Glyphs for space-time Jacobians of time-dependent vector fields

Glyphs have proven to be a powerful visualization technique for general tensor fields modeling physical phenomena such as diffusion or the derivative of flow fields. Most glyph constructions, however, do not provide a way of considering the temporal derivative, which is generally nonzero in non-stationary vector fields. This derivative offers a deeper understanding of features in time-dependent vector fields. We introduce an extension to 2D and 3D tensor glyph design that additionally encodes the temporal information of velocities, and thus makes it possible to represent time-dependent Jacobians. At the same time, a certain set of requirements for general tensor glyphs is fulfilled, such that the new method provides a visualization of the steadiness or unsteadiness of a vector field at a given instance of time

Probing white-matter microstructure with higher-order diffusion tensors and susceptibility tensor MRI.

Diffusion MRI has become an invaluable tool for studying white matter microstructure and brain connectivity. The emergence of quantitative susceptibility mapping and susceptibility tensor imaging (STI) has provided another unique tool for assessing the structure of white matter. In the highly ordered white matter structure, diffusion MRI measures hindered water mobility induced by various tissue and cell membranes, while susceptibility sensitizes to the molecular composition and axonal arrangement. Integrating these two methods may produce new insights into the complex physiology of white matter. In this study, we investigated the relationship between diffusion and magnetic susceptibility in the white matter. Experiments were conducted on phantoms and human brains in vivo. Diffusion properties were quantified with the diffusion tensor model and also with the higher order tensor model based on the cumulant expansion. Frequency shift and susceptibility tensor were measured with quantitative susceptibility mapping and susceptibility tensor imaging. These diffusion and susceptibility quantities were compared and correlated in regions of single fiber bundles and regions of multiple fiber orientations. Relationships were established with similarities and differences identified. It is believed that diffusion MRI and susceptibility MRI provide complementary information of the microstructure of white matter. Together, they allow a more complete assessment of healthy and diseased brains

Visualization of Tensor Fields in Mechanics

Tensors 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

Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold

Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can lead to the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis

Nlcviz: Tensor Visualization And Defect Detection In Nematic Liquid Crystals

Visualization and exploration of nematic liquid crystal (NLC) data is a challenging task due to the multidimensional and multivariate nature of the data. Simulation study of an NLC consists of multiple timesteps, where each timestep computes scalar, vector, and tensor parameters on a geometrical mesh. Scientists developing an understanding of liquid crystal interaction and physics require tools and techniques for effective exploration, visualization, and analysis of these data sets. Traditionally, scientists have used a combination of different tools and techniques like 2D plots, histograms, cut views, etc. for data visualization and analysis. However, such an environment does not provide the required insight into NLC datasets. This thesis addresses two areas of the study of NLC data---understanding of the tensor order field (the Q-tensor) and defect detection in this field. Tensor field understanding is enhanced by using a new glyph (NLCGlyph) based on a new design metric which is closely related to the underlying physical properties of an NLC, described using the Q-tensor. A new defect detection algorithm for 3D unstructured grids based on the orientation change of the director is developed. This method has been used successfully in detecting defects for both structured and unstructured models with varying grid complexity

Analysis and Visualization of Higher-Order Tensors: Using the Multipole Representation

Materialien wie Kristalle, biologisches Gewebe oder elektroaktive Polymere kommen häufig in verschiedenen Anwendung, wie dem Prothesenbau oder der Simulation von künstlicher Muskulatur vor. Diese und viele weitere Materialien haben gemeinsam, dass sie unter gewissen Umständen ihre Form und andere Materialeigenschaften ändern. Um diese Veränderung beschreiben zu können, werden, abhängig von der Anwendung, verschiedene Tensoren unterschiedlicher Ordnung benutzt. Durch die Komplexität und die starke Abhängigkeit der Tensorbedeutung von der Anwendung, gibt es bisher kein Verfahren Tensoren höherer Ordnung darzustellen, welches standardmäßig benutzt wird. Auch bezogen auf einzelne Anwendungen gibt es nur sehr wenig Arbeiten, die sich mit der visuellen Darstellung dieser Tensoren auseinandersetzt. Diese Arbeit beschäftigt sich mit diesem Problem. Es werden drei verschiedene Methoden präsentiert, Tensoren höherer Ordnung zu analysieren und zu visualisieren. Alle drei Methoden basieren auf der sogenannte deviatorischen Zerlegung und der Multipoldarstellung. Mit Hilfe der Multipole können die Symmetrien des Tensors und damit des beschriebenen Materials bestimmt werden. Diese Eigenschaft wird in für die Visualisierung des Steifigkeitstensors benutzt. Die zweite Methode basiert direkt auf den Multipolen und kann damit beliebige Tensoren in drei Dimensionen darstellen. Dieses Verfahren wird anhand des Kopplungs Tensors, ein Tensor dritter Ordnung, vorgestellt. Die ersten zwei Verfahren sind lokale Glyph-basierte Verfahren. Das dritte Verfahren ist ein erstes globales Tensorvisualisierungsverfahren, welches Tensoren beliebiger Ordnung und Symmetry in drei Dimensionen mit Hilfe eines linienbasierten Verfahrens darstellt.Materials like crystals, biological tissue or electroactive polymers are frequently used in applications like prosthesis construction or the simulation of artificial musculature. These and many other materials have in common that they change their shape and other material properties under certain circumstances. To describe these changes, different tensors of different order, dependent of the application, are used. Due to the complexity and the strong dependency of the tensor meaning of the application, there is, by now, no visualization method that is used by default. Also for specific applications there are only a few methods that address the visual analysis of higher-order tensors. This work adresses this problem. Three different methods to analyse and visualize tensors of higher order will be provided. All three methods are based on the so called deviatoric decomposition and the multipole representation. Using the multipoles the symmetries of a tensor and, therefore, of the described material, can be calculated. This property is used to visualize the stiffness tensor. The second method uses the multipoles directly and can be used for each tensor of any order in three dimensions. This method is presented by analysing the third-order coupling tensor. These two techniques are glyph-based visualization methods. The third one, a line-based method, is, according to our knowledge, a first global visualization method that can be used for an arbitrary tensor in three dimensions

A Visual Approach to Analysis of Stress Tensor Fields

We present a visual approach for the exploration of stress tensor fields. In contrast to common tensor visualization methods that only provide a single view to the tensor field, we pursue the idea of providing various perspectives onto the data in attribute and object space. Especially in the context of stress tensors, advanced tensor visualization methods have a young tradition. Thus, we propose a combination of visualization techniques domain experts are used to with statistical views of tensor attributes. The application of this concept to tensor fields was achieved by extending the notion of shape space. It provides an intuitive way of finding tensor invariants that represent relevant physical properties. Using brushing techniques, the user can select features in attribute space, which are mapped to displayable entities in a three-dimensional hybrid visualization in object space. Volume rendering serves as context, while glyphs encode the whole tensor information in focus regions. Tensorlines can be included to emphasize directionally coherent features in the tensor field. We show that the benefit of such a multi-perspective approach is manifold. Foremost, it provides easy access to the complexity of tensor data. Moreover, including well-known analysis tools, such as Mohr diagrams, users can familiarize themselves gradually with novel visualization methods. Finally, by employing a focus-driven hybrid rendering, we significantly reduce clutter, which was a major problem of other three-dimensional tensor visualization methods