Abstract An important part of critical thinking is analyzing the logical structure of inferences made in text, such as checking whether if–then rules appearing in legal regulations or scientific arguments are applied correctly. Metacognitive sensitivity, defined as feeling confident about one's correct answers and uncertain about incorrect ones, is an indispensable component of such assessments because it supports the reliable identification of unstated assumptions and flawed conclusions. In this paper, we explore how visual cues can foster metacognitive sensitivity during the different phases of reasoning in logical inference tasks. We propose a visual cueing framework based on findings from deductive reasoning psychology and multimedia learning. The cues highlight premises and conclusions of problems stated in natural language, and visualize the extracted sequences of conditional statements as diagrams to guide viewers' attention to common invalid inferences. Results from an empirical study (N =164) demonstrate that users exhibit significantly greater metacognitive sensitivity with such visual cues than without.

In the visual analysis of hierarchical data, a main challenge is comparing data attributes both within and between different levels of the hierarchy. Such tasks are typically addressed using aggregated data, where the attribute of a parent node is calculated from the attributes of its children. Our review of existing literature on visualization methods that encode the hierarchy implicitly (e.g., icicle plots, treemaps) shows that most approaches rely on additive aggregation. Proportional aggregation, where a parent is assigned a weighted average of the values of its children, remains unexplored although relevant in practice. We introduce stalactite plots, a visualization technique that displays proportional aggregation and supports visual value comparison. Our empirical evaluation (N = 148, N = 50) shows that, with some explanation, stalactite plots are as easily understood as established visualization techniques for hierarchical data. Furthermore, for large datasets, participants are faster and more accurate using our approach.

This paper presents a particle-based visualization approach for finite-time analysis of the connectivity of fluid portions in multiphase flow, i.e., the evolution of the droplets in volume of fluid simulations. We address the Lagrangian inconsistency between the interpolated flow field and the interpolated volume of fluid field by a correction approach, and complement that with an uncertainty measure that provides an estimate of the involved inconsistency. We demonstrate the utility and versatility of our approach using different multiphase flow simulations, exemplify its application in physics-based assessment of droplet formation processes, and discuss its limitations and benefits.

This January 2025 issue of the IEEE Transactions on Visualization and Computer Graphics (TVCG) contains the proceedings of IEEE VIS 2024, held on October 1318 October, 2024 in St. Pete Beach, Florida, USA, with the three General Chairs Paul Rosen (University of Utah), Kristi Potter (U.S. National Renewable Energy Laboratory), and Remco Chang (Tufts University). With IEEE VIS 2024, the conference series is in its 35th year.

We present a readily accessible web-based interactive system for ray casting-based flow visualization in quotient spaces and ${\mathbb{S}^3}$. It is the result of a long-term close collaboration with researchers from symplectic geometry, and focuses on accessibility, flexibility, and provenance. By storing its state in URLs and providing a version-aware mechanism to link from images in PDFs to the online tool in the state that produced the image, we ensure reproducibility, explorability, and collaboration. We exemplify the utility of our approach with examples from mathematics and symplectic geometry.

This paper extends the concept and the visualization of vector field topology to vector fields with discontinuities. We address the non-uniqueness of flow in such fields by introduction of a time-reversible concept of equivalence. This concept generalizes streamlines to streamsets and thus vector field topology to discontinuous vector fields in terms of invariant streamsets. We identify respective novel critical structures as well as their manifolds, investigate their interplay with traditional vector field topology, and detail the application and interpretation of our approach using specifically designed synthetic cases and a simulated case from physics.

Abstract We present a novel approach for rendering volumetric data including the Doppler effect of light. Similar to the acoustic Doppler effect, which is caused by relative motion between a sound emitter and an observer, light waves also experience compression or expansion when emitter and observer exhibit relative motion. We account for this by employing spectral volume rendering in an emission–absorption model, with the volumetric matter moving according to an accompanying vector field, and emitting and attenuating light at wavelengths subject to the Doppler effect. By introducing a novel piecewise linearear representation of the involved light spectra, we achieve accurate volume rendering at interactive frame rates. We compare our technique to rendering with traditional point‐based spectral representation, and demonstrate its utility using a simulation of galaxy formation.

In this paper, we study the visual design of hierarchical multivariate data analysis. We focus on the extension of four hierarchical univariate concepts-the sunburst chart, the icicle plot, the circular treemap, and the bubble treemap-to the multivariate domain. Our study identifies several advantageous design variants, which we discuss with respect to previous approaches, and whose utility we evaluate with a user study and demonstrate for different analysis purposes and different types of data.

Abstract We present an approach to local extraction of 3D time‐dependent vector field topology. In this concept, Lagrangian coherent structures, which represent the separating manifolds in time‐dependent transport, correspond to generalized streak manifolds seeded along hyperbolic path surfaces (HPSs). Instead of expensive and numerically challenging direct computation of the HPSs by intersection of ridges in the forward and backward finite‐time Lyapunov exponent (FTLE) fields, our approach employs local extraction of respective candidates in the four‐dimensional space‐time domain. These candidates are subsequently refined toward the hyperbolic path surfaces, which provides unsteady equivalents of saddle‐type critical points, periodic orbits, and bifurcation lines from steady, traditional vector field topology. In contrast to FTLE‐based methods, we obtain an explicit geometric representation of the topological skeleton of the flow, which for steady flows coincides with the hyperbolic invariant manifolds of vector field topology. We evaluate our approach on analytical flows, as well as data from computational fluid dynamics, using the FTLE as a ground truth superset, i.e., we also show that FTLE ridges exhibit several types of false positives.

In this paper, we introduce uncertainty to continuous scatterplots and continuous parallel coordinates. We derive respective models, validate them with sampling-based brute-force schemes, and present acceleration strategies for their computation. At the same time, we show that our approach lends itself as well for introducing uncertainty into the definition of fibers in bivariate data. Finally, we demonstrate the properties and the utility of our approach using specifically designed synthetic cases and simulated data.

Abstract This paper does two main contributions to 2D time‐dependent vector field topology. First, we present a technique for robust, accurate, and efficient extraction of distinguished hyperbolic trajectories (DHT), the generative structures of 2D time‐dependent vector field topology. It is based on refinement of initial candidate curves. In contrast to previous approaches, it is robust because the refinement converges for reasonably close initial candidates, it is accurate due to its adaptive scheme, and it is efficient due to its high convergence speed. Second, we provide a detailed evaluation and discussion of previous approaches for the extraction of DHTs and time‐dependent vector field topology in general. We demonstrate the utility of our approach using analytical flows, as well as data from computational fluid dynamics.

Abstract In this paper, we present an integrated visual analytics approach to support the parametrization and exploration of flow visualization based on the finite‐time Lyapunov exponent. Such visualization of time‐dependent flow faces various challenges, including the choice of appropriate advection times, temporal regions of interest, and spatial resolution. Our approach eases these challenges by providing the user with context by means of parametric aggregations, with support and guidance for a more directed exploration, and with a set of derived measures for better qualitative assessment. We demonstrate the utility of our approach with examples from computation fluid dynamics and time‐dependent dynamical systems.

Abstract In this paper, we generalize the parallel vectors operator due to Peikert and Roth to arbitrary dimension, i.e., to four‐dimensional fields and beyond. Whereas the original operator tested for parallelism of two (derived) 2D or 3D vector fields, we reformulate the concept in terms of linear dependency of sets of vector fields, and propose a generic technique to extract and filter the solution manifolds. We exemplify our approach for vortex cores, bifurcations, and ridges as well as valleys in higher dimensions.

Abstract In this paper, we show how the equivalence property leads to the novel concept of equivalent regions in mappings from ℝn to ℝn. We present a technique for obtaining these regions both in the domain and the codomain of such a mapping, and determine their correspondence. This enables effective investigation of variation equivalence within mappings, and between mappings in terms of comparative visualization. We implement our approach for n = 2, and demonstrate its utility using different examples.

In this paper, we present Gaia Sky, a free and open-source multiplatform 3D Universe system, developed since 2014 in the Data Processing and Analysis Consortium framework of ESA's Gaia mission. Gaia's data release 2 represents the largest catalog of the stars of our Galaxy, comprising 1.3 billion star positions, with parallaxes, proper motions, magnitudes, and colors. In this mission, Gaia Sky is the central tool for off-the-shelf visualization of these data, and for aiding production of outreach material. With its capabilities to effectively handle these data, to enable seamless navigation along the high dynamic range of distances, and at the same time to provide advanced visualization techniques including relativistic aberration and gravitational wave effects, currently no actively maintained cross-platform, modern, and open alternative exists.

In this paper, we present a novel visualization approach for the analysis of fragmentation of molecules, with a particular focus on fullerenes. Our approach consists of different components at different levels of detail. Whereas one component is geometric but invariant to rotations, two other components are based on the topological structure of the molecules and thus additionally invariant to deformations. By combining these three components, which aim at the analysis of simulation ensembles of such molecules, and complementing them with a space-time representation that enables detailed interactive inspection of individual simulations, we obtain a versatile tool for the analysis of the fragmentation of structured, symmetrical molecules such as fullerenes. We exemplify the utility of our approach using a tightly coupled simulation approach for the dynamics of fullerenes.

Abstract In this paper, we present an approach to the topological analysis of four‐dimensional vector fields. In analogy to traditional 2D and 3D vector field topology, we provide a classification and visual representation of critical points, together with a technique for extracting their invariant manifolds. For effective exploration of the resulting four‐dimensional structures, we present a 4D camera that provides concise representation by exploiting projection degeneracies, and a 4D clipping approach that avoids self‐intersection in the 3D projection. We exemplify the properties and the utility of our approach using specific synthetic cases.

In single-phase flow visualization, research focuses on the analysis of vector field properties. In two-phase flow, in contrast, analysis of the phase components is typically of major interest. So far, visualization research of two-phase flow concentrated on proper interface reconstruction and the analysis thereof. In this paper, we present a novel visualization technique that enables the investigation of complex two-phase flow phenomena with respect to the physics of breakup and coalescence of inclusions. On the one hand, we adapt dimensionless quantities for a localized analysis of phase instability and breakup, and provide detailed inspection of breakup dynamics with emphasis on oscillation and its interplay with rotational motion. On the other hand, we present a parametric tightly linked space-time visualization approach for an effective interactive representation of the overall dynamics. We demonstrate the utility of our approach using several two-phase CFD datasets.

Jet-streams, their core lines and their role in atmospheric dynamics have been subject to considerable meteorological research since the first half of the twentieth century. Yet, until today no consistent automated feature detection approach has been proposed to identify jet-stream core lines from 3D wind fields. Such 3D core lines can facilitate meteorological analyses previously not possible. Although jet-stream cores can be manually analyzed by meteorologists in 2D as height ridges in the wind speed field, to the best of our knowledge no automated ridge detection approach has been applied to jet-stream core detection. In this work, we -a team of visualization scientists and meteorologists-propose a method that exploits directional information in the wind field to extract core lines in a robust and numerically less involved manner than traditional 3D ridge detection. For the first time, we apply the extracted 3D core lines to meteorological analysis, considering real-world case studies and demonstrating our method's benefits for weather forecasting and meteorological research.

Traditional vector field visualization has a close focus on velocity, and is typically constrained to the dynamics of massless particles. In this paper, we present a novel approach to the analysis of the force-induced dynamics of inertial particles. These forces can arise from acceleration fields such as gravitation, but also be dependent on the particle dynamics itself, as in the case of magnetism. Compared to massless particles, the velocity of an inertial particle is not determined solely by its position and time in a vector field. In contrast, its initial velocity can be arbitrary and impacts the dynamics over its entire lifetime. This leads to a four-dimensional problem for 2D setups, and a six-dimensional problem for the 3D case. Our approach avoids this increase in dimensionality and tackles the visualization by an integrated topological analysis approach. We demonstrate the utility of our approach using a synthetic time-dependent acceleration field, a system of magnetic dipoles, and N-body systems both in 2D and 3D.

Abstract We present a novel and efficient technique to extract Lagrangian coherent structures in two‐dimensional time‐dependent vector fields. We show that this can be achieved by employing bifurcation line extraction in the space‐time representation of the vector field, and generating space‐time bifurcation manifolds therefrom. To show the utility and applicability of our approach, we provide an evaluation of existing extraction techniques for Lagrangian coherent structures, and compare them to our approach.

Abstract We present a novel physically‐based method to visualize stress tensor fields. By incorporating photoelasticity into traditional raycasting and extending it with reflection and refraction, taking into account polarization, we obtain the virtual counterpart to traditional experimental polariscopes. This allows us to provide photoelastic analysis of stress tensor fields in arbitrary domains. In our model, the optical material properties, such as stress‐optic coefficient and refractive index, can either be chosen in compliance with the subject under investigation, or, in case of stress problems that do not model optical properties or that are not transparent, be chosen according to known or even new transparent materials. This enables direct application of established polariscope methodology together with respective interpretation. Using a GPU‐based implementation, we compare our technique to experimental data, and demonstrate its utility with several simulated datasets.

Abstract Visualization of pathlines is common and highly relevant for the analysis of unsteady flow. However, pathlines can intersect, leading to visual clutter and perceptual issues. This makes it intrinsically difficult to provide expressive visualizations of the entire domain by an arrangement of multiple pathlines, in contrast to well‐established streamline placement techniques. We present an approach to reduce these problems. It is inspired by glyph‐based visualization and small multiples: we partition the domain into cells, each corresponding to a downscaled version of the entire domain. Inside these cells, a single downscaled pathline is drawn. On the overview scale, our pathline glyphs lead to emergent visual patterns that provide insight into time‐dependent flow behavior. Zooming‐in allows us to analyze individual pathlines in detail and compare neighboring lines. The overall approach is complemented with a context‐preserving zoom lens and interactive pathline‐based exploration. While we primarily target the visualization of 2D flow, we also address the extension to 3D. Our evaluation includes several examples, comparison to other flow visualization techniques, and a user study with domain experts.

Abstract For the rendering of multiple scattering effects in participating media, methods based on the diffusion approximation are an extremely efficient alternative to Monte Carlo path tracing. However, in sufficiently transparent regions, classical diffusion approximation suffers from non‐physical radiative fluxes which leads to a poor match to correct light transport. In particular, this prevents the application of classical diffusion approximation to heterogeneous media, where opaque material is embedded within transparent regions. To address this limitation, we introduce flux‐limited diffusion, a technique from the astrophysics domain. This method provides a better approximation to light transport than classical diffusion approximation, particularly when applied to heterogeneous media, and hence broadens the applicability of diffusion‐based techniques. We provide an algorithm for flux‐limited diffusion, which is validated using the transport theory for a point light source in an infinite homogeneous medium. We further demonstrate that our implementation of flux‐limited diffusion produces more accurate renderings of multiple scattering in various heterogeneous datasets than classical diffusion approximation, by comparing both methods to ground truth renderings obtained via volumetric path tracing.

We present a novel and efficient method to compute volumetric soft shadows for interactive direct volume visualization to improve the perception of spatial depth. By direct control of the softness of volumetric shadows, disturbing visual patterns due to hard shadows can be avoided and users can adapt the illumination to their personal and application-specific requirements. We compute the shadowing of a point in the data set by employing spatial filtering of the optical depth over a finite area patch pointing toward each light source. Conceptually, the area patch spans a volumetric region that is sampled with shadow rays; afterward, the resulting optical depth values are convolved with a low-pass filter on the patch. In the numerical computation, however, to avoid expensive shadow ray marching, we show how to align and set up summed area tables for both directional and point light sources. Once computed, the summed area tables enable efficient evaluation of soft shadows for each point in constant time without shadow ray marching and the softness of the shadows can be controlled interactively. We integrated our method in a GPU-based volume renderer with ray casting from the camera, which offers interactive control of the transfer function, light source positions, and viewpoint, for both static and time-dependent data sets. Our results demonstrate the benefit of soft shadows for visualization to achieve user-controlled illumination with many-point lighting setups for improved perception combined with high rendering speed.

We present a novel scheme for progressive rendering in interactive visualization. Static settings with respect to a certain image quality or frame rate are inherently incapable of delivering both high frame rates for rapid changes and high image quality for detailed investigation. Our novel technique flexibly adapts by steering the visualization process in three major degrees of freedom: when to terminate the refinement of a frame in the background and start a new one, when to display a frame currently computed, and how much resources to consume. We base these decisions on the correlation of the errors due to insufficient sampling and response delay, which we estimate separately using fast yet expressive heuristics. To automate the configuration of the steering behavior, we employ offline video quality analysis. We provide an efficient implementation of our scheme for the application of volume raycasting, featuring integrated GPU-accelerated image reconstruction and error estimation. Our implementation performs an integral handling of the changes due to camera transforms, transfer function adaptations, as well as the progression of the data to in time. Finally, the overall technique is evaluated with an expert study.

We present a technique to visualize the streamline-based mapping between the boundary of a simply-connected subregion of arbitrary 3D vector fields. While the streamlines are seeded on one part of the boundary, the remaining part serves as escape border. Hence, the seeding part of the boundary represents a map of streamline behavior, indicating if streamlines reach the escape border or not. Since the resulting maps typically exhibit a very fine and complex structure and are thus not amenable to direct sampling, our approach instead aims at topologically consistent extraction of their boundary. We show that isocline surfaces of the projected vector field provide a robust basis for stream-surface-based extraction of these boundaries. The utility of our technique is demonstrated in the context of transport processes using vector field data from different domains.

Piecewise linear interface calculation (PLIC) is one of the most widely employed reconstruction schemes for the simulation of multiphase flow. In this visualization paper we focus on the reconstruction from the simulation point of view, i.e., we present a framework for the analysis of this reconstruction scheme together with its implications on the overall simulation. By interpreting PLIC reconstruction as an isosurface extraction problem from the first-order Taylor approximation of the underlying volume of fluid field, we obtain a framework for error analysis and geometric representation of the reconstruction including the fluxes involved in the simulation. At the same time this generalizes PLIC to higher-order approximation. We exemplify the utility and versatility of our visualization approach on several multiphase CFD examples.

Abstract It is difficult to create appropriate bar charts for data that cover large value ranges. The usual approach for these cases employs a logarithmic scale, which, however, suffers from issues inherent to its non‐linear mapping: for example, a quantitative comparison of different values is difficult. We present a new approach for bar charts that combines the advantages of linear and logarithmic scales, while avoiding their drawbacks. Our scale‐stack bar charts use multiple scales to cover a large value range, while the linear mapping within each scale preserves the ability to visually compare quantitative ratios. Scale‐stack bar charts can be used for the same applications as classic bar charts; in particular, they can readily handle stacked bar representations and negative values. Our visualization technique is demonstrated with results for three different application areas and is assessed by an expert review and a quantitative user study confirming advantages of our technique for quantitative comparisons.

We present ambient scattering as a preintegration method for scattering on mesoscopic scales in direct volume rendering. Far-range scattering effects usually provide negligible contributions to a given location due to the exponential attenuation with increasing distance. This motivates our approach to preintegrating multiple scattering within a finite spherical region around any given sample point. To this end, we solve the full light transport with a Monte-Carlo simulation within a set of spherical regions, where each region may have different material parameters regarding anisotropy and extinction. This precomputation is independent of the data set and the transfer function, and results in a small preintegration table. During rendering, the look-up table is accessed for each ray sample point with respect to the viewing direction, phase function, and material properties in the spherical neighborhood of the sample. Our rendering technique is efficient and versatile because it readily fits in existing ray marching algorithms and can be combined with local illumination and volumetric ambient occlusion. It provides interactive volumetric scattering and soft shadows, with interactive control of the transfer function, anisotropy parameter of the phase function, lighting conditions, and viewpoint. A GPU implementation demonstrates the benefits of ambient scattering for the visualization of different types of data sets, with respect to spatial perception, high-quality illumination, translucency, and rendering speed.

It was shown recently how the 2D vector field topology concept, directly applicable to stationary vector fields only, can be generalized to time-dependent vector fields by replacing the role of stream lines by streak lines. The present paper extends this concept to 3D vector fields. In traditional 3D vector field topology separatrices can be obtained by integrating stream lines from 0D seeds corresponding to critical points. We show that in our new concept, in contrast, 1D seeding constructs are required for computing streak-based separatrices. In analogy to the 2D generalization we show that invariant manifolds can be obtained by seeding streak surfaces along distinguished path surfaces emanating from intersection curves between codimension-1 ridges in the forward and reverse finite-time Lyapunov exponent (FTLE) fields. These path surfaces represent a time-dependent generalization of critical points and convey further structure in time-dependent topology of vector fields. Compared to the traditional approach based on FTLE ridges, the resulting streak manifolds ease the analysis of Lagrangian coherent structures (LCS) with respect to visual quality and computational cost, especially when time series of LCS are computed. We exemplify validity and utility of the new approach using both synthetic examples and computational fluid dynamics results.

High performance computing still typically disregards heterogeneous environments and focuses on homogeneous clusters. While providing advantages and easing development, this fails to address the recent change toward heterogeneous computing infrastructure. With DIANA we presented an abstraction layer for unified access to local compute hardware including different hardware APIs. In this paper we extend DIANA, enabling transparent access to remote hardware. In this context, we demonstrate how to use DIANA for distributed computation and large-scale visualization of the finite-time Lyapunov exponent field in heterogeneous environments.

Abstract Magnetic fields exhibit higher‐order, nonlinear singularities in the form of point‐dipole singularities. In addition, due to absence of divergence, they feature only a subset of invariant structures from traditional vector field topology. For magnetic fields of sets of point dipoles—widely present in physics and often used as an approximation—we present a technique revealing the topology of magnetic flux. The flux topology is identified with areas covered by field lines that directly connect pairs of dipoles. We introduce the dipole connectrix as a reduced one‐manifold representation of those areas. The set of connectrices serves as our concise visualization of the global structure of magnetic flux. In addition, the quantitative values of flux are displayed by the thickness of the connectrices. We evaluate our technique for simulations of ferroparticle monolayers and magnetic gels.

Abstract Advection has been the standard transport mechanism in flow visualization. Diffusion, in contrast, has not been considered important in visual flow field analysis so far, although it is inherent to many physical processes. We present a novel technique that allows for interactive 3D visualization of both advection and diffusion in unsteady fluid flow. We extend texture‐based flow visualization, which is advection‐oriented, by diffusion. Our finite volume approach based on WENO (weighted essentially non‐oscillatory) reconstruction is well parallelizable and features low numerical diffusion at interactive rates. Our scheme contributes to three different applications: (a) high‐quality dye advection at low numerical diffusion, (b) physically‐based dye advection accounting for diffusivity of virtual media, and (c) visualization of advection‐diffusion fluxes in physical media where the velocity field is accompanied by a concentration field. Interactive rendering of the virtual dye is accomplished by ray casting. We apply our GPU implementation to CFD examples of thermal convection and evaporation phenomena.

This paper presents a visualization approach for detecting and exploring similarity in the temporal variation of field data. We provide an interactive technique for extracting correlations from similarity matrices which capture temporal similarity of univariate functions. We make use of the concept to extract periodic and quasiperiodic behavior at single (spatial) points as well as similarity between different locations within a field and also between different data sets. The obtained correlations are utilized for visual exploration of both temporal and spatial relationships in terms of temporal similarity. Our entire pipeline offers visual interaction and inspection, allowing for the flexibility that in particular time-dependent data analysis techniques require. We demonstrate the utility and versatility of our approach by applying our implementation to data from both simulation and measurement.

Abstract The parallel vectors (PV) operator is a feature extraction approach for defining line‐type features such as creases (ridges and valleys) in scalar fields, as well as separation, attachment, and vortex core lines in vector fields. In this work, we extend PV feature extraction to higher‐order data represented by piecewise analytical functions defined over grid cells. The extraction uses PV in two distinct stages. First, seed points on the feature lines are placed by evaluating the inclusion form of the PV criterion with reduced affine arithmetic. Second, a feature flow field is derived from the higher‐order PV expression where the features can be extracted as streamlines starting at the seeds. Our approach allows for guaranteed bounds regarding accuracy with respect to existence, position, and topology of the features obtained. The method is suitable for parallel implementation and we present results obtained with our GPU‐based prototype. We apply our method to higher‐order data obtained from discontinuous Galerkin fluid simulations.

This paper presents an acceleration scheme for the numerical computation of sets of trajectories in vector fields or iterated solutions in maps, possibly with simultaneous evaluation of quantities along the curves such as integrals or extrema. It addresses cases with a dense evaluation on the domain, where straightforward approaches are subject to redundant calculations. These are avoided by first calculating short solutions for the whole domain. From these, longer solutions are then constructed in a hierarchical manner until the designated length is achieved. While the computational complexity of the straightforward approach depends linearly on the length of the solutions, the computational cost with the proposed scheme grows only logarithmically with increasing length. Due to independence of subtasks and memory locality, our algorithm is suitable for parallel execution on many-core architectures like GPUs. The trade-offs of the method--lower accuracy and increased memory consumption--are analyzed, including error order as well as numerical error for discrete computation grids. The usefulness and flexibility of the scheme are demonstrated with two example applications: line integral convolution and the computation of the finite-time Lyapunov exponent. Finally, results and performance measurements of our GPU implementation are presented for both synthetic and simulated vector fields from computational fluid dynamics.

This paper generalizes the concept of Lagrangian coherent structures, which is known for its potential to visualize coherent regions in vector fields and to distinguish them from each other. In particular, we extend the concept of the flow map to generic mappings of coordinates. As the major application of this generalization, we present a semiglobal method for visualizing coherent structures in symmetric second order tensor fields. We demonstrate the usefulness by examples from DT-MRI, uncovering anatomical structures in linearly anisotropic regions not amenable to local feature criteria. To further exemplify the suitability of our concept, we also present its application to stress tensor fields. Last, an accelerated implementation utilizing GPUs is presented.

Abstract In this paper we present an extended critical point concept which allows us to apply vector field topology in the case of unsteady flow. We propose a measure forunsteadinesswhich describes the rate of change of the velocities in a fluid element over time. This measure allows us to select particles for which topological properties remain intact inside a finite spatio‐temporal neighborhood. One benefit of this approach is that the classification of critical points based on the eigenvalues of the Jacobian remains meaningful. In the steady case the proposed criterion reduces to the classical definition of critical points. As a first step we show that finding an optimal Galilean frame of reference can be obtained implicitly by analyzing the acceleration field. In a second step we show that this can be extended by switching to the Lagrangian frame of reference. This way the criterion can detect critical points moving along intricate trajectories. We analyze the behavior of the proposed criterion based on two analytical vector fields for which a correct solution is defined by their inherent symmetries and present results for numerical vector fields.

Abstract This paper presents an approach to a time‐dependent variant of the concept of vector field topology for 2‐D vector fields. Vector field topology is defined for steady vector fields and aims at discriminating the domain of a vector field into regions of qualitatively different behaviour. The presented approach represents a generalization for saddle‐type critical points and their separatrices to unsteady vector fields based on generalized streak lines, with the classical vector field topology as its special case for steady vector fields. The concept is closely related to that of Lagrangian coherent structures obtained as ridges in the finite‐time Lyapunov exponent field. The proposed approach is evaluated on both 2‐D time‐dependent synthetic and vector fields from computational fluid dynamics.

Motivated by the growing interest in the use of ridges in scientific visualization, we analyze the two height ridge definitions by Eberly and Lindeberg. We propose a raw feature definition leading to a superset of the ridge points as obtained by these two definitions. The set of raw feature points has the correct dimensionality, and it can be narrowed down to either Eberly's or Lindeberg's ridges by using Boolean filters which we formulate. While the straight-forward computation of height ridges requires explicit eigenvalue calculation, this can be avoided by using an equivalent definition of the raw feature set, for which we give a derivation. We describe efficient algorithms for two special cases, height ridges of dimension one and of co-dimension one. As an alternative to the aforementioned filters, we propose a new criterion for filtering raw features based on the distance between contours which generally makes better decisions, as we demonstrate on a few synthetic fields, a topographical dataset, and a fluid flow simulation dataset. The same set of test data shows that it is unavoidable to use further filters to eliminate false positives. For this purpose, we use the angle between feature tangent and slope line as a quality measure and, based on this, formalize a previously published filter.

Feature-based flow visualization is naturally dependent on feature extraction. To extract flow features, often higher-order properties of the flow data are used such as the Jacobian or curvature properties, implicitly describing the flow features in terms of their inherent flow characteristics (e.g., collinear flow and vorticity vectors). In this paper we present recent research which leads to the (not really surprising) conclusion that feature extraction algorithms need to be extended to a time-dependent analysis framework (in terms of time derivatives) when dealing with unsteady flow data. Accordingly, we present two extensions of the parallel vectors based vortex extraction criteria to the time-dependent domain and show the improvements of feature-based flow visualization in comparison to the steady versions of this extraction algorithm both in the context of a high-resolution dataset, i.e., a simulation specifically designed to evaluate our new approach, as well as for a real-world dataset from a concrete application.

Vortex breakdown bubbles are a subject which is of interest in many disciplines such as aeronautics, mixing, and combustion. Existing visualization methods are based on stream surfaces, direct volume rendering, tensor field visualization, and vector field topology. This paper presents a topological approach which is more closely oriented at the underlying theory of continuous dynamical systems. Algorithms are described for the detection of vortex rings and vortex breakdown bubbles, and for visualization of their characteristic properties such as the boundary, the chaotic dynamics, and possible islands of stability. Since some of these require very long streamlines, the effect of numerically introduced divergence has to be considered. From an existing subdivision scheme, a novel method for divergence conserving interpolation of cuboid cells is derived, and results are compared with those from standard trilinear interpolation. Also a comparison of results obtained with and without divergence cleaning is given.

This paper presents a method for filtered ridge extraction based on adaptive mesh refinement. It is applicable in situations where the underlying scalar field can be refined during ridge extraction. This requirement is met by the concept of Lagrangian coherent structures which is based on trajectories started at arbitrary sampling grids that are independent of the underlying vector field. The Lagrangian coherent structures are extracted as ridges in finite Lyapunov exponent fields computed from these grids of trajectories. The method is applied to several variants of finite Lyapunov exponents, one of which is newly introduced. High computation time due to the high number of required trajectories is a main drawback when computing Lyapunov exponents of 3-dimensional vector fields. The presented method allows a substantial speed-up by avoiding the seeding of trajectories in regions where no ridges are present or do not satisfy the prescribed filter criteria such as a minimum finite Lyapunov exponent.

Vortices are undesirable in many applications while indispensable in others. It is therefore of common interest to understand their mechanisms of creation. This paper aims at analyzing the transport of vorticity inside incompressible flow. The analysis is based on the vorticity equation and is performed along pathlines which are typically started in upstream direction from vortex regions. Different methods for the quantitative and explorative analysis of vorticity transport are presented and applied to CFD simulations of water turbines. Simulation quality is accounted for by including the errors of meshing and convergence into analysis and visualization. The obtained results are discussed and interpretations with respect to engineering questions are given.

For the rendering of vector and tensor fields, several texture-based volumetric rendering methods were presented in recent years. While they have indisputable merits, the classical vertex-based rendering of integral curves has the advantage of better zooming capabilities as it is not bound to a fixed resolution. It has been shown that lighting can improve spatial perception of lines significantly, especially if lines appear in bundles. Although OpenGL does not directly support lighting of lines, fast rendering of illuminated lines can be achieved by using basic texture mapping. This existing technique is based on a maximum principle which gives a good approximation of specular reflection. Diffuse reflection however is essentially limited to bidirectional lights at infinity. We show how the realism can be further increased by improving diffuse reflection. We present simplified expressions for the Phong/Blinn lighting of infinitesimally thin cylindrical tubes. Based on these, we propose a fast rendering technique with diffuse and specular reflection for orthographic and perspective views and for multiple local and infinite lights. The method requires commonly available programmable vertex and fragment shaders and only two-dimensional lookup textures.

Vorticity is the quantity used to describe the creation, transformation and extinction of vortices. It is present not only in vortices but also in shear flow. Especially in ducted flows, most of the overall vorticity is usually contained in the boundary layer. When a vortex develops from the boundary layer, this can be described by transport of vorticity. For a better understanding of a flow it is therefore of interest to examine vorticity in all of its different roles. The goal of this application study was not primarily the visualization of vortices but of vorticity distribution and its role in vortex phenomena. The underlying industrial case is a design optimization for a Pelton turbine. An important industrial objective is to improve the quality of the water jets driving the runner. Jet quality is affected mostly by vortices originating in the distributor ring. For a better understanding of this interrelation, it is crucial to not only visualize these vortices but also to analyze the mechanisms of their creation. We used various techniques for the visualization of vorticity, including field lines and modified isosurfaces. For field line based visualization, we extended the image-guided streamline placement algorithm of Turk and Banks to data-guided field line placement on three-dimensional unstructured grids.