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Ingrid Hotz - One of the best experts on this subject based on the ideXlab platform.
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visualizing symmetric inDefinite 2d Tensor fields using the heat kernel signature
Visualization and Processing of Higher Order Descriptors for Multi-Valued Data, 2015Co-Authors: Valentin Zobel, Jan Reininghaus, Ingrid HotzAbstract:The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric Positive Definite Tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given Tensor data. In this paper, we propose a generalization of this approach that enables the treatment of inDefinite Tensor fields, like the stress Tensor, by interpreting them as a generator of a Positive Definite Tensor field. To investigate the usefulness of this approach we consider the stress Tensor from the two-point-load model example and from a mechanical work piece.
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visualization of two dimensional symmetric Positive Definite Tensor fields using the heat kernel signature
Topological Methods in Data Analysis and Visualization, 2014Co-Authors: Valentin Zobel, Jan Reininghaus, Ingrid HotzAbstract:We propose a method for visualizing two-dimensional symmetric Positive Definite Tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. Each Positive Definite Tensor field defines a Riemannian manifold by considering the Tensor field as a Riemannian metric. On this Riemmanian manifold we can apply the definition of the HKS. The resulting scalar quantity is used for the visualization of Tensor fields. The HKS is closely related to the Gaussian curvature of the Riemannian manifold and the time parameter of the heat kernel allows a multiscale analysis in a natural way. In this way, the HKS represents field related scale space properties, enabling a level of detail analysis of Tensor fields. This makes the HKS an interesting new scalar quantity for Tensor fields, which differs significantly from usual Tensor invariants like the trace or the determinant. A method for visualization and a numerical realization of the HKS for Tensor fields is proposed in this chapter. To validate the approach we apply it to some illustrating simple examples as isolated critical points and to a medical diffusion Tensor data set.
Subotnik, Joseph E. - One of the best experts on this subject based on the ideXlab platform.
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An Antisymmetric Berry Frictional Force At Equilibrium in the Presence of Spin-Orbit Coupling
2021Co-Authors: Teh Hung-hsuan, Dou Wenjie, Subotnik, Joseph E.Abstract:We analytically calculate the electronic friction Tensor for a molecule near a metal surface in the case that the electronic Hamiltonian is complex-valued, e.g. the case that there is spin-orbit coupling and/or an external magnetic field. In such a case, {\em even at equilibrium}, we show that the friction Tensor is not symmetric. Instead, the Tensor is the real-valued sum of one Positive Definite Tensor (corresponding to dissipation) plus one antisymmetric Tensor (corresponding to a Berry pseudomagnetic force). Moreover, we find that this Berry force can be much larger than the dissipational force, suggesting the possibility of strongly spin-polarized chemicurrents or strongly spin-dependent rate constants for systems with spin-orbit coupling.Comment: 15 pages, 2 figures (main text, preprint
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An Antisymmetric Berry Frictional Force At Equilibrium in the Presence of Spin-Orbit Coupling
'American Physical Society (APS)', 2021Co-Authors: Teh Hung-hsuan, Dou Wenjie, Subotnik, Joseph E.Abstract:We analytically calculate the electronic friction Tensor for a molecule near a metal surface in the case that the electronic Hamiltonian is complex-valued, e.g. the case that there is spin-orbit coupling and/or an external magnetic field. In such a case, even at equilibrium, we show that the friction Tensor is not symmetric. Instead, the Tensor is the real-valued sum of one Positive Definite Tensor (corresponding to dissipation) plus one antisymmetric Tensor (corresponding to a Berry pseudomagnetic force). Moreover, we find that this Berry force can be much larger than the dissipational force, suggesting the possibility of strongly spin-polarized chemicurrents or strongly spin-dependent rate constants for systems with spin-orbit coupling.Comment: 6 pages, 2 figures (main text); 15 pages, 15 figures (supplemental material
Anish Roy - One of the best experts on this subject based on the ideXlab platform.
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size effects and idealized dislocation microstructure at small scales predictions of a phenomenological model of mesoscopic field dislocation mechanics part i
Journal of The Mechanics and Physics of Solids, 2006Co-Authors: Amit Acharya, Anish RoyAbstract:Abstract A Phenomenological Mesoscopic Field Dislocation Mechanics (PMFDM) model is developed, extending continuum plasticity theory for studying initial-boundary value problems of small-scale plasticity. PMFDM results from an elementary space-time averaging of the equations of Field Dislocation Mechanics (FDM), followed by a closure assumption from any strain-gradient plasticity model that attempts to account for effects of geometrically necessary dislocations (GNDs) only in work hardening. The specific lower-order gradient plasticity model chosen to substantiate this work requires one additional material parameter compared to its conventional continuum plasticity counterpart. The further addition of dislocation mechanics requires no additional material parameters. The model (a) retains the constitutive dependence of the free-energy only on elastic strain as in conventional continuum plasticity with no explicit dependence on dislocation density, (b) does not require higher-order stresses, and (c) does not require a constitutive specification of a ‘back-stress’ in the expression for average dislocation velocity/plastic strain rate. However, long-range stress effects of average dislocation distributions are predicted by the model in a mechanistically rigorous sense. Plausible boundary conditions (with obvious implication for corresponding interface conditions) are discussed in some detail from a physical point of view. Energetic and dissipative aspects of the model are also discussed. The developed framework is a continuous-time model of averaged dislocation plasticity, without having to rely on the notion of incremental work functions, their convexity properties, or their minimization. The tangent modulus relating stress rate and total strain rate in the model is the Positive-Definite Tensor of linear elasticity, and this is not an impediment to the development of idealized microstructure in the theory and computations, even when such a convexity property is preserved in a computational scheme. A model of finite deformation, mesoscopic single crystal plasticity is also presented, motivated by the above considerations. Lower-order gradient plasticity appears as a constitutive limit of PMFDM, and the development suggests a plausible boundary condition on the plastic strain rate for this limit that is appropriate for the modeling of constrained plastic flow in three-dimensional situations.
Arthur W Toga - One of the best experts on this subject based on the ideXlab platform.
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Comparison of fractional and geodesic anisotropy in diffusion Tensor images of 90 monozygotic and dizygotic twins
2008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2008Co-Authors: Natasha Lepore, Marina Barysheva, Yiyu Chou, Caroline Brun, Sarah K Madsen, Katie L Mcmahon, Greig I De Zubicaray, Margaret J Wright, Matthew Meredith, Arthur W TogaAbstract:We used diffusion Tensor magnetic resonance imaging (DTI) to reveal the extent of genetic effects on brain fiber microstructure, based on Tensor-derived measures, in 22 pairs of monozygotic (MZ) twins and 23 pairs of dizygotic (DZ) twins (90 scans). After Log-Euclidean denoising to remove rank-deficient Tensors, DTI volumes were fluidly registered by high-dimensional mapping of co-registered MP-RAGE scans to a geometrically- centered mean neuroanatomical template. After Tensor reorientation using the strain of the 3D fluid transformation, we computed two widely used scalar measures of fiber integrity: fractional anisotropy (FA), and geodesic anisotropy (GA), which measures the geodesic distance between Tensors in the symmetric Positive-Definite Tensor manifold. Spatial maps of intraclass correlations (r) between MZ and DZ twins were compared to compute maps of Falconer's heritability statistics, i.e. the proportion of population variance explainable by genetic differences among individuals. Cumulative distribution plots (CDF) of effect sizes showed that the manifold measure, GA, comparably the Euclidean measure, FA, in detecting genetic correlations. While maps were relatively noisy, the CDFs showed promise for detecting genetic influences on brain fiber integrity as the current sample expands.
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gene effects mapped using fractional and geodesic anisotropy in diffusion Tensor images of 92 monozygotic and dizygotic twins
MICCAI workshop on computational diffusion MRI, 2008Co-Authors: Agatha D Lee, Natasha Lepore, Marina Barysheva, Yiyu Chou, Caroline Brun, Sarah K Madsen, Katie L Mcmahon, Greig I De Zubicaray, Margaret J Wright, Arthur W TogaAbstract:We used Tensor-derived measures to map the extent of genetic effects on brain fiber microstructure, in 23 monozygotic and 23 dizygotic twin pairs. All 92 DTI volumes were fluidly registered to a geometrically-centered template via a high-dimensional mapping of co-registered structural-MRI. After Tensor re-orientation, we computed three scalar DTI measures: the fractional anisotropy (FA), geodesic anisotropy (GA), and the hyperbolic tangent of GA (tGA); GA measures the geodesic distance between Tensors on the symmetric Positive-Definite Tensor manifold. Spatial maps of intraclass correlations between MZ and DZ twins were compared to compute maps of Falconer’s heritability statistics. We also performed a maximum likelihood estimation of genetic influences using path analysis. The manifold-based measure, tGA, was marginally more powerful than FA for detecting genetic influences, and improved the fit of quantitative genetic models relative to FA and GA. The pattern of genetic influences was remarkably consistent with the neurodevelopmental sequence, with strong occipital genetic effects and strong frontal environmental effects.
Valentin Zobel - One of the best experts on this subject based on the ideXlab platform.
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visualizing symmetric inDefinite 2d Tensor fields using the heat kernel signature
Visualization and Processing of Higher Order Descriptors for Multi-Valued Data, 2015Co-Authors: Valentin Zobel, Jan Reininghaus, Ingrid HotzAbstract:The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric Positive Definite Tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given Tensor data. In this paper, we propose a generalization of this approach that enables the treatment of inDefinite Tensor fields, like the stress Tensor, by interpreting them as a generator of a Positive Definite Tensor field. To investigate the usefulness of this approach we consider the stress Tensor from the two-point-load model example and from a mechanical work piece.
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visualization of two dimensional symmetric Positive Definite Tensor fields using the heat kernel signature
Topological Methods in Data Analysis and Visualization, 2014Co-Authors: Valentin Zobel, Jan Reininghaus, Ingrid HotzAbstract:We propose a method for visualizing two-dimensional symmetric Positive Definite Tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. Each Positive Definite Tensor field defines a Riemannian manifold by considering the Tensor field as a Riemannian metric. On this Riemmanian manifold we can apply the definition of the HKS. The resulting scalar quantity is used for the visualization of Tensor fields. The HKS is closely related to the Gaussian curvature of the Riemannian manifold and the time parameter of the heat kernel allows a multiscale analysis in a natural way. In this way, the HKS represents field related scale space properties, enabling a level of detail analysis of Tensor fields. This makes the HKS an interesting new scalar quantity for Tensor fields, which differs significantly from usual Tensor invariants like the trace or the determinant. A method for visualization and a numerical realization of the HKS for Tensor fields is proposed in this chapter. To validate the approach we apply it to some illustrating simple examples as isolated critical points and to a medical diffusion Tensor data set.