- "1 Introduction" -- "2 Previous Work" -- "3 Background on Tensors and Tensor Fields" -- "3.1 Tensors" -- "3.2 Tensor Fields" -- "4 Our Approach" -- "5 Tensor Classification" -- "6 Conclusion" -- "References" -- "Moment Invariants for Multi-Dimensional Data" -- "1 Introduction" -- "2 Related Work" -- "3 Theory" -- "3.1 Tensors and Transformations" -- "3.2 Moment Tensors" -- "4 Algorithm and Complexity" -- "5 Results" -- "5.1 3D Scalar" -- "5.2 3D Vector" -- "5.3 2D Tensor" -- "5.4 3D Tensor" -- "6 Discussion" -- "References" -- "Visualizing Gradients of Stress Tensor Fields" -- "1 Introduction" -- "2 Related Work" -- "3 Theoretical Background" -- "3.1 Tensor Fields" -- "3.2 Gradients of Tensor Fields" -- "3.3 Stress Tensor" -- "4 Glyphs for Gradients of Stress Tensor Fields" -- "4.1 Visualizing Gradients of Stress Vectors" -- "4.1.1 Partial Derivatives of Stress Vectors".
  • "4.1.2 Envelope of Stress Vectors under Linear Approximation" -- "4.2 Reducing Directional Derivatives of Stress Vectors to Scalar Quantities" -- "4.2.1 Magnitude of Directional Derivatives of Stress Vectors" -- "4.2.2 Directional Derivatives of Normal Stress" -- "5 Application to Tensile Bars" -- "6 Conclusion and Future Work" -- "References" -- "Part II Image Processing and Analysis" -- "Geometries and Interpolations for Symmetric Positive DefiniteMatrices" -- "1 Introduction" -- "2 The Positive Definite Cone and the Frobenius Metric" -- "2.1 Acknowledgement" -- "3 Classical Riemannian Metrics on Sym+(3)" -- "3.1 The Wasserstein Metric" -- "3.2 The Affine-Invariant Metric" -- "3.3 The Log-Euclidean Metric" -- "4 Avoiding the Swelling, But Not the Fattening, Effect" -- "5 Decoupling Shape and Rotation" -- "5.1 The Shape-and-Orientation Rotation Metric" -- "5.2 Scaling-Rotation Curves" -- "5.2.1 The Scaling-Rotation Distance is Not a Metric" -- "5.3 Linear Invariant Tensor Interpolation" -- "5.4 Further Simulations" -- "5.5 Tensor Statistics and Tensor Decomposition for Visualization Purposes" -- "6 Discussion and Conclusion" -- "6.1 Further Related Work" -- "6.2 Geometry Versus Shape Preservation" -- "6.3 Why Are Second Order Tensors Still Interesting?" -- "References" -- "Towards Processing Fields of General Real-Valued Square Matrices" -- "1 Introduction" -- "2 The R-Vector Space of Hermitian Matrices" -- "3 Rudimentary Calculus for Fields of Hermitian Matrices" -- "3.1 Basic Functions for Hermitian Matrices" -- "3.2 Examples for Basic Hermitian Matrix Calculus" -- "4 An Isomorphism Between MR(n) and H(n)" -- "4.1 Embedding SO(n) in H(n)" -- "4.2 Embedding Gl(n) in H(n)" -- "5 Approximate psup, pinf and Averaging" -- "5.1 The p-Supremum/p-Infimum of Finitely Many Hermitian Matrices" -- "Iteration
  • "6 Application 1: Elementary Image Processing for Invertible Real Matrices" -- "6.1 p-Supremum and p-Infimum in Gl(2)" -- "7 Application 2 : Elementary Image Processing for Orthogonal Matrices" -- "7.1 A Glance at SO(3) and the Cayley Transform" -- "7.2 Linear Averaging in SO(3)" -- "7.3 p-Supremum and p-Infimum in SO(3)" -- "8 Concluding Remarks and Outlook" -- "References" -- "Towards Grey Scale-Based Tensor Voting for Blood Vessel Analysis" -- "1 Introduction" -- "2 Adapting Tensor Voting to Grey Scale Image Processing" -- "2.1 Interpretation of Tensor Shape and Orientation" -- "2.2 Voting Formulation" -- "2.3 Simplified Approach in 3D" -- "2.4 Differences Between 4D and 3D Approach" -- "3 Tube Detection and Orientation Estimation" -- "4 Experimental Results" -- "4.1 4D Tensor Voting" -- "4.1.1 Relationship Between Intensity Weight σI and Noise Level" -- "4.1.2 Estimation of Vessel Centre Candidates" -- "4.2 Comparison of 4D and 3D Tensor Voting Approaches" -- "5 Discussion" -- "6 Conclusion" -- "References" -- "Local Geometric Descriptors for Multi-Scale Probabilistic Point Classification of Airborne LiDAR Point Clouds" -- "1 Introduction" -- "2 Related Work" -- "3 Local Geometric Descriptors" -- "4 Multi-Scale Probabilistic Point Classification" -- "5 Comparison of Local Geometry Descriptors" -- "5.1 Juxtaposed Views" -- "5.2 Classification Matrix Visualization" -- "6 Experiments and Results" -- "7 Conclusions" -- "References" -- "Part III Diffusion Modeling and Microstructure" -- "Diffusion MRI Anisotropy: Modeling, Analysis and Interpretation" -- "1 Introduction" -- "2 Diffusion Anisotropy: The Phenomenon" -- "2.1 Diffusion and the Ensemble Average Propagator" -- "2.2 Microstructure of the Brain: The Complicated Reality" -- "3 Measurements of Diffusion with Diffusion-Weighted MRI " -- "4 The Inter-Model Variability of Diffusion Anisotropy
  • "4.1 Data Set Description and Adopted Notation" -- "4.2 Diffusion Anisotropy as a Signal Property" -- "4.3 Anisotropy as Orientation Dispersion of Micro-Environments" -- "4.4 Anisotropy as a Property of Micro-Environments" -- "5 Sensitivity to Diffusion Time" -- "5.1 Anisotropy Due to Axon Packing" -- "6 Discussion" -- "7 Conclusion" -- "References" -- "Measuring Microscopic Anisotropy with Diffusion Magnetic Resonance: From Material Science to Biomedical Imaging" -- "1 Introduction" -- "2 Microscopic Anisotropy in the Gaussian Regime" -- "2.1 Single Diffusion Encoding" -- "Probability Distribution of Diffusion Coefficients" -- "2.2 Double Diffusion Encoding" -- "2.3 Isotropic Diffusion Encoding and q-Space Trajectories" -- "Isotropic Encoding and the Trace of Diffusion Tensor" -- "A Probability Distribution Perspective" -- "3 Restricted Diffusion" -- "3.1 Restricted Signal Model for SDE Sequences" -- "3.2 DDE Sequences" -- "Signal Model" -- "Estimating μA with Angular DDE" -- "Rotationally Invariant Metrics of μA" -- "3.3 Double Oscillating Diffusion Encoding" -- "3.4 Comparison of Two μA Metrics" -- "3.5 Model-Based Estimation of Pore Size and Shape" -- "3.6 Direct Estimation of the Pore Shape Function" -- "4 Summary" -- "References" -- "Bayesian Heteroscedastic Regression for Diffusion Tensor Imaging" -- "1 Introduction" -- "2 The Heteroscedastic Diffusion Tensor Model" -- "2.1 The Homoscedastic DTI Model" -- "2.2 The Heteroscedastic DTI Model" -- "2.3 Prior Distribution and Variable Selection" -- "2.3.1 Priors on the Regression Coefficients" -- "2.3.2 Variable Selection Priors" -- "3 Inference Methods" -- "3.1 Traditional Estimation Methods" -- "3.2 Bayesian Inference Using MCMC Sampling with Variable Selection" -- "4 Data" -- "5 Results" -- "5.1 The Effect of Allowing for Heteroscedasticity and Comparison to Traditional Estimation Methods
  • "5.2 Properties of the Log-Cholesky Parametrization" -- "6 Discussion" -- "References" -- "Multi-Fiber Reconstruction Using Probabilistic Mixture Models for Diffusion MRI Examinations of the Brain" -- "1 Introduction" -- "2 Theory" -- "2.1 Mixture of Gaussian Distributions" -- "2.2 Mixture of Von Mises-Fisher Distributions" -- "2.3 Mixture of Central Wishart Distributions (MoCW)" -- "2.4 Proposed Model of Mixture of Non-central Wishart Distributions (MoNCW)" -- "2.4.1 Estimation of Model Parameters" -- "2.5 Visualization of Estimation Results with Displacement Probabilities" -- "3 Results" -- "4 Conclusion" -- "References" -- "Part IV Tractography" -- "Edge Detection in Diffusion Weighted MRI Using a Tangent Curve Similarity Metric" -- "1 Introduction" -- "2 Related Work" -- "2.1 Tractography in DTI and HARDI" -- "2.2 Diffusion Weighted MRI Analysis" -- "3 Method" -- "3.1 Edges as Ridges" -- "3.2 Fiber-Valued Image Computation" -- "3.3 Feature Encoding" -- "3.4 Edge Strength in Vector-Valued Images" -- "3.5 Extension to HARDI" -- "4 Visualization" -- "5 Results and Discussions" -- "5.1 Phantom Data" -- "5.2 IIT2 Human Brain DTI Template" -- "5.3 Human Brain with DTI and HARDI" -- "6 Conclusion and Future Work" -- "References" -- "Repeated Tractography of a Single Subject: How HighIs the Variance?" -- "1 Introduction" -- "2 Data" -- "3 Methods" -- "3.1 FSL" -- "3.2 Dipy" -- "4 Results" -- "4.1 FSL" -- "4.2 Dipy" -- "5 Discussion and Conclusion" -- "References" -- "Automatic Atlas-Based Segmentation of Brain White Matter in Neonates at Risk for Neurodevelopmental Disorders" -- "1 Introduction" -- "2 Material and Methods" -- "2.1 Subjects" -- "2.2 Data Acquisition" -- "2.3 Tractography" -- "2.3.1 Masking" -- "2.3.2 Tractography" -- "2.4 Atlas Creation" -- "2.5 Data Processing" -- "2.5.1 Cluster Based Sampling" -- "2.5.2 Registration" -- "2.5.3 Labeling
  • "2.5.4 Label Propagation