scholarly journals Visualizing High-Order Symmetric Tensor Field Structure with Differential Operators

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Tim McGraw ◽  
Takamitsu Kawai ◽  
Inas Yassine ◽  
Lierong Zhu
Keyword(s):  
Feature Extraction ◽  
Vector Field ◽  
Tensor Field ◽  
Differential Operators ◽  
Synthetic Data ◽  
Symmetric Tensor ◽  
Field Structure ◽  
Data Sets ◽  
Tensor Fields ◽  
New Techniques

The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionallyandspatially. We explore the use of differential operators to extract features from tensor fields. These features can be used to generate skeleton representations of the data that accurately characterize the global field structure. Previously, vector field operators such as gradient, divergence, and curl have previously been used to visualize of flow fields. In this paper, we use generalizations of these operators to locate and classify tensor field degenerate points and to partition the field into regions of homogeneous behavior. We describe the implementation of our feature extraction and demonstrate our new techniques on synthetic data sets of order 2, 3 and 4.

scholarly journals Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Venkateswaran P. Krishnan ◽  
Vladimir A. Sharafutdinov
Keyword(s):  
Fourier Transform ◽  
Sobolev Spaces ◽  
Tensor Field ◽  
Differential Operators ◽  
Symmetric Tensor ◽  
Higher Order ◽  
Tensor Fields ◽  
Ray Transform ◽  
The Fourier Transform ◽  
Derivatives Of

<p style='text-indent:20px;'>For an integer <inline-formula><tex-math id="M1">\begin{document}$ r\ge0 $\end{document}</tex-math></inline-formula>, we prove the <inline-formula><tex-math id="M2">\begin{document}$ r^{\mathrm{th}} $\end{document}</tex-math></inline-formula> order Reshetnyak formula for the ray transform of rank <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula> symmetric tensor fields on <inline-formula><tex-math id="M4">\begin{document}$ {{\mathbb R}}^n $\end{document}</tex-math></inline-formula>. Roughly speaking, for a tensor field <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula>, the order <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula> refers to <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-integrability of higher order derivatives of the Fourier transform <inline-formula><tex-math id="M8">\begin{document}$ \widehat f $\end{document}</tex-math></inline-formula> over spheres centered at the origin. Certain differential operators <inline-formula><tex-math id="M9">\begin{document}$ A^{(m,r,l)}\ (0\le l\le r) $\end{document}</tex-math></inline-formula> on the sphere <inline-formula><tex-math id="M10">\begin{document}$ {{\mathbb S}}^{n-1} $\end{document}</tex-math></inline-formula> are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any <inline-formula><tex-math id="M11">\begin{document}$ r $\end{document}</tex-math></inline-formula> although the volume of calculations grows fast with <inline-formula><tex-math id="M12">\begin{document}$ r $\end{document}</tex-math></inline-formula>. The algorithm is realized for small values of <inline-formula><tex-math id="M13">\begin{document}$ r $\end{document}</tex-math></inline-formula> and Reshetnyak formulas of orders <inline-formula><tex-math id="M14">\begin{document}$ 0,1,2 $\end{document}</tex-math></inline-formula> are presented in an explicit form.</p>

scholarly journals Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 830
Author(s):  
Evgeniya V. Goloveshkina ◽  
Leonid M. Zubov
Keyword(s):  
Nonlinear Theory ◽  
Hollow Sphere ◽  
Tensor Field ◽  
Exact Analytical Solution ◽  
Symmetric Tensor ◽  
Symmetric Distribution ◽  
Spherically Symmetric ◽  
Tensor Fields ◽  
Spherically Symmetric Distribution ◽  
Nonlinear Elastic

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i. e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

scholarly journals Using control charts for on-line video summarisation

2019 ◽  
Vol 277 ◽  
pp. 01012 ◽  
Author(s):  
Clare E. Matthews ◽  
Paria Yousefi ◽  
Ludmila I. Kuncheva
Keyword(s):  
Feature Extraction ◽  
Control Chart ◽  
Control Charts ◽  
State Of The Art ◽  
Synthetic Data ◽  
New Method ◽  
Data Sets ◽  
On Line ◽  
Memory Constraints ◽  
Video Summarisation

Many existing methods for video summarisation are not suitable for on-line applications, where computational and memory constraints mean that feature extraction and frame selection must be simple and efficient. Our proposed method uses RGB moments to represent frames, and a control-chart procedure to identify shots from which keyframes are then selected. The new method produces summaries of higher quality than two state-of-the-art on-line video summarisation methods identified as the best among nine such methods in our previous study. The summary quality is measured against an objective ideal for synthetic data sets, and compared to user-generated summaries of real videos.

scholarly journals A Yang–Mills Theory in Loop Space and Chapline–Manton Coupling

1997 ◽  
Vol 12 (02) ◽  
pp. 111-119 ◽  
Author(s):  
Shinichi Deguchi ◽  
Tadahito Nakajima
Keyword(s):  
Field Theory ◽  
Local Field ◽  
Loop Space ◽  
Tensor Field ◽  
Symmetric Tensor ◽  
Tensor Fields ◽  
Yang Mills ◽  
Local Field Theory ◽  
Antisymmetric Tensor Field ◽  
Chern Simons

We consider a Yang–Mills theory in loop space with the affine gauge group. From this theory, we derive a local field theory with Yang–Mills fields and Abelian antisymmetric and symmetric tensor fields of the second rank. The Chapline–Manton coupling, i.e. coupling of Yang–Mills fields and a second-rank antisymmetric tensor field via the Chern–Simons three-form is obtained systematically.

A NEW APPROACH FOR SPACE-TIME-MATTER THEORY

2013 ◽  
Vol 10 (04) ◽  
pp. 1350004 ◽  
Author(s):  
AUREL BEJANCU
Keyword(s):  
Tensor Field ◽  
Differential Operators ◽  
Space Time ◽  
Horizontal Gradient ◽  
Tensor Fields ◽  
Tensor Calculus ◽  
New Approach ◽  
Horizontal Connection ◽  
Klein Theory ◽  
Kaluza Klein

This is the first paper in a series of three papers on a new approach for space-time-matter (STM) theory. The main purpose of this approach is to replace the Levi-Civita connection on the space-time from the classical Kaluza–Klein theory by what we call the Riemannian horizontal connection on the general Kaluza–Klein space. This is done by a development of a 4D tensor calculus whose geometrical objects live in a 5D space. The 4D tensor calculus and the Riemannian horizontal connection enable us to define in a 5D space some 4D differential operators: horizontal differential, horizontal gradient, horizontal divergence and horizontal Laplacian, which have a great role in the presentation of the STM theory in a covariant form. Finally, we introduce and study the horizontal electromagnetic tensor field, the horizontal Ricci tensor and the horizontal Einstein gravitational tensor field, which replace the well-known tensor fields from the classical Kaluza–Klein theory.

scholarly journals Robust Extraction and Simplification of 2D Symmetric Tensor Field Topology

2021 ◽  
Author(s):  
Jochen Jankowai ◽  
Bei Wang ◽  
Ingrid Hotz
Keyword(s):  
Structural Stability ◽  
Real World ◽  
Tensor Field ◽  
Symmetric Tensor ◽  
Tensor Fields ◽  
Real World Data ◽  
World Data ◽  
Labeling Algorithm ◽  
Edge Labeling

In this work, we propose a controlled simplification strategy for degenerated points in symmetric 2D tensor fields that is based on the topological notion of robustness. Robustness measures the structural stability of the degenerate points with respect to variation in the underlying field. We consider an entire pipeline for generating a hierarchical set of degenerate points based on their robustness values. Such a pipeline includes the following steps: the stable extraction and classification of degenerate points using an edge labeling algorithm, the computation and assignment of robustness values to the degenerate points, and the construction of a simplification hierarchy. We also discuss the challenges that arise from the discretization and interpolation of real world data.

Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in ℝn

2020 ◽  
Vol 28 (2) ◽  
pp. 173-184 ◽  
Author(s):  
Rohit Kumar Mishra
Keyword(s):  
Vector Field ◽  
Integral Geometry ◽  
Tensor Field ◽  
Vector Fields ◽  
First Integral ◽  
Tensor Fields ◽  
Line Complex ◽  
Full Reconstruction ◽  
Ill Posed ◽  
Fixed Curve

AbstractWe show that a vector field in {\mathbb{R}^{n}} can be reconstructed uniquely from the knowledge of restricted Doppler and first integral moment transforms. The line complex we consider consists of all lines passing through a fixed curve {\gamma\subset\mathbb{R}^{n}}. The question of reconstruction of a symmetric m-tensor field from the knowledge of the first {m+1} integral moments was posed by Sharafutdinov [Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser. 1, De Gruyter, Berlin, 1994, p. 78]. In this work, we provide an answer to Sharafutdinov’s question for the case of vector fields from restricted data comprising of the first two integral moment transforms.

scholarly journals Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus

2021 ◽  
Vol 14 (1) ◽  
pp. 1
Author(s):  
Vladimir A. Sharafutdinov
Keyword(s):  
Tensor Field ◽  
Killing Vector ◽  
First Integral ◽  
Symmetric Tensor ◽  
Two Dimensional ◽  
Killing Tensor ◽  
Tensor Fields ◽  
Killing Field ◽  
Quadratic Equations ◽  
Isothermal Coordinates

A rank m symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero. Such a field determines the first integral of the geodesic flow which is a degree m homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form ds^2= &lambda;(z)|dz|^2 in the coordinates. The torus admits a third rank Killing tensor field if and only if the function &lambda; satisfies the equation R(&part;/&part;z（&lambda;(c∆^-1&lambda;_zz+a))= 0 with some complex constants a and c&ne;0. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function &lambda;. If the functions &lambda; and &lambda; + &lambda;_0 satisfy the equation for a real constant &lambda;0, 0, then there exists a non-zero Killing vector field on the torus.

scholarly journals Tensor volume exploration using attribute space representatives

2020 ◽  
Author(s):  
Jochen Jankowai ◽  
Robin Skånberg ◽  
Daniel Jönsson ◽  
Anders Ynnerman ◽  
Ingrid Hotz
Keyword(s):  
Transfer Function ◽  
Volume Rendering ◽  
Tensor Field ◽  
Complex Geometry ◽  
Scalar Fields ◽  
Data Sets ◽  
Tensor Fields ◽  
Directional Information ◽  
3D Volume ◽  
Attribute Space

While volume rendering for scalar fields has been advanced into a powerful visualisation method, similar volumetric representations for tensor fields are still rare. The complexity of the data challenges not only the rendering but also the design of the transfer function. In this paper we propose an interface using glyph widgets to design a transfer function for the rendering of tensor data sets. Thereby the transfer function (TF) controls a volume rendering which represents sought after tensor-features and a texture that conveys directional information. The basis of the design interface is a two-dimensional projection of the attribute space. Characteristicrepresentatives in the form of glyphs support an intuitive navigation through the attribute space. We provide three different options to select the representatives: automatic selection based on attribute space clustering, uniform sampling of the attribute space, or manually selected representatives. In contrast to glyphs placed into the 3D volume, we use glyphs with complex geometry as widgets to control the shape and extent of the representatives. In the final rendering the glyphs with their assigned colors play a similar role as a legend in an atlas like representation. The method provides an overview of the tensor field in the 3D volume at the same time as it allows the user to explore the tensor field in an attribute space. We demonstrate the flexibility of our approach on tensor fields for selected data sets with very different characteristics.

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