Visualization Of Tensor Fields Research Articles

Global Topology of 3D Symmetric Tensor Fields.

There have been recent advances in the analysis and visualization of 3D symmetric tensor fields, with a focus on the robust extraction of tensor field topology. However, topological features such as degenerate curves and neutral surfaces do not live in isolation. Instead, they intriguingly interact with each other. In this paper, we introduce the notion of topological graph for 3D symmetric tensor fields to facilitate global topological analysis of such fields. The nodes of the graph include degenerate curves and regions bounded by neutral surfaces in the domain. The edges in the graph denote the adjacency information between the regions and degenerate curves. In addition, we observe that a degenerate curve can be a loop and even a knot and that two degenerate curves (whether in the same region or not) can form a link. We provide a definition and theoretical analysis of individual degenerate curves in order to help understand why knots and links may occur. Moreover, we differentiate between wedges and trisectors, thus making the analysis more detailed about degenerate curves. We incorporate this information into the topological graph. Such a graph can not only reveal the global structure in a 3D symmetric tensor field but also allow two symmetric tensor fields to be compared. We demonstrate our approach by applying it to solid mechanics and material science data sets.

Feature Curves and Surfaces of 3D Asymmetric Tensor Fields.

3D asymmetric tensor fields have found many applications in science and engineering domains, such as fluid dynamics and solid mechanics. 3D asymmetric tensors can have complex eigenvalues, which makes their analysis and visualization more challenging than 3D symmetric tensors. Existing research in tensor field visualization focuses on 2D asymmetric tensor fields and 3D symmetric tensor fields. In this paper, we address the analysis and visualization of 3D asymmetric tensor fields. We introduce six topological surfaces and one topological curve, which lead to an eigenvalue space based on the tensor mode that we define. In addition, we identify several non-topological feature surfaces that are nonetheless physically important. Included in our analysis are the realizations that triple degenerate tensors are structurally stable and form curves, unlike the case for 3D symmetric tensors fields. Furthermore, there are two different ways of measuring the relative strengths of rotation and angular deformation in the tensor fields, unlike the case for 2D asymmetric tensor fields. We extract these feature surfaces using the A-patches algorithm. However, since three of our feature surfaces are quadratic, we develop a method to extract quadratic surfaces at any given accuracy. To facilitate the analysis of eigenvector fields, we visualize a hyperstreamline as a tree stem with the other two eigenvectors represented as thorns in the real domain or the dual-eigenvectors as leaves in the complex domain. To demonstrate the effectiveness of our analysis and visualization, we apply our approach to datasets from solid mechanics and fluid dynamics.

Visualization of Tensor Fields in Mechanics

AbstractTensors are used to describe complex physical processes in many applications. Examples include the distribution of stresses in technical materials, acting forces during seismic events, or remodeling of biological tissues. While tensors encode such complex information mathematically precisely, the semantic interpretation of a tensor is challenging. Visualization can be beneficial here and is frequently used by domain experts. Typical strategies include the use of glyphs, color plots, lines, and isosurfaces. However, data complexity is nowadays accompanied by the sheer amount of data produced by large‐scale simulations and adds another level of obstruction between user and data. Given the limitations of traditional methods, and the extra cognitive effort of simple methods, more advanced tensor field visualization approaches have been the focus of this work. This survey aims to provide an overview of recent research results with a strong application‐oriented focus, targeting applications based on continuum mechanics, namely the fields of structural, bio‐, and geomechanics. As such, the survey is complementing and extending previously published surveys. Its utility is twofold: (i) It serves as basis for the visualization community to get an overview of recent visualization techniques. (ii) It emphasizes and explains the necessity for further research for visualizations in this context.

Anisotropic osmosis filtering for shadow removal in images

We present an anisotropic extension of the isotropic osmosis model that has been introduced by Weickert et al (2013 Energy Minimization Methods in Computer Vision and Pattern Recognition (Berlin: Springer)) for visual computing applications, and we adapt it specifically to shadow removal applications. We show that in the integrable setting, linear anisotropic osmosis minimises an energy that involves a suitable quadratic form which models local directional structures. In our shadow removal applications we estimate the local structure via a modified tensor voting approach (Moreno et al 2012 New Developments in the Visualization and Processing of Tensor Fields (Berlin: Springer)) and use this information within an anisotropic diffusion inpainting that resembles edge-enhancing anisotropic diffusion inpainting (Galić et al 2008 J. Math. Imaging Vis. 31 255–69; Weickert and Welk 2006 Visualization and Processing of Tensor Fields (Berlin: Springer)). Our numerical scheme combines the nonnegativity preserving stencil of Fehrenbach and Mirebeau (2014 J. Math. Imaging Vis. 49 123–47) with an exact time stepping based on highly accurate polynomial approximations of the matrix exponential. The resulting anisotropic model is tested on several synthetic and natural images corrupted by constant shadows. We show that it outperforms isotropic osmosis, since it does not suffer from blurring artefacts at the shadow boundaries.

A survey on visualization of tensor field

Tensor field has been widely used in various applications, such as medical imaging, industrial manufacturing, high-dimensional data analysis, and so forth. However, it is a challenging task to understand tensor field intuitively. Therefore, tensor field visualization has become an important research topic. In this survey, we present a comprehensive survey for two kinds of visualization methods for tensor fields: glyphs and streamlines. For glyphs, the eigenvalues of tensor fields will be used to classify existing visualization methods. There are mainly three types of eigenvalues: diffusion tensor fields with all positive real eigenvalues; the tensor field with negative real eigenvalues; the tensor field with imaginary eigenvalues. The methods showing the difference between two tensors (glyphs) are also introduced. For streamlines, there are mainly three important issues: the selection of seed points (streamlines), interpolation of tensor fields, the singularity problem around isotropic tensors. Finally, we discuss challenges and open questions for future studies.

Multi-Scale Topological Analysis of Asymmetric Tensor Fields on Surfaces.

Asymmetric tensor fields have found applications in many science and engineering domains, such as fluid dynamics. Recent advances in the visualization and analysis of 2D asymmetric tensor fields focus on pointwise analysis of the tensor field and effective visualization metaphors such as colors, glyphs, and hyperstreamlines. In this paper, we provide a novel multi-scale topological analysis framework for asymmetric tensor fields on surfaces. Our multi-scale framework is based on the notions of eigenvalue and eigenvector graphs. At the core of our framework are the identification of atomic operations that modify the graphs and the scale definition that guides the order in which the graphs are simplified to enable clarity and focus for the visualization of topological analysis on data of different sizes. We also provide efficient algorithms to realize these operations. Furthermore, we provide physical interpretation of these graphs. To demonstrate the utility of our system, we apply our multi-scale analysis to data in computational fluid dynamics.

A stable tensor-based method for controlled fluid simulations

The association between fluids and tensors can be observed in some practical situations, such as diffusion tensor imaging and permeable flow. For simulation purposes, tensors may be used to constrain the fluid flow along specific directions. This requires a customized mathematical model for describing fluid motion influenced by tensor. In this work, we propose a formulation for fluid dynamics to locally change momentum, deflecting the fluid along intended paths. Building upon classical computer graphics approaches for fluid simulation, the numerical method is altered to accommodate the new formulation. Gaining control over fluid diffusion can also aid on visualization of tensor fields, where the detection and highlighting of paths of interest is often desired. Experiments show that the fluid adequately follows meaningful paths induced by the underlying tensor field, resulting in a method that is numerically stable and suitable for visualization and animation purposes.

Визуализация тензорных полей на основе геометрического представления тензоров

Визуализация тензорных полей на основе геометрического представления тензоров

Extremal curves and surfaces in symmetric tensor fields

The visualization of symmetric second-order tensor fields in two or three dimensions is still a challenging task, particularly if global structures of the data are desired. One approach is tensor field topology which provides structures characterizing the behavior of the eigenvector fields. Another widely used approach is analyzing tensor fields by means of scalar invariants, i.e., quantities invariant with respect to changes of the coordinate system. In this case, the selection of the relevant invariants might be difficult. Thus, we propose an approach which analyzes the complete invariant part of the tensor. We define extremal points for tensor fields in a mathematically rigorous way, which form curves for two-dimensional and surfaces for three-dimensional tensor fields. We propose a way to compute extremal curves or surfaces from a suitable set of two or three invariants, respectively. We also show that commonly used sets of invariants lead to the same extremal points. Consequently, extremal points contain minima and maxima of most invariants used in tensor field analysis and they are linked to the tensor field topology by containing the degenerate points. Moreover, we show that each extremal point is an extremum or a saddle of a certain invariant. The method is demonstrated on synthetic datasets as well as on stress tensor fields from structure simulations.

Interactive Design and Visualization of Branched Covering Spaces.

Branched covering spaces are a mathematical concept which originates from complex analysis and topology and has applications in tensor field topology and geometry remeshing. Given a manifold surface and an -way rotational symmetry field, a branched covering space is a manifold surface that has an -to-1 map to the original surface except at the ramification points, which correspond to the singularities in the rotational symmetry field. Understanding the notion and mathematical properties of branched covering spaces is important to researchers in tensor field visualization and geometry processing, and their application areas. In this paper, we provide a framework to interactively design and visualize the branched covering space (BCS) of an input mesh surface and a rotational symmetry field defined on it. In our framework, the user can visualize not only the BCSs but also their construction process. In addition, our system allows the user to design the geometric realization of the BCS using mesh deformation techniques as well as connecting tubes. This enables the user to verify important facts about BCSs such as that they are manifold surfaces around singularities, as well as the Riemann-Hurwitz formula which relates the Euler characteristic of the BCS to that of the original mesh. Our system is evaluated by student researchers in scientific visualization and geometry processing as well as faculty members in mathematics at our university who teach topology. We include their evaluations and feedback in the paper.

Functional Decomposition for Bundled Simplification of Trail Sets.

Bundling visually aggregates curves to reduce clutter and help finding important patterns in trail-sets or graph drawings. We propose a new approach to bundling based on functional decomposition of the underling dataset. We recover the functional nature of the curves by representing them as linear combinations of piecewise-polynomial basis functions with associated expansion coefficients. Next, we express all curves in a given cluster in terms of a centroid curve and a complementary term, via a set of so-called principal component functions. Based on the above, we propose a two-fold contribution: First, we use cluster centroids to design a new bundling method for 2D and 3D curve-sets. Secondly, we deform the cluster centroids and generate new curves along them, which enables us to modify the underlying data in a statistically-controlled way via its simplified (bundled) view. We demonstrate our method by applications on real-world 2D and 3D datasets for graph bundling, trajectory analysis, and vector field and tensor field visualization.

Fast Visualization of Stress and Strain Tensor Fields on Streamsurfaces with LIC Texture

Fast Visualization of Stress and Strain Tensor Fields on Streamsurfaces with LIC Texture

A Topologically-Informed Hyperstreamline Seeding Method for Alignment Tensor Fields.

A topologically-informed hyperstreamline seeding method is presented for visualization of alignment tensor fields. The method is inspired by and applied to visualization of nematic liquid crystal (LC) orientation dynamics simulations. The method distributes hyperstreamlines along domain boundaries and edges of a nearest-neighbor graph whose vertices are degenerate regions of the alignment tensor field, which correspond to orientational defects in a nematic LC domain. This is accomplished without iteration while conforming to a user-specified spacing between hyperstreamlines and avoids possible failure modes associated with hyperstreamline integration in the vicinity of degeneracies in alignment (orientational defects). It is shown that the presented seeding method enables automated hyperstreamline-based visualization of a broad range of alignment tensor fields which enhances the ability of researchers to interpret these fields and provides an alternative to using glyph-based techniques.

A NEW ALGORITHM FOR COMPUTING RIEMANNIAN GEODESIC DISTANCE IN RECTANGULAR 2-D AND 3-D GRIDS

We present a novel way to efficiently compute Riemannian geodesic distance over a two- or three-dimensional domain. It is based on a previously presented method for computation of geodesic distances on surface meshes. Our method is adapted for rectangular grids, equipped with a variable anisotropic metric tensor. Processing and visualization of such tensor fields is common in certain applications, for instance structure tensor fields in image analysis and diffusion tensor fields in medical imaging. The included benchmark study shows that our method provides significantly better results in anisotropic regions in 2-D and 3-D and is faster than current stat-of-the-art solvers in 2-D grids. Additionally, our method is straightforward to code; the test implementation is less than 150 lines of C++ code. The paper is an extension of a previously presented conference paper and includes new sections on 3-D grids in particular.

Visualization and Analysis of Second‐Order Tensors: Moving Beyond the Symmetric Positive‐Definite Case

AbstractTensors provide a powerful language to describe physical phenomena. Consequently, they have a long tradition in physics and appear in various application areas, either as the final result of simulations or as intermediate product. Due to their complexity, tensors are hard to interpret. This motivates the development of well‐conceived visualization methods. As a sub‐branch of scientific visualization, tensor field visualization has been especially pushed forward by diffusion tensor imaging. In this review, we focus on second‐order tensors that are not diffusion tensors. Until now, these tensors, which might be neither positive‐definite nor symmetric, are under‐represented in visualization and existing visualization tools are often not appropriate for these tensors. Hence, we discuss the strengths and limitations of existing methods when dealing with such tensors as well as challenges introduced by them. The goal of this paper is to reveal the importance of the field and to encourage the development of new visualization methods for tensors from various application fields.

Asymmetric Tensor Field Visualization for Surfaces

Asymmetric tensor field visualization can provide important insight into fluid flows and solid deformations. Existing techniques for asymmetric tensor fields focus on the analysis, and simply use evenly-spaced hyperstreamlines on surfaces following eigenvectors and dual-eigenvectors in the tensor field. In this paper, we describe a hybrid visualization technique in which hyperstreamlines and elliptical glyphs are used in real and complex domains, respectively. This enables a more faithful representation of flow behaviors inside complex domains. In addition, we encode tensor magnitude, an important quantity in tensor field analysis, using the density of hyperstreamlines and sizes of glyphs. This allows colors to be used to encode other important tensor quantities. To facilitate quick visual exploration of the data from different viewpoints and at different resolutions, we employ an efficient image-space approach in which hyperstreamlines and glyphs are generated quickly in the image plane. The combination of these techniques leads to an efficient tensor field visualization system for domain scientists. We demonstrate the effectiveness of our visualization technique through applications to complex simulated engine fluid flow and earthquake deformation data. Feedback from domain expert scientists, who are also co-authors, is provided.

Visualizing High‐Order Symmetric Tensor Field Structure with Differential Operators

The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionally and spatially. 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.

Stress Tensor Field Visualization for Implant Planning in Orthopedics

We demonstrate the application of advanced 3D visualization techniques to determine the optimal implant design and position in hip joint replacement planning. Our methods take as input the physiological stress distribution inside a patient's bone under load and the stress distribution inside this bone under the same load after a simulated replacement surgery. The visualization aims at showing principal stress directions and magnitudes, as well as differences in both distributions. By visualizing changes of normal and shear stresses with respect to the principal stress directions of the physiological state, a comparative analysis of the physiological stress distribution and the stress distribution with implant is provided, and the implant parameters that most closely replicate the physiological stress state in order to avoid stress shielding can be determined. Our method combines volume rendering for the visualization of stress magnitudes with the tracing of short line segments for the visualization of stress directions. To improve depth perception, transparent, shaded, and antialiased lines are rendered in correct visibility order, and they are attenuated by the volume rendering. We use a focus+context approach to visually guide the user to relevant regions in the data, and to support a detailed stress analysis in these regions while preserving spatial context information. Since all of our techniques have been realized on the GPU, they can immediately react to changes in the simulated stress tensor field and thus provide an effective means for optimal implant selection and positioning in a computational steering environment.

Colliding galaxies, rotating neutron stars and merging black holes—visualizing high dimensional datasets on arbitrary meshes

Visualization of datasets stemming from diverse sources is challenged by the large variety of substantial differences in topology, geometry and nature of the associated data fields. Since there is no standard on how to formulate and treat data for scientific visualization, algorithms are frequently implemented in a highly domain-specific way. Here, we explore the potential of point-wise rendering as a generic way to represent single or multiple fields instantaneously on arbitrary mesh types. This approach is discussed within the terminology of fiber bundles as a general mathematical concept to model scalar-, vector- and tensorfields given on topological spaces (with manifolds as a particular case). We give application examples based on datasets originating from astrophysics and show first results of a tensor field visualization of a recently produced complex dataset of colliding black holes in their final orbit. We finally propose a data layout representing the mathematical concept of a ‘field’ generic enough to handle all cases involved.

Anisotropic Noise Samples

We present a practical approach to generate stochastic anisotropic samples with Poisson-disk characteristic over a two-dimensional domain. In contrast to isotropic samples, we understand anisotropic samples as non-overlapping ellipses whose size and density match a given anisotropic metric. Anisotropic noise samples are useful for many visualization and graphics applications. The spot samples can be used as input for texture generation, e.g., line integral convolution (LIC), but can also be used directly for visualization. The definition of the spot samples using a metric tensor makes them especially suitable for the visualization of tensor fields that can be translated into a metric. Our work combines ideas from sampling theory and mesh generation. To generate these samples with the desired properties we construct a first set of non-overlapping ellipses whose distribution closely matches the underlying metric. This set of samples is used as input for a generalized anisotropic Lloyd relaxation to distribute noise samples more evenly. Instead of computing the Voronoi tessellation explicitly, we introduce a discrete approach which combines the Voronoi cell and centroid computation in one step. Our method supports automatic packing of the elliptical samples, resulting in textures similar to those generated by anisotropic reaction-diffusion methods. We use Fourier analysis tools for quality measurement of uniformly distributed samples. The resulting samples have nice sampling properties, for example, they satisfy a blue noise property where low frequencies in the power spectrum are reduced to a minimum.