The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Constantin Călin - One of the best experts on this subject based on the ideXlab platform.
-
on the 1 3 threading of spacetime with respect to an arbitrary timelike vector Field
European Physical Journal C, 2015Co-Authors: Aurel Bejancu, Constantin CălinAbstract:We develop a new approach on the (\(1+3\)) threading of spacetime \((M, g)\) with respect to a congruence of curves defined by an arbitrary timelike vector Field. The study is based on spatial Tensor Fields and on the Riemannian spatial connection \(\nabla ^{\star }\), which behave as 3D geometric objects. We obtain new formulas for local components of the Ricci Tensor Field of \((M, g)\) with respect to the threading frame Field, in terms of the Ricci Tensor Field of \(\nabla ^{\star }\) and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri’s equation enables us to prove Lemma 6.3, which completes a well-known lemma used in the proof of the Penrose–Hawking singularity theorems. Finally, we apply the new \((1+3)\) formalism to the study of the dynamics of a Kerr–Newman black hole.
-
on the 1 3 threading of spacetime with respect to an arbitrary timelike vector Field
arXiv: Differential Geometry, 2015Co-Authors: Aurel Bejancu, Constantin CălinAbstract:We develop a new approach on the (1+3) threading of spacetime $(M, g)$ with respect to a congruence of curves defined by an arbitrary timelike vector Field. The study is based on spatial Tensor Fields and on the Riemannian spatial connection $\nabla^{\star}$, which behave as $3D$ geometric objects. We obtain new formulas for local components of the Ricci Tensor Field of $(M, g)$ with respect to the threading frame Field, in terms of the Ricci Tensor Field of $\nabla^{\star}$ and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri's equation enables us to prove Lemma 6.2, which completes a well known lemma used in the proof of Penrose-Hawking singularity theorems.Finally, we apply the new $(1+3)$ formalism to the study of the dynamics of a Kerr-Newman black hole.
Eugene Zhang - One of the best experts on this subject based on the ideXlab platform.
-
robust and fast extraction of 3d symmetric Tensor Field topology
IEEE Transactions on Visualization and Computer Graphics, 2019Co-Authors: Lawrence Roy, Prashant Kumar, Yue Zhang, Eugene ZhangAbstract:3D symmetric Tensor Fields appear in many science and engineering Fields, and topology-driven analysis is important in many of these application domains, such as solid mechanics and fluid dynamics. Degenerate curves and neutral surfaces are important topological features in 3D symmetric Tensor Fields. Existing methods to extract degenerate curves and neutral surfaces often miss parts of the curves and surfaces, respectively. Moreover, these methods are computationally expensive due to the lack of knowledge of structures of degenerate curves and neutral surfaces. In this paper, we provide theoretical analysis on the geometric and topological structures of degenerate curves and neutral surfaces of 3D linear Tensor Fields. These structures lead to parameterizations for degenerate curves and neutral surfaces that can not only provide more robust extraction of these features but also incur less computational cost. We demonstrate the benefits of our approach by applying our degenerate curve and neutral surface detection techniques to solid mechanics simulation data sets.
-
interactive Tensor Field design and visualization on surfaces
IEEE Transactions on Visualization and Computer Graphics, 2007Co-Authors: Eugene Zhang, James Hays, Greg TurkAbstract:Designing Tensor Fields in the plane and on surfaces is a necessary task in many graphics applications, such as painterly rendering, pen-and-ink sketching of smooth surfaces, and anisotropic remeshing. In this article, we present an interactive design system that allows a user to create a wide variety of symmetric Tensor Fields over 3D surfaces either from scratch or by modifying a meaningful input Tensor Field such as the curvature Tensor. Our system converts each user specification into a basis Tensor Field and combines them with the input Field to make an initial Tensor Field. However, such a Field often contains unwanted degenerate points which cannot always be eliminated due to topological constraints of the underlying surface. To reduce the artifacts caused by these degenerate points, our system allows the user to move a degenerate point or to cancel a pair of degenerate points that have opposite Tensor indices. These operations provide control over the number and location of the degenerate points in the Field. We observe that a Tensor Field can be locally converted into a vector Field so that there is a one-to-one correspondence between the set of degenerate points in the Tensor Field and the set of singularities in the vector Field. This conversion allows us to effectively perform degenerate point pair cancellation and movement by using similar operations for vector Fields. In addition, we adapt the image-based flow visualization technique to Tensor Fields, therefore allowing interactive display of Tensor Fields on surfaces. We demonstrate the capabilities of our Tensor Field design system with painterly rendering, pen-and-ink sketching of surfaces, and anisotropic remeshing
Aurel Bejancu - One of the best experts on this subject based on the ideXlab platform.
-
on the 1 3 threading of spacetime with respect to an arbitrary timelike vector Field
European Physical Journal C, 2015Co-Authors: Aurel Bejancu, Constantin CălinAbstract:We develop a new approach on the (\(1+3\)) threading of spacetime \((M, g)\) with respect to a congruence of curves defined by an arbitrary timelike vector Field. The study is based on spatial Tensor Fields and on the Riemannian spatial connection \(\nabla ^{\star }\), which behave as 3D geometric objects. We obtain new formulas for local components of the Ricci Tensor Field of \((M, g)\) with respect to the threading frame Field, in terms of the Ricci Tensor Field of \(\nabla ^{\star }\) and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri’s equation enables us to prove Lemma 6.3, which completes a well-known lemma used in the proof of the Penrose–Hawking singularity theorems. Finally, we apply the new \((1+3)\) formalism to the study of the dynamics of a Kerr–Newman black hole.
-
on the 1 3 threading of spacetime with respect to an arbitrary timelike vector Field
arXiv: Differential Geometry, 2015Co-Authors: Aurel Bejancu, Constantin CălinAbstract:We develop a new approach on the (1+3) threading of spacetime $(M, g)$ with respect to a congruence of curves defined by an arbitrary timelike vector Field. The study is based on spatial Tensor Fields and on the Riemannian spatial connection $\nabla^{\star}$, which behave as $3D$ geometric objects. We obtain new formulas for local components of the Ricci Tensor Field of $(M, g)$ with respect to the threading frame Field, in terms of the Ricci Tensor Field of $\nabla^{\star}$ and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri's equation enables us to prove Lemma 6.2, which completes a well known lemma used in the proof of Penrose-Hawking singularity theorems.Finally, we apply the new $(1+3)$ formalism to the study of the dynamics of a Kerr-Newman black hole.
Baba C Vemuri - One of the best experts on this subject based on the ideXlab platform.
-
groupwise registration and atlas construction of 4th order Tensor Fields using the r riemannian metric
Medical Image Computing and Computer-Assisted Intervention, 2009Co-Authors: Angelos Barmpoutis, Baba C VemuriAbstract:Registration of Diffusion-Weighted MR Images (DW-MRI) can be achieved by registering the corresponding 2nd-order Diffusion Tensor Images (DTI). However, it has been shown that higher-order diffusion Tensors (e.g. order-4) outperform the traditional DTI in approximating complex fiber structures such as fiber crossings. In this paper we present a novel method for unbiased group-wise non-rigid registration and atlas construction of 4th-order diffusion Tensor Fields. To the best of our knowledge there is no other existing method to achieve this task. First we define a metric on the space of positive-valued functions based on the Riemannian metric of real positive numbers (denoted by R + ). Then, we use this metric in a novel functional minimization method for non-rigid 4th-order Tensor Field registration. We define a cost function that accounts for the 4th-order Tensor re-orientation during the registration process and has analytic derivatives with respect to the transformation parameters. Finally, the Tensor Field atlas is computed as the minimizer of the variance defined using the Riemannian metric. We quantitatively compare the proposed method with other techniques that register scalar-valued or diffusion Tensor (rank-2) representations of the DWMRI.
-
regularized positive definite fourth order Tensor Field estimation from dw mri
NeuroImage, 2009Co-Authors: Angelos Barmpoutis, Min Sig Hwang, Dena R Howland, John R Forder, Baba C VemuriAbstract:In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing, a 2nd order Tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. From this Tensor approximation, one can compute useful scalar quantities (e.g. anisotropy, mean diffusivity) which have been clinically used for monitoring encephalopathy, sclerosis, ischemia and other brain disorders. It is now well known that this 2nd-order Tensor approximation fails to capture complex local tissue structures, e.g. crossing fibers, and as a result, the scalar quantities derived from these Tensors are grossly inaccurate at such locations. In this paper we employ a 4th order symmetric positive-definite (SPD) Tensor approximation to represent the diffusivity function and present a novel technique to estimate these Tensors from the DW-MRI data guaranteeing the SPD property. Several articles have been reported in literature on higher order Tensor approximations of the diffusivity function but none of them guarantee the positivity of the estimates, which is a fundamental constraint since negative values of the diffusivity are not meaningful. In this paper we represent the 4th-order Tensors as ternary quartics and then apply Hilbert's theorem on ternary quartics along with the Iwasawa parametrization to guarantee an SPD 4th-order Tensor approximation from the DW-MRI data. The performance of this model is depicted on synthetic data as well as real DW-MRIs from a set of excised control and injured rat spinal cords, showing accurate estimation of scalar quantities such as generalized anisotropy and trace as well as fiber orientations.
-
Tensor Field segmentation using region based active contour model
European Conference on Computer Vision, 2004Co-Authors: Zhizhou Wang, Baba C VemuriAbstract:Tensor Fields (matrix valued data sets) have recently attracted increased attention in the Fields of image processing, computer vision, visualization and medical imaging. Tensor Field segmentation is an important problem in Tensor Field analysis and has not been addressed adequately in the past. In this paper, we present an effective region-based active contour model for Tensor Field segmentation and show its application to diffusion Tensor magnetic resonance images (MRI) as well as for the texture segmentation problem in computer vision. Specifically, we present a variational principle for an active contour using the Euclidean difference of Tensors as a discriminant. The variational formulation is valid for piecewise smooth regions, however, for the sake of simplicity of exposition, we present the piecewise constant region model in detail. This variational principle is a generalization of the region-based active contour to matrix valued functions. It naturally leads to a curve evolution equation for Tensor Field segmentation, which is subsequently expressed in a level set framework and solved numerically. Synthetic and real data experiments involving the segmentation of diffusion Tensor MRI as well as structure Tensors obtained from real texture data are shown to depict the performance of the proposed model.
-
A constrained variational principle for direct estimation and smoothing of the diffusion Tensor Field from complex DWI
IEEE Transactions on Medical Imaging, 2004Co-Authors: Zhizhou Wang, Baba C Vemuri, Thomas H. MareciAbstract:In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion Tensor Field from complex valued diffusion-weighted images (DWI). The constrained variational principle involves the minimization of a regularization term of L/sup p/ norms, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The complex valued nonlinear form leads to a more accurate (when compared to the linearized version) estimate of the Tensor Field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion Tensor, it is expressed in terms of Cholesky factors and estimated. The constrained variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Experiments with complex-valued synthetic and real data are shown to depict the performance of our Tensor Field estimation and smoothing algorithm.
Zhizhou Wang - One of the best experts on this subject based on the ideXlab platform.
-
Tensor Field segmentation using region based active contour model
European Conference on Computer Vision, 2004Co-Authors: Zhizhou Wang, Baba C VemuriAbstract:Tensor Fields (matrix valued data sets) have recently attracted increased attention in the Fields of image processing, computer vision, visualization and medical imaging. Tensor Field segmentation is an important problem in Tensor Field analysis and has not been addressed adequately in the past. In this paper, we present an effective region-based active contour model for Tensor Field segmentation and show its application to diffusion Tensor magnetic resonance images (MRI) as well as for the texture segmentation problem in computer vision. Specifically, we present a variational principle for an active contour using the Euclidean difference of Tensors as a discriminant. The variational formulation is valid for piecewise smooth regions, however, for the sake of simplicity of exposition, we present the piecewise constant region model in detail. This variational principle is a generalization of the region-based active contour to matrix valued functions. It naturally leads to a curve evolution equation for Tensor Field segmentation, which is subsequently expressed in a level set framework and solved numerically. Synthetic and real data experiments involving the segmentation of diffusion Tensor MRI as well as structure Tensors obtained from real texture data are shown to depict the performance of the proposed model.
-
A constrained variational principle for direct estimation and smoothing of the diffusion Tensor Field from complex DWI
IEEE Transactions on Medical Imaging, 2004Co-Authors: Zhizhou Wang, Baba C Vemuri, Thomas H. MareciAbstract:In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion Tensor Field from complex valued diffusion-weighted images (DWI). The constrained variational principle involves the minimization of a regularization term of L/sup p/ norms, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The complex valued nonlinear form leads to a more accurate (when compared to the linearized version) estimate of the Tensor Field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion Tensor, it is expressed in terms of Cholesky factors and estimated. The constrained variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Experiments with complex-valued synthetic and real data are shown to depict the performance of our Tensor Field estimation and smoothing algorithm.