Abstract Integral lines of the gradient flow are standard features in continuously differentiable scalar fields that enjoy some useful properties: They cover the domain densely, do not split, merge, or intersect, and are therefore invertible. For widely used discretizations of scalar fields, the corresponding polygonal approximations of integral lines do not enjoy these properties anymore. We analyze conditions for integral lines in 2D piecewise linear (PL) scalar fields to be invertible by identifying and classifying critical edges in the underlying triangulation. We show that under mild conditions, every 2D PL scalar field can be transformed into an arbitrarily close PL field with invertible integral lines. We present an algorithm that computes this transformation and apply it to a number of test data sets.

Abstract Flow maps are a fundamental concept for the visualization and analysis of time‐dependent flows. A common approach is to apply sampling and reconstruction – which comes with a number of problems: flow maps are high‐dimensional functions with locally large gradients which makes approximations very challenging. In general, one must apply large sampling resolutions and more advanced methods to achieve sufficiently good results. Existing methods for efficient, high‐quality sampling and approximation restrict to subsets of flow maps, e.g., with fixed start and end times, or at least fixed differences. In this paper, we propose the first method to handle the entire flow map by decreasing the dimensionality of the space‐time‐time domain. Essentially, it is a framework which splits the high‐dimensional flow map into few lower‐dimensional functions. This enables to achieve much larger resolutions than traditional sampling with the same memory consumption. We use our method in established applications and show that it can significantly decrease errors in the reconstruction by using these larger resolutions.

Principal Component Analysis (PCA) is perhaps the most popular linear projection technique for dimensionality reduction. We consider PCA under the assumption that the high-dimensional data points are equipped with Gaussian uncertainty. Several approaches to such uncertainty-aware PCA have been developed recently in the visualization community. Since PCA is a discontinuous map, a small uncertainty in the data points can result in a huge uncertainty in the projected points. We show that the uncertainty of the data points also creates uncertainty in the eigenvectors of the covariance matrix that defines the PCA projection. We present a closed-form expression to quantify eigenvector uncertainty. Based on this, we propose a 3D glyph that supports the decision whether existing solutions for uncertainty-aware PCA are sufficient, or whether a more expensive sampling-based approach is required. We apply our approach to several test data sets.

Aortic dissection is a life-threatening cardiovascular disease characterized by blood entering the media layer of the aortic vessel wall. This creates a second flow channel, known as the false lumen, which weakens the aortic wall and can potentially lead to fatal aortic rupture. Current risk stratification of aortic dissections is primarily based on morphological features of the aorta. However, hemodynamics also play a significant role in disease progression, though their investigation and visualization remain challenging. Common flow visualizations often experience visual clutter, especially when dealing with the intricate morphologies of aortic dissections. In this work, we implement and evaluate different approaches to visualizing the flow in aortic dissections effectively. We employ three techniques, namely streaklines with depth-dependent halos, transparent streaklines, and smoke surfaces. The latter is a technique based on streak surfaces, enhanced with opacity modulations, to produce a smoke-like appearance that improves visual clarity. We adapt the original opacity modulation of smoke surfaces to visualize flow even within the complex geometries of aortic dissections, thereby enhancing visual fidelity. To effectively capture dissection hemodynamics, we developed customized seeding structures that adapt to the shape of the surrounding lumen. Our evaluation, conducted via an online questionnaire, included medical professionals, fluid simulation experts, and visualization specialists. By analyzing results across these groups, we highlight differences in preference and interpretability, offering insight into domain-specific needs. No single visualization technique emerged as the best overall. Smoke surfaces provide the best overall clarity and visual realism. However, participants found streaklines with halos to be the best for quantifying flow, dispite them introducing significant visual clutter. Transparent streaklines serve as a middle ground, offering improved clarity over halos while maintaining some level of detail. Across all participant groups, smoke surfaces were rated as the most visually appealing and lifelike, with medical professionals highlighting their resemblance to contrast-agent injections used in clinical practice.

Abstract Finding projections of multidimensional data domains to the 2D screen space is a well‐known problem. Multidimensional data often comes with the property that the dimensions are measured in different physical units, which renders the ratio between dimensions, i.e., their scale, arbitrary. The result of common projections, like PCA, t‐SNE, or MDS, depends on this ratio, i.e., these projections are variant to scaling. This results in an undesired subjective view of the data, and thus, their projection. Simple solutions like normalization of each dimension are widely used, but do not always give high‐quality results. We propose to visually analyze the space of all scalings and to find optimal scalings w.r.t. the quality of the visualization. For this, we evaluate different quality criteria on scatter plots. Given a quality criterion, our approach finds scalings that yield good visualizations with little to no user input using numerical optimization. Simultaneously, our method results in a scaling invariant projection, proposing an objective view to the projected data. We show for several examples that such an optimal scaling can significantly improve the visualization quality.

Abstract Lagrangian coherent structures (LCS) is an important concept for the visualization of unsteady flows. They describe the boundaries of regions for which material transport stays mostly coherent over time which can help for a better understanding of dynamical systems. One of the most common techniques for their computation is the extraction of ridges from the finite‐time Lyapunov exponent (FTLE) field. FTLE ridges are challenging to extract, both in terms of accuracy and performance, because they expose strong gradients of the underlying field, tend to come close to each other and are dynamic with respect to different time parameters. We present a new method for extracting FTLE ridges for series of integration times which is able to show how coherent regions and their borders evolve over time. Our techniques mainly build on a particle system which is used for sampling the ridges uniformly. This system is highly optimized for the challenges of FTLE ridge extraction. Further, it is able to take advantage of the continuous evolvement of the ridges which makes their sampling for multiple integration times much faster. We test our method on multiple 3D datasets and compare it to the standard Marching Ridges technique. For the extraction examples our method is 13 to over 300 times faster, suggesting a significant advantage.

The numerical extraction of vortex cores from time-dependent fluid flow attracted much attention over the past decades. A commonly agreed upon vortex definition remained elusive since a proper vortex core needs to satisfy two hard constraints: it must be objective and Lagrangian. Recent methods on objectivization met the first but not the second constraint, since there was no formal guarantee that the resulting vortex coreline is indeed a pathline of the fluid flow. In this paper, we propose the first vortex core definition that is both objective and Lagrangian. Our approach restricts observer motions to follow along pathlines, which reduces the degrees of freedoms: we only need to optimize for an observer rotation that makes the observed flow as steady as possible. This optimization succeeds along Lagrangian vortex corelines and will result in a non-zero time-partial everywhere else. By performing this optimization at each point of a spatial grid, we obtain a residual scalar field, which we call vortex deviation error. The local minima on the grid serve as seed points for a gradient descent optimization that delivers sub-voxel accurate corelines. The visualization of both 2D and 3D vortex cores is based on the separation of the movement of the vortex core and the swirling flow behavior around it. While the vortex core is represented by a pathline, the swirling motion around it is visualized by streamlines in the correct frame. We demonstrate the utility of the approach on several 2D and 3D time-dependent vector fields.

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.

Abstract Interpolation between camera positions is a standard problem in computer graphics and can be considered the foundation of camera path planning. As the basis for a new interpolation method, we introduce a new Riemannian metric in camera space, which measures the 3D image flow under a small movement of the camera. Building on this, we define a linear interpolation between two cameras as shortest geodesic in camera space, for which we provide a closed‐form solution after a mild simplification of the metric. Furthermore, we propose a geodesic Catmull‐Rom interpolant for keyframe camera animation. We compare our approach with several standard camera interpolation methods and obtain consistently better camera paths especially for cameras with extremely varying scales.

Abstract The model of a Double Gyre flow by Shadden et al. is a standard benchmark data set for the computation of hyperbolic Lagrangian Coherent Structures (LCS) in flow data. While structurally extremely simple, it generates hyperbolic LCS of arbitrary complexity. Unfortunately, the Double Gyre does not come with a well‐defined ground truth: the location of hyperbolic LCS boundaries can only be approximated by numerical methods that usually involve the gradient of the flow map. We present a new benchmark data set that is a small but carefully designed modification of the Double Gyre, which comes with ground truth closed‐form hyperbolic trajectories. This allows for computing hyperbolic LCS boundaries by a simple particle integration without the consideration of the flow map gradient. We use these hyperbolic LCS as a ground truth solution for testing an existing numerical approach for extracting hyperbolic trajectories. In addition, we are able to construct hyperbolic LCS curves that are significantly longer than in existing numerical methods.

This special section of the IEEE Transactions on Visualization and Computer Graphics (IEEE TVCG) presents the five most highly rated papers from the 2021 IEEE Pacific Visualization Symposium (IEEE PacificVis). This year, IEEE PacificVis was scheduled to be hosted by Tianjin University and held in Tianjin, China, from April 19 to 22, 2021. IEEE PacificVis, sponsored by the IEEE Visualization and Graphics Technical Committee (VGTC), aims to foster greater exchange between visualization researchers and practitioners, especially in the Asia-Pacific region. This forum has grown to be a truly international event, attracting submissions and attendees from many countries in the Asia-Pacific and Europe, America, and beyond. Thus, IEEE PacificVis is serving the additional purposes of sharing the latest advances in visualization with researchers and practitioners in the region and introducing research developments in the region to the broader international visualization research community.

and vorticity change under arbitrary movements of the observer. This is undesirable since measurements of physical properties should ideally not depend on the way the (virtual) measurement device moves. There are some properties that are invariant under certain types of reference frame transformations: Galilean invariance (invariance under equal-speed translation) and objectivity (invariance under any smooth rotation and translation of the reference frame). In this paper, we introduce even harder conditions than objectivity: we demand invariance under any smooth similarity transformation (rotation, translation and uniform scale) as well as invariance under any smooth affine transformation of the reference frame. We show that these new hyper-objective measures allow the extraction of vortices that change their volume or deform. Further, we present a generic approach that transforms almost any vortex measure into a hyper-objective one. We apply our methods to vortex extraction in 2D and 3D vector fields, and analyze the numerical robustness, extraction time and the minimization residuals for the Galilean invariant, objective, and the two new hyper-objective approaches.

Abstract Measured data often incorporates some amount of uncertainty, which is generally modeled as a distribution of possible samples. In this paper, we consider second‐order symmetric tensors with uncertainty. In the 3D case, this means the tensor data consists of 6 coefficients – uncertainty, however, is encoded by 21 coefficients assuming a multivariate Gaussian distribution as model. The high dimension makes the direct visualization of tensor data with uncertainty a difficult problem, which was until now unsolved. The contribution of this paper consists in the design of glyphs for uncertain second‐order symmetric tensors in 2D and 3D. The construction consists of a standard glyph for the mean tensor that is augmented by a scalar field that represents uncertainty. We show that this scalar field and therefore the displayed glyph encode the uncertainty comprehensively, i.e., there exists a bijective map between the glyph and the parameters of the distribution. Our approach can extend several classes of existing glyphs for symmetric tensors to additionally encode uncertainty and therefore provides a possible foundation for further uncertain tensor glyph design. For demonstration, we choose the well‐known superquadric glyphs, and we show that the uncertainty visualization satisfies all their design constraints.

Vortices are one of the most-frequently studied phenomena in fluid flows. The center of the rotating motion is called the vortex coreline and its successful detection strongly depends on the choice of the reference frame. The optimal frame moves with the center of the vortex, which incidentally makes the observed fluid flow steady and thus standard vortex coreline extractors such as Sujudi-Haimes become applicable. Recently, an objective optimization framework was proposed that determines a near-steady reference frame for tracer particles. In this paper, we extend this technique to the detection of vortex corelines of inertial particles. An inertial particle is a finite-sized object that is carried by a fluid flow. In contrast to the usual tracer particles, they do not move tangentially with the flow, since they are subject to gravity and exhibit mass-dependent inertia. Their particle state is determined by their position and own velocity, which makes the search for the optimal frame a high-dimensional problem. We demonstrate in this paper that the objective detection of an inertial vortex coreline can be reduced in 2D to a critical point search in 2D. For 3D flows, however, the vortex coreline criterion remains a parallel vectors condition in 6D. To detect the vortex corelines we propose a recursive subdivision approach that is tailored to the underlying structure of the 6D vectors. The resulting algorithm is objective, and we demonstrate the vortex coreline extraction in a number of 2D and 3D vector fields.

We present a formal approach to the visual analysis of recirculation in flows by introducing recirculation surfaces for 3D unsteady flow fields. Recirculation surfaces are the loci where massless particle integration returns to its starting point after some variable, finite integration. We give a rigorous definition of recirculation surfaces as 2-manifolds embedded in 5D space and study their properties. Based on this we construct an algorithm for their extraction, which searches for intersections of a recirculation surface with lines defined in 3D. This reduces the problem to a repeated search for critical points in 3D vector fields. We provide a uniform sampling of the search space paired with a surface reconstruction and visualize results. This way, we present the first algorithm for a comprehensive feature extraction in the 5D flow map of a 3D flow. The problem of finding isolated closed orbits in steady vector fields occurs as a special case of recirculation surfaces. This includes isolated closed orbits with saddle behavior. We show recirculation surfaces for a number of artificial and real flow data sets.

Abstract The Parallel Vectors (PV) Operator extracts the locations of points where two vector fields are parallel. In general, these features are line structures. The PV operator has been used successfully for a variety of problems, which include finding vortex‐core lines or extremum lines. We present a new generic feature extraction method for multiple 3D vector fields: TheApproximate Parallel Vectors(APV) Operator extracts lines where all fields are approximately parallel. The definition of the APV operator is based on the application of PV for two vector fields that are derived from the given set of fields. The APV operator enables the direct visualization of features of vector field ensembles without processing fields individually and without causing visual clutter. We give a theoretical analysis of the APV operator and demonstrate its utility for a number of ensemble data.

Abstract Vortices are commonly understood as rotating motions in fluid flows. The analysis of vortices plays an important role in numerous scientific applications, such as in engineering, meteorology, oceanology, medicine and many more. The successful analysis consists of three steps: vortex definition, extraction and visualization. All three have a long history, and the early themes and topics from the 1970s survived to this day, namely, the identification of vortex cores, their extent and the choice of suitable reference frames. This paper provides an overview over the advances that have been made in the last 40 years. We provide sufficient background on differential vector field calculus, extraction techniques like critical point search and the parallel vectors operator, and we introduce the notion of reference frame invariance. We explain the most important region‐based and line‐based methods, integration‐based and geometry‐based approaches, recent objective techniques, the selection of reference frames by means of flow decompositions, as well as a recent local optimization‐based technique. We point out relationships between the various approaches, classify the literature and identify open problems and challenges for future work.

Abstract The visual analysis of combustion processes is one of the challenges of modern flow visualization. In turbulent combustion research, the behaviour of the flame surface contains important information about the interactions between turbulence and chemistry. The extraction and tracking of this surface is crucial for understanding combustion processes. This is impossible to realize as a post‐process because of the size of the involved datasets, which are too large to be stored on disk. We present an on‐the‐fly method for tracking the flame surface directly during simulation and computing the local tangential surface deformation for arbitrary time intervals. In a massively parallel simulation, the data are distributed over many processes and only a single time step is in memory at any time. To satisfy the demands on parallelism and accuracy posed by this situation, we track the surface with independent micro‐patches and adapt their distribution as needed to maintain numerical stability. With our method, we enable combustion researchers to observe the detailed movement and deformation of the flame surface over extended periods of time and thus gain novel insights into the mechanisms of turbulence–chemistry interactions. We validate our method on analytic ground truth data and show its applicability on two real‐world simulations.

Abstract Vortices are important features in vector fields that show a swirling behavior around a common core. The concept of a vortex core line describes the center of this swirling behavior. In this work, we examine the extension of this concept to 3D second‐order tensor fields. Here, a behavior similar to vortices in vector fields can be observed for trajectories of the eigenvectors. Vortex core lines in vector fields were defined by Sujudi and Haimes to be the locations where stream lines are parallel to an eigenvector of the Jacobian. We show that a similar criterion applied to the eigenvector trajectories of a tensor field yields structurally stable lines that we calltensor core lines. We provide a formal definition of these structures and examine their mathematical properties. We also present a numerical algorithm for extracting tensor core lines in piecewise linear tensor fields. We find all intersections of tensor core lines with the faces of a dataset using a simple and robust root finding algorithm. Applying this algorithm to tensor fields obtained from structural mechanics simulations shows that it is able to effectively detect and visualize regions of rotational or hyperbolic behavior of eigenvector trajectories.

Star Coordinates are a popular projection technique in order to analyze and to disclose characteristic patterns of multidimensional data. Unfortunately, the shape, appearance, and distribution of such patterns are strongly affected by the given scaling of the data and can mislead the projection-based data analysis. In an extreme case, patterns might be more related to the choice of scaling than to the data themselves. Thus, we present the LloydRelaxer : a tool to minimize scaling-based effects in Star Coordinates. Our algorithm enforces a scaling configuration for which the data explains the observed patterns better than any scaling of them could do. It does so by an iterative minimizing and optimization process based on Voronoi diagrams and on the Lloyd relaxation within the projection space. We evaluate and test our approach by real benchmark multidimensional data of the UCI data repository.

Displaying geometry inflow visualization is often accompanied by occlusion problems, making it difficult to perceive information that is relevant in the respective application. In a recent technique, named opacity optimization, the balance of occlusion avoidance and the selection of meaningful geometry was recognized to be a view-dependent, global optimization problem. The method solves a bounded-variable least-squares problem, which minimizes energy terms for the reduction of occlusion, background clutter, adding smoothness and regularization. The original technique operates on an object-space discretization and was shown for line and surface geometry. Recently, it has been extended to volumes, where it was solved locally per ray by dropping the smoothness energy term and replacing it by pre-filtering the importance measure. In this paper, we pick up the idea of splitting the opacity optimization problem into two smaller problems. The first problem is a minimization with analytic solution, and the second problem is a smoothing of the obtained minimizer in object-space. Thereby, the minimization problem can be solved locally per pixel, making it possible to combine all geometry types (points, lines and surfaces) consistently in a single optimization framework. We call this decoupled opacity optimization and apply it to a number of steady 3D vector fields.

Abstract One main task for domain experts in analysing their nD data is to detect and interpret class/cluster separations and outliers. In fact, an important question is, which features/dimensions separate classes best or allow a cluster‐based data classification. Common approaches rely on projections from nD to 2D, which comes with some challenges, such as: The space of projection contains an infinite number of items. How to find the right one? The projection approaches suffers from distortions and misleading effects. How to rely to the projected class/cluster separation? The projections involve the complete set of dimensions/features. How to identify irrelevant dimensions? Thus, to address these challenges, we introduce a visual analytics concept for the feature selection based on linear discriminative star coordinates (DSC), which generate optimal cluster separating views in a linear sense for both labeled and unlabeled data. This way the user is able to explore how each dimension contributes to clustering. To support to explore relations between clusters and data dimensions, we provide a set of cluster‐aware interactions allowing to smartly iterate through subspaces of both records and features in a guided manner. We demonstrate our features selection approach for optimal cluster/class separation analysis with a couple of experiments on real‐life benchmark high‐dimensional data sets.

In flow visualization, vortex extraction is a long-standing and unsolved problem. For decades, scientists developed numerous definitions that characterize vortex regions and their corelines in different ways, but none emerged as ultimate solution. One reason is that almost all techniques have a fundamental weakness: they are not invariant under changes of the reference frame, i.e., they are not objective. This has two severe implications: First, the result depends on the movement of the observer, and second, they cannot track vortices that are moving on arbitrary paths, which limits their reliability and usefulness in practice. Objective measures are rare, but recently gained more attention in the literature. Instead of only introducing a new objective measure, we show in this paper how all existing measures that are based on velocity and its derivatives can be made objective. We achieve this by observing the vector field in optimal local reference frames, in which the temporal derivative of the flow vanishes, i.e., reference frames in which the flow appears steady. The central contribution of our paper is to show that these optimal local reference frames can be found by a simple and elegant linear optimization. We prove that in the optimal frame, all local vortex extraction methods that are based on velocity and its derivatives become objective. We demonstrate our approach with objective counterparts to λ 2 , vorticity and Sujudi-Haimes.

Glyphs are a powerful tool for visualizing second-order tensors in a variety of scientic data as they allow to encode physical behavior in geometric properties. Most existing techniques focus on symmetric tensors and exclude non-symmetric tensors where the eigenvectors can be non-orthogonal or complex. We present a new construction of 2d and 3d tensor glyphs based on piecewise rational curves and surfaces with the following properties: invariance to (a) isometries and (b) scaling, (c) direct encoding of all real eigenvalues and eigenvectors, (d) one-to-one relation between the tensors and glyphs, (e) glyph continuity under changing the tensor. We apply the glyphs to visualize the Jacobian matrix fields of a number of 2d and 3d vector fields.

Inertial particles are finite-sized objects that are carried by fluid flows and in contrast to massless tracer particles they are subject to inertia effects. In unsteady flows, the dynamics of tracer particles have been extensively studied by the extraction of Lagrangian coherent structures (LCS), such as hyperbolic LCS as ridges of the Finite-Time Lyapunov Exponent (FTLE). The extension of the rich LCS framework to inertial particles is currently a hot topic in the CFD literature and is actively under research. Recently, backward FTLE on tracer particles has been shown to correlate with the preferential particle settling of small inertial particles. For larger particles, inertial trajectories may deviate strongly from (massless) tracer trajectories, and thus for a better agreement, backward FTLE should be computed on inertial trajectories directly. Inertial backward integration, however, has not been possible until the recent introduction of the influence curve concept, which - given an observation and an initial velocity - allows to recover all sources of inertial particles as tangent curves of a derived vector field. In this paper, we show that FTLE on the influence curve vector field is in agreement with preferential particle settling and more importantly it is not only valid for small (near-tracer) particles. We further generalize the influence curve concept to general equations of motion in unsteady spatio-velocity phase spaces, which enables backward integration with more general equations of motion. Applying the influence curve concept to tracer particles in the spatio-velocity domain emits streaklines in massless flows as tangent curves of the influence curve vector field. We demonstrate the correlation between inertial backward FTLE and the preferential particle settling in a number of unsteady vector fields.

Abstract Traditionally, Lagrangian fields such as finite‐time Lyapunov exponents (FTLE) are precomputed on a discrete grid and are ray casted afterwards. This, however, introduces both grid discretization errors and sampling errors during ray marching. In this work, we apply a progressive, view‐dependent Monte Carlo‐based approach for the visualization of such Lagrangian fields in time‐dependent flows. Our approach avoids grid discretization and ray marching errors completely, is consistent, and has a low memory consumption. The system provides noisy previews that converge over time to an accurate high‐quality visualization. Compared to traditional approaches, the proposed system avoids explicitly predefined fieldline seeding structures, and uses a Monte Carlo sampling strategy named Woodcock tracking to distribute samples along the view ray. An acceleration of this sampling strategy requires local upper bounds for the FTLE values, which we progressively acquire during the rendering. Our approach is tailored for high‐quality visualizations of complex FTLE fields and is guaranteed to faithfully represent detailed ridge surface structures as indicators for Lagrangian coherent structures (LCS). We demonstrate the effectiveness of our approach by using a set of analytic test cases and real‐world numerical simulations.

Abstract In recent years, much work was devoted to the design of light editing methods such as relighting and light path editing. So far, little work addressed the target‐based manipulation and animation of caustics, for instance to a differently‐shaped caustic, text or an image. The aim of this work is the animation of caustics by blending towards a given target irradiance distribution. This enables an artist to coherently change appearance and style of caustics, e.g., for marketing applications and visual effects. Generating a smooth animation is nontrivial, as photon density and caustic structure may change significantly. Our method is based on the efficient solution of a discrete assignment problem that incorporates constraints appropriate to make intermediate blends plausibly resemble caustics. The algorithm generates temporally coherent results that are rendered with stochastic progressive photon mapping. We demonstrate our system in a number of scenes and show blends as well as a key frame animation.

Abstract Vector field topology is a powerful and matured tool for the study of the asymptotic behavior of tracer particles in steady flows. Yet, it does not capture the behavior of finite‐sized particles, because they develop inertia and do not move tangential to the flow. In this paper, we use the fact that the trajectories of inertial particles can be described as tangent curves of a higher dimensional vector field. Using this, we conduct a full classification of the first‐order critical points of this higher dimensional flow, and devise a method to their efficient extraction. Further, we interactively visualize the asymptotic behavior of finite‐sized particles by a glyph visualization that encodes the outcome of any initial condition of the governing ODE, i.e., for a varying initial position and/or initial velocity. With this, we present a first approach to extend traditional vector field topology to the inertial case.

Abstract Inertial particles are finite‐sized objects traveling with a certain velocity that differs from the underlying carrying flow, i.e., they are mass‐dependent and subject to inertia. Their backward integration is in practice infeasible, since a slight change in the initial velocity causes extreme changes in the recovered position. Thus, if an inertial particle is observed, it is difficult to recover where it came from. This is known as the source inversion problem, which has many practical applications in recovering the source of airborne or waterborne pollutions. Inertial trajectories live in a higher dimensional spatio‐velocity space. In this paper, we show that this space is only sparsely populated. Assuming that inertial particles are released with a given initial velocity (e.g., from rest), particles may reach a certain location only with a limited set of possible velocities. In fact, with increasing integration duration and dependent on the particle response time, inertial particles converge to a terminal velocity. We show that the set of initial positions that lead to the same location form a curve. We extract these curves by devising a derived vector field in which they appear as tangent curves. Most importantly, the derived vector field only involves forward integrated flow map gradients, which are much more stable to compute than backward trajectories. After extraction, we interactively visualize the curves in the domain and display the reached velocities using glyphs. In addition, we encode the rate of change of the terminal velocity along the curves, which gives a notion for the convergence to the terminal velocity. With this, we present the first solution to the source inversion problem that considers actual inertial trajectories. We apply the method to steady and unsteady flows in both 2D and 3D domains.

Abstract To project high‐dimensional data to a 2D domain, there are two well‐established classes of approaches: RadViz and Star Coordinates. Both are well‐explored in terms of accuracy, completeness, distortions, and interaction issues. We present a generalization of both RadViz and Star Coordinates such that it unifies both approaches. We do so by considering the space of all projective projections. This gives additional degrees of freedom, which we use for three things: Firstly, we define a smooth transition between RadViz and Star Coordinates allowing the user to exploit the advantages of both approaches. Secondly, we define a data‐dependent magic lens to explore the data. Thirdly, we optimize the new degrees of freedom to minimize distortion. We apply our approach to a number of high‐dimensional benchmark datasets.

We propose a new class of vortex definitions for flows that are induced by rotating mechanical parts, such as stirring devices, helicopters, hydrocyclones, centrifugal pumps, or ventilators. Instead of a Galilean invariance, we enforce a rotation invariance, i.e., the invariance of a vortex under a uniform-speed rotation of the underlying coordinate system around a fixed axis. We provide a general approach to transform a Galilean invariant vortex concept to a rotation invariant one by simply adding a closed form matrix to the Jacobian. In particular, we present rotation invariant versions of the well-known Sujudi-Haimes, Lambda-2, and Q vortex criteria. We apply them to a number of artificial and real rotating flows, showing that for these cases rotation invariant vortices give better results than their Galilean invariant counterparts.

Finding good projections of n-dimensional datasets into a 2D visualization domain is one of the most important problems in Information Visualization. Users are interested in getting maximal insight into the data by exploring a minimal number of projections. However, if the number is too small or improper projections are used, then important data patterns might be overlooked. We propose a data-driven approach to find minimal sets of projections that uniquely show certain data patterns. For this we introduce a dissimilarity measure of data projections that discards affine transformations of projections and prevents repetitions of the same data patterns. Based on this, we provide complete data tours of at most n/2 projections. Furthermore, we propose optimal paths of projection matrices for an interactive data exploration. We illustrate our technique with a set of state-of-the-art real high-dimensional benchmark datasets.

Abstract The visual analysis of flows with inertial particle trajectories is a challenging problem because time‐dependent particle trajectories additionally depend on mass, which gives rise to an infinite number of possible trajectories passing through every point in space‐time. This paper presents an approach to a comparative visualization of the inertial particles’ separation behavior. For this, we define the Finite‐Time Mass Separation (FTMS), a scalar field that measures at each point in the domain how quickly inertial particles separate that were released from the same location but with slightly different mass. Extracting and visualizing the mass that induces the largest separation provides a simplified view on the critical masses. By using complementary coordinated views, we additionally visualize corresponding inertial particle trajectories in space‐time by integral curves and surfaces. For a quantitative analysis, we plot Euclidean and arc length‐based distances to a reference particle over time, which allows to observe the temporal evolution of separation events. We demonstrate our approach on a number of analytic and one real‐world unsteady 2D field.

Abstract The visual analysis of multivariate projections is a challenging task, because complex visual structures occur. This causes fatigue or misinterpretations, which distorts the analysis. In fact, the same projection can lead to different analysis results. We provide visual guidance pictograms to improve objectivity of the visual search. A visual guidance pictogram is an iconic visual density map encoding the visual structure of certain data properties. By using them to guide the analysis, structures in the projection can be better understood and mentally linked to properties in the data. We introduce a systematic scheme for designing such pictograms and provide a set of pictograms for standard visual tasks, such as correlation and distribution analysis, for standard projections like scatterplots, RadVis, and Star Coordinates. We conduct a study that compares the visual analysis of real data with and without the support of guidance pictograms. Our tests show that the training effort for a visual search can be decreased and the analysis bias can be reduced by supporting the user's visual search with guidance pictograms.

Abstract The selection of meaningful lines for 3D line data visualization has been intensively researched in recent years. Most approaches focus on single line fields where one line passes through each domain point. This paper presents a selection approach for sets of line fields which is based on a global optimization of the opacity of candidate lines. For this, existing approaches for single line fields are modified such that significantly larger amounts of line representatives are handled. Furthermore, time coherence is addressed for animations, making this the first approach that solves the line selection problem for 3D time‐dependent flow. We apply our technique to visualize dense sets of pathlines, sets of magnetic field lines, and animated sets of pathlines, streaklines and masslines.

Abstract In flow visualization, integral surfaces rapidly tend to expand, fold and produce vast amounts of occlusion. While silhouette enhancements and local transparency mappings proved useful for semi‐transparent depictions, they still introduce visual clutter when surfaces grow more complex. An effective visualization of the flow requires a balance between the presentation of interesting surface parts and the avoidance of occlusions that hinder the view. In this paper, we extend the concept of opacity optimization to surfaces to obtain a global approach to the occlusion problem. Starting with a partition of the surfaces into patches, we compute per‐patch opacity as minimizer of a bounded‐variable least‐squares problem. For the final rendering, opacity is interpolated on the surfaces. The resulting visualization technique is interactive, frame‐coherent, view‐dependent and driven by domain knowledge.

Abstract Lagrangian coherent structures (LCSs) have become a widespread and powerful method to describe dynamic motion patterns in time‐dependent flow fields. The standard way to extract LCS is to compute height ridges in the finite‐time Lyapunov exponent field. In this work, we present an alternative method to approximate Lagrangian features for 2D unsteady flow fields that achieve subgrid accuracy without additional particle sampling. We obtain this by a geometric reconstruction of the flow map using additional material constraints for the available samples. In comparison to the standard method, this allows for a more accurate global approximation of LCS on sparse grids and for long integration intervals. The proposed algorithm works directly on a set of given particle trajectories and without additional flow map derivatives. We demonstrate its application for a set of computational fluid dynamic examples, as well as trajectories acquired by Lagrangian methods, and discuss its benefits and limitations.

Abstract Direct Numerical Simulations of premixed combustion produce terabytes of raw data, which are prohibitively large to be stored, and have to be analyzed and visualized. A simultaneous and integrated treatment of data storage, data analysis and data visualization is required. For this, we introduce a sparse representation tailored to DNS data which can directly be used for both analysis and visualization. The method is based on the observation that most information is located in narrow‐band regions where the chemical reactions take place, but these regions are not well defined. An approach for the visual investigation of feature surfaces of the scalar fields involved in the simulation is shown as a possible application. We demonstrate our approach on multiple real datasets.

Abstract Stream surfaces are a well‐studied and widely used tool for the visualization of 3D flow fields. Usually, stream surface seeding is carried out manually in time‐consuming trial and error procedures. Only recently automatic selection methods were proposed. Local methods support the selection of a set of stream surfaces, but, contrary to global selection methods, they evaluate only thequalityof the seeding lines but not the quality of the whole stream surfaces. Global methods, on the other hand, only support the selection of asingleoptimal stream surface until now. However, for certain flow fields a single stream surface is not sufficient to represent all flow features. In our work, we overcome this limitation by introducing a global selection technique for asetof stream surfaces. All selected surfaces optimize global stream surface quality measures and are guaranteed to be mutually distant, such that they can convey different flow features. Our approach is an efficient extension of the most recent global selection method for single stream surfaces. We illustrate its effectiveness on a number of analytical and simulated flow fields and analyze the quality of the results in a user study.

In this article, we study the regularization of quadratic energies that are integrated over discrete domains. This is a fairly general setting, often found in, but not limited to, geometry processing. The standard Tikhonov regularization is widely used such that, for instance, a low-pass filter enforces smoothness of the solution. This approach, however, is independent of the energy and the concrete problem, which leads to artifacts in various applications. Instead, we propose a regularization that enforces a low variation of the energy and is problem specific by construction. Essentially, this approach corresponds to minimization with respect to a different norm. Our construction is generic and can be plugged into any quadratic energy minimization, is simple to implement, and has no significant runtime overhead. We demonstrate this for a number of typical problems and discuss the expected benefits.

The cores of massless, swirling particle motion are an indicator for vortex-like behavior in vector fields and to this end, a number of coreline extractors have been proposed in the literature. Though, many practical applications go beyond the study of the vector field. Instead, engineers seek to understand the behavior of inertial particles moving therein, for instance in sediment transport, helicopter brownout and pulverized coal combustion. In this paper, we present two strategies for the extraction of the corelines that inertial particles swirl around, which depend on particle density, particle diameter, fluid viscosity and gravity. The first is to deduce the local swirling behavior from the autonomous inertial motion ODE, which eventually reduces to a parallel vectors operation. For the second strategy, we use a particle density estimation to locate inertial attractors. With this, we are able to extract the cores of swirling inertial particle motion for both steady and unsteady 3D vector fields. We demonstrate our techniques in a number of benchmark data sets, and elaborate on the relation to traditional massless corelines.

Understanding the hemodynamics of blood flow in vascular pathologies such as intracranial aneurysms is essential for both their diagnosis and treatment. Computational fluid dynamics (CFD) simulations of blood flow based on patient-individual data are performed to better understand aneurysm initiation and progression and more recently, for predicting treatment success. In virtual stenting, a flow-diverting mesh tube (stent) is modeled inside the reconstructed vasculature and integrated in the simulation. We focus on steady-state simulation and the resulting complex multiparameter data. The blood flow pattern captured therein is assumed to be related to the success of stenting. It is often visualized by a dense and cluttered set of streamlines.We present a fully automatic approach for reducing visual clutter and exposing characteristic flow structures by clustering streamlines and computing cluster representatives. While individual clustering techniques have been applied before to streamlines in 3D flow fields, we contribute a general quantitative and a domain-specific qualitative evaluation of three state-of-the-art techniques. We show that clustering based on streamline geometry as well as on domain-specific streamline attributes contributes to comparing and evaluating different virtual stenting strategies. With our work, we aim at supporting CFD engineers and interventional neuroradiologists.

This paper applies Poisson-based methods to assist in interactive exploration of steady flow fields. Using data-driven deformations we obtain flow-orthogonal and flow-tangential surfaces by a flux-based optimization. Surfaces are positioned interactively and deformed in real-time according to local flow. The deformed surfaces are particularly useful for defining seed structures. We show how the same gradient-based computational framework can be applied to obtain parametrizations of flow-aligned surfaces. This way it is easy to define nontrivial seed structures for integration-based flow visualization methods. Additionally, the flow-aligned parametrizations are employed for view-independent surface-based LIC visualizations. We apply our method to a number of data sets to show the effectiveness of our deformations and parametrization-based seed extraction methods for interactive flow exploration.

Abstract Stream surfaces are well‐known and widely‐used structures for 3D flow visualization. A single surface can be sufficient to represent important global flow characteristics. Unfortunately, due to the huge space of possible stream surfaces, finding the globally most representative stream surface turns out to be a hard task that is usually performed by time‐consuming manual trial and error exploration using slight modifications of seed geometries. To assist users we propose a new stream surface selection method that acts as an automatic preprocessing step before data analysis. We measure stream surface relevance by a novel surface‐based quality measure that prefers surfaces where the flow is aligned with principal curvature directions. The problem of seed structure selection can then be reduced to the computation of simple minimal paths in a weighted graph spanning the domain. We apply a simulated annealing‐based optimization method to find smooth seed curves of globally near‐optimal stream surfaces. We illustrate the effectiveness of our method on a series of synthetic and real‐world data sets.

Abstract Recent research in flow visualization is focusing on the analysis of time‐dependent, but mass‐less particles. However, in many application scenarios, the mass of particles – and their resulting inertia – is essential in understanding fluid mechanics. This includes critical processes, such as dust particles interacting with aircraft (e.g., brown‐or white‐out effects) and particle separation based on density variation. In this paper, we contribute a generalized description of mass‐dependent particle trajectories and apply existing unsteady flow visualization methods to the mass‐dependent case. This comprises the extension of common concepts, i.e., path lines, streak lines, and time lines. Furthermore, we introduce a new class of integral curves, called mass lines that effectively visualizes mass separation and captures mass‐related features in unsteady flow fields that are inaccessible using traditional methods. We demonstrate the applicability of our method, using a number of real‐world and artificial data sets, in which mass is a crucial parameter. In particular, we focus on the analysis of brown‐out conditions, introduced by a helicopter in forward flight close to the ground.

For the visualization of dense line fields, the careful selection of lines to be rendered is a vital aspect. In this paper, we present a global line selection approach that is based on an optimization process. Starting with an initial set of lines that covers the domain, all lines are rendered with a varying opacity, which is subject to the minimization of a bounded-variable least-squares problem. The optimization strives to keep a balance between information presentation and occlusion avoidance. This way, we obtain view-dependent opacities of the line segments, allowing a real-time free navigation while minimizing the danger of missing important structures in the visualization. We compare our technique with existing local and greedy approaches and apply it to data sets in flow visualization, medical imaging, physics, and computer graphics.

Cardiovascular diseases (CVD) are the leading cause of death worldwide. Their initiation and evolution depends strongly on the blood flow characteristics. In recent years, advances in 4D PC-MRI acquisition enable reliable and time-resolved 3D flow measuring, which allows a qualitative and quantitative analysis of the patient-specific hemodynamics. Currently, medical researchers investigate the relation between characteristic flow patterns like vortices and different pathologies. The manual extraction and evaluation is tedious and requires expert knowledge. Standardized, (semi-)automatic and reliable techniques are necessary to make the analysis of 4D PC-MRI applicable for the clinical routine. In this work, we present an approach for the extraction of vortex flow in the aorta and pulmonary artery incorporating line predicates. We provide an extensive comparison of existent vortex extraction methods to determine the most suitable vortex criterion for cardiac blood flow and apply our approach to ten datasets with different pathologies like coarctations, Tetralogy of Fallot and aneurysms. For two cases we provide a detailed discussion how our results are capable to complement existent diagnosis information. To ensure real-time feedback for the domain experts we implement our method completely on the GPU.

Star coordinates is a popular projection technique from an nD data space to a 2D/3D visualization domain. It is defined by setting n coordinate axes in the visualization domain. Since it generally defines an affine projection, strong distortions can occur: an nD sphere can be mapped to an ellipse of arbitrary size and aspect ratio. We propose to restrict star coordinates to orthographic projections which map an nD sphere of radius r to a 2D circle of radius r. We achieve this by formulating conditions for the coordinate axes to define orthographic projections, and by running a repeated non-linear optimization in the background of every modification of the coordinate axes. This way, we define a number of orthographic interaction concepts as well as orthographic data tour sequences: a scatterplot tour, a principle component tour, and a grand tour. All concepts are illustrated and evaluated with synthetic and real data.

The Finite Time Lyapunov Exponent (FTLE) has become a widespread tool for analyzing unsteady flow behavior. For its computation, several numerical methods have been introduced, which provide trade-offs between performance and accuracy. In order to decide which methods and parameter settings are suitable for a particular application, an evaluation of the different FTLE methods is necessary. We propose a general benchmark for FTLE computation, which consists of a number of 2D time-dependent flow fields and error measures. Evaluating the accuracy of a numerically computed FTLE field requires a ground truth, which is not available for realistic flow data sets, since such fields can generally not be described in a closed form. To overcome this, we introduce approaches to create non-trivial vector fields with a closed-form formulation of the FTLE field. Using this, we introduce a set of benchmark flow data sets that resemble relevant geometric aspects of Lagrangian structures, but have an analytic solution for FTLE. Based on this ground truth, we perform a comparative evaluation of three standard FTLE concepts. We suggest error measures based on the variance of both, the fields and the extracted ridge structures.

We define APAP surfaces, surfaces that are as perpendicular as possible to steady 3D vector fields, and present a method to construct discrete representations of them. Since, in general, a perfectly perpendicular surface to a vector field does not exist, we propose and minimize an error metric to enforce perpendicularity as much as possible. Our algorithm constructs an APAP surface by deforming a seed surface anchored in a domain point. In the discrete setting this minimization results in iteratively solving linear least-squares problems and integrating a locally scaled version of the vector field. The definition of the error metric and its numerical minimization guarantee that the minimum zero is attained for the perfectly perpendicular surface if it exists. Otherwise, the minimization converges to the same local minimum independent of the seed configuration, and the resulting surface is - in a least-squares sense - as perpendicular as possible to the flow. We apply these APAP surfaces as an interactive flow visualization tool which we demonstrate on a number of synthetic and real flow data sets.

The availability of dense motion information in computer vision domain allows for the effective application of Lagrangian techniques that have their origin in fluid flow analysis and dynamical systems theory. A well established technique that has been proven to be useful in image-based crowd analysis are Finite Time Lyapunov Exponents (FTLE). Based on this, we present a method to detect people carrying object and describe a methodology how to apply established flow field methods onto the problem of describing individuals. Further, we reinterpret Lagrangian features in relation to the underlying motion process and show their applicability towards the appearance modeling of pedestrians. This definition allows to increase performance of state-of-the-art methods and is shown to be robust under varying parameter settings and different optical flow extraction approaches.

Human action recognition requires the description of complex motion patterns in image sequences. In general, these patterns span varying temporal scales. In this context, Lagrangian methods have proven to be valuable for crowd analysis tasks such as crowd segmentation. In this paper, we show that, besides their potential in describing large scale motion patterns, Lagrangian methods are also well suited to model complex individual human activities over variable time intervals. We use Finite Time Lyapunov Exponents and time-normalized arc length measures in a linear SVM classification scheme. We evaluated our method on the Weizmann and KTH datasets. The results demonstrate that our approach is promising and that human action recognition performance is improved by fusing Lagrangian measures.

The extraction of motion patterns from image sequences based on the optical flow methodology is an important and timely topic among visual multi media applications. In this work we will present a novel framework that combines the optical flow methodology from image processing with methods developed for the Lagrangian analysis of time-dependent vector fields. The Lagrangian approach has been proven to be a valuable and powerful tool to capture the complex dynamic motion behavior within unsteady vector fields. To come up with a compact and applicable framework, this paper will provide concepts on how to compute trajectory-based Lagrangian measures in series of optical flow fields, a set of basic measures to capture the essence of the motion behavior within the image, and a compact hierarchical, feature-based description of the resulting motion features. The resulting framework will bee shown to be suitable for an automated image analysis as well as compact visual analysis of image sequences in its spatio-temporal context. We show its applicability for the task of motion feature description and extraction on different temporal scales, crowd motion analysis, and automated detection of abnormal events within video sequences.

Abstract The scatterplot matrix (SPLOM) is a well‐established technique to visually explore high‐dimensional data sets. It is characterized by the number of scatterplots (plots) of which it consists of. Unfortunately, this number quadratically grows with the number of the data set’s dimensions. Thus, an SPLOM scales very poorly. Consequently, the usefulness of SPLOMs is restricted to a small number of dimensions. For this, several approaches already exist to explore such ‘small’ SPLOMs. Those approaches address the scalability problem just indirectly and without solving it. Therefore, we introduce a new greedy approach to manage ‘large’ SPLOMs with more than 100 dimensions. We establish a combined visualization and interaction scheme that produces intuitively interpretable SPLOMs by combining known quality measures, a pre‐process reordering and a perception‐based abstraction. With this scheme, the user can interactively find large amounts of relevant plots in large SPLOMs.

Abstract We present an approach to extract and visualize vortex structures in uncertain vector fields. For this, we generalize the concepts of the most common vortex detectors to uncertain vector fields, namely the λ2‐criterion, Q‐criterion, and the concept of parallel vectors at the example of the method by Sujudi and Haimes. All these methods base on the computation of derivatives of the uncertain vector field which are uncertain fields as well. Since they generally cannot be computed in a closed form, we provide a Monte Carlo algorithm to compute the respective probability distributions. Based on this, uncertain versions of vortex regions and core structures are introduced. We present results of our approach on three real world data sets in order to give a proof of concept.

Abstract The generation of discrete stream surfaces is an important and challenging task in scientific visualization, which can be considered a particular instance of geometric modeling. The quality of numerically integrated stream surfaces depends on a number of parameters that can be controlledlocally, such as time step or distance of adjacent vertices on the front line. In addition there is a parameter that cannot be controlled locally: stream surface meshes tend to show high quality, well‐shaped elements only if the current front line is “globally” approximately perpendicular to the flow direction. We analyze the impact of this geometric property and present a novel solution – a stream surface integrator that forces the front line to be perpendicular to the flow and that generates quad‐dominant meshes with well‐shaped and well‐aligned elements. It is based on the integration of a scaled version of the flow field, and requires repeated minimization of an error functional along the current front line. We show that this leads to computing the 1‐dimensional kernel of a bidiagonal matrix: a linear problem that can be solved efficiently. We compare our method with existing stream surface integrators and apply it to a number of synthetic and real world data sets.

Abstract Two‐dimensional transfer functions are an effective and well‐accepted tool in volume classification. The design of them mostly depends on the user's experience and thus remains a challenge. Therefore, we present an approach in this paper to automate the transfer function design based on 2D density plots. By exploiting their smoothness, we adopted the Morse theory to automatically decompose the feature space into a set of valley cells. We design a simplification process based on cell separability to eliminate cells which are mainly caused by noise in the original volume data. Boundary persistence is first introduced to measure the separability between adjacent cells and to suitably merge them. Afterward, a reasonable classification result is achieved where each cell represents a potential feature in the volume data. This classification procedure is automatic and facilitates an arbitrary number and shape of features in the feature space. The opacity of each feature is determined by its persistence and size. To further incorporate the user's prior knowledge, a hierarchical feature representation is created by successive merging of the cells. With this representation, the user is allowed to merge or split features of interest and set opacity and color freely. Experiments on various volumetric data sets demonstrate the effectiveness and usefulness of our approach in transfer function generation.

Abstract We present the first general scheme to describe all four types of characteristic curves of flow fields – stream, path, streak, and time lines – as tangent curves of a derived vector field. Thus, all these lines can be obtained by a simple integration of an autonomous ODE system. Our approach draws on the principal ideas of the recently introduced tangent curve description of streak lines. We provide the first description of time lines as tangent curves of a derived vector field, which could previously only be constructed in a geometric manner. Furthermore, our scheme gives rise to new types of curves. In particular, we introduce advected stream lines as a parameter‐free variant of the time line metaphor. With our novel mathematical description of characteristic curves, a large number of feature extraction and analysis tools becomes available for all types of characteristic curves, which were previously only available for stream and path lines. We will highlight some of these possible applications including the computation of time line curvature fields and the extraction of cores of swirling advected stream lines.

Cerebral aneurysms are a pathological vessel dilatation that bear a high risk of rupture. For the understanding and evaluation of the risk of rupture, the analysis of hemodynamic information plays an important role. Besides quantitative hemodynamic information, also qualitative flow characteristics, e.g., the inflow jet and impingement zone are correlated with the risk of rupture. However, the assessment of these two characteristics is currently based on an interactive visual investigation of the flow field, obtained by computational fluid dynamics (CFD) or blood flow measurements. We present an automatic and robust detection as well as an expressive visualization of these characteristics. The detection can be used to support a comparison, e.g., of simulation results reflecting different treatment options. Our approach utilizes local streamline properties to formalize the inflow jet and impingement zone. We extract a characteristic seeding curve on the ostium, on which an inflow jet boundary contour is constructed. Based on this boundary contour we identify the impingement zone. Furthermore, we present several visualization techniques to depict both characteristics expressively. Thereby, we consider accuracy and robustness of the extracted characteristics, minimal visual clutter and occlusions. An evaluation with six domain experts confirms that our approach detects both hemodynamic characteristics reasonably.

We propose a new technique for visual exploration of streamlines in 3D vector fields. We construct a map from the space of all streamlines to points in IR(n) based on the preservation of the Hausdorff metric in streamline space. The image of a vector field under this map is a set of 2-manifolds in IR(n) with characteristic geometry and topology. Then standard clustering methods applied to the point sets in IR(n) yield a segmentation of the original vector field. Our approach provides a global analysis of 3D vector fields which incorporates the topological segmentation but yields additional information. In addition to a pure segmentation, the established map provides a natural "parametrization” visualized by the manifolds. We test our approach on a number of synthetic and real-world data sets.

We present a technique to visualize global uncertainty in stationary 3D vector fields by a topological approach. We start from an existing approach for 2D uncertain vector field topology and extend this into 3D space. For this a number of conceptional and technical challenges in performance and visual representation arise. In order to solve them, we develop an acceleration for finding sink and source distributions. Having these distributions we use overlaps of their corresponding volumes to find separating structures and saddles. As part of the approach, we introduce uncertain saddle and boundary switch connectors and provide algorithms to extract them. For the visual representation, we use multiple direct volume renderings. We test our method on a number of synthetic and real data sets.

Abstract Given an unsteady flow field, one common way to compute Lagrangian Coherent Structures (LCS) is to extract extremal structures of the Finite Time Lyapunov Exponent (FTLE). Experience has shown that the resulting structures are often close to material structures (i.e., material lines or material surfaces). Moreover, it has been proven that for an integration time converging to infinity, they converge to exact material structures. However, due to the finite integration time in FTLE, they are generally not exact material structures. In this paper we introduce a modification of the FTLE method which is guaranteed to produce separating material structures as features of a scalar field. We achieve this by incorporating the complete available integration time both in forward and backward direction, and by choosing an appropriate definition for separating structures. We apply our method to two test data sets and show the differences to classical FTLE.

Abstract Vector fields are a common concept for the representation of many different kinds of flow phenomena in science and engineering. Methods based on vector field topology are known for their convenience for visualizing and analysing steady flows, but a counterpart for unsteady flows is still missing. However, a lot of good and relevant work aiming at such a solution is available. We give an overview of previous research leading towards topology‐based and topology‐inspired visualization of unsteady flow, pointing out the different approaches and methodologies involved as well as their relation to each other, taking classical (i.e. steady) vector field topology as our starting point. Particularly, we focus on Lagrangian methods, space–time domain approaches, local methods and stochastic and multifield approaches. Furthermore, we illustrate our review with practical examples for the different approaches.

Continuous Parallel Coordinates (CPC) are a contemporary visualization technique in order to combine several scalar fields, given over a common domain. They facilitate a continuous view for parallel coordinates by considering a smooth scalar field instead of a finite number of straight lines. We show that there are feature curves in CPC which appear to be the dominant structures of a CPC. We present methods to extract and classify them and demonstrate their usefulness to enhance the visualization of CPCs. In particular, we show that these feature curves are related to discontinuities in Continuous Scatterplots (CSP). We show this by exploiting a curve-curve duality between parallel and Cartesian coordinates, which is a generalization of the well-known point-line duality. Furthermore, we illustrate the theoretical considerations. Concluding, we discuss relations and aspects of the CPC's/CSP's features concerning the data analysis.

Visual exploration of multivariate data typically requires projection onto lower dimensional representations. The number of possible representations grows rapidly with the number of dimensions, and manual exploration quickly becomes ineffective or even unfeasible. This paper proposes automatic analysis methods to extract potentially relevant visual structures from a set of candidate visualizations. Based on features, the visualizations are ranked in accordance with a specified user task. The user is provided with a manageable number of potentially useful candidate visualizations, which can be used as a starting point for interactive data analysis. This can effectively ease the task of finding truly useful visualizations and potentially speed up the data exploration task. In this paper, we present ranking measures for class-based as well as non-class-based scatterplots and parallel coordinates visualizations. The proposed analysis methods are evaluated on different data sets.

Feature Flow Fields are a well-accepted approach for extracting and tracking features. In particular, they are often used to track critical points in time-dependent vector fields and to extract and track vortex core lines. The general idea is to extract the feature or its temporal evolution using a stream line integration in a derived vector field-the so-called Feature Flow Field (FFF). Hence, the desired feature line is a stream line of the FFF. As we will carefully analyze in this paper, the stream lines around this feature line may diverge from it. This creates an unstable situation: if the integration moves slightly off the feature line due to numerical errors, then it will be captured by the diverging neighborhood and carried away from the real feature line. The goal of this paper is to define a new FFF with the guarantee that the neighborhood of a feature line has always converging behavior. This way, we have an automatic correction of numerical errors: if the integration moves slightly off the feature line, it automatically moves back to it during the ongoing integration. This yields results which are an order of magnitude more accurate than the results from previous schemes. We present new stable FFF formulations for the main applications of tracking critical points and solving the Parallel Vectors operator. We apply our method to a number of data sets.

Abstract We introduce an approach to visualize stationary 2D vector fields with global uncertainty obtained by considering the transport of local uncertainty in the flow. For this, we extend the concept of vector field topology to uncertain vector fields by considering the vector field as a density distribution function. By generalizing the concepts of stream lines and critical points we obtain a number of density fields representing an uncertain topological segmentation. Their visualization as height surfaces gives insight into both the flow behavior and its uncertainty. We present a Monte Carlo approach where we integrate probabilistic particle paths, which lead to the segmentation of topological features. Moreover, we extend our algorithms to detect saddle points and present efficient implementations. Finally, we apply our technique to a number of real and synthetic test data sets.

Streak surfaces are among the most important features to support 3D unsteady flow exploration, but they are also among the computationally most demanding. Furthermore, to enable a feature driven analysis of the flow, one is mainly interested in streak surfaces that show separation profiles and thus detect unstable manifolds in the flow. The computation of such separation surfaces requires to place seeding structures at the separation locations and to let the structures move correspondingly to these locations in the unsteady flow. Since only little knowledge exists about the time evolution of separating streak surfaces, at this time, an automated exploration of 3D unsteady flows using such surfaces is not feasible. Therefore, in this paper we present an interactive approach for the visual analysis of separating streak surfaces. Our method draws upon recent work on the extraction of Lagrangian coherent structures (LCS) and the real-time visualization of streak surfaces on the GPU. We propose an interactive technique for computing ridges in the finite time Lyapunov exponent (FTLE) field at each time step, and we use these ridges as seeding structures to track streak surfaces in the time-varying flow. By showing separation surfaces in combination with particle trajectories, and by letting the user interactively change seeding parameters such as particle density and position, visually guided exploration of separation profiles in 3D is provided. To the best of our knowledge, this is the first time that the reconstruction and display of semantic separable surfaces in 3D unsteady flows can be performed interactively, giving rise to new possibilities for gaining insight into complex flow phenomena.

The concept of continuous scatterplot (CSP) is a modern visualization technique. The idea is to define a scalar density value based on the map between an n-dimensional spatial domain and an m-dimensional data domain, which describe the CSP space. Usually the data domain is two-dimensional to visually convey the underlying, density coded, data. In this paper we investigate kinds of map-based discontinuities, especially for the practical cases n = m = 2 and n = 3 | m = 2, and we depict relations between them and attributes of the resulting CSP itself. Additionally, we show that discontinuities build critical line structures, and we introduce algorithms to detect them. Further, we introduce a discontinuity-based visualization approach—called contribution map (CM)—which establishes a relationship between the CSP's data domain and the number of connected components in the spatial domain. We show that CMs enhance the CSP-based linking & brushing interaction. Finally, we apply our approaches to a number of synthetic as well as real data sets.

Crease surfaces are two-dimensional manifolds along which a scalar field assumes a local maximum (ridge) or a local minimum (valley) in a constrained space. Unlike isosurfaces, they are able to capture extremal structures in the data. Creases have a long tradition in image processing and computer vision, and have recently become a popular tool for visualization. When extracting crease surfaces, degeneracies of the Hessian (i.e., lines along which two eigenvalues are equal) have so far been ignored. We show that these loci, however, have two important consequences for the topology of crease surfaces: First, creases are bounded not only by a side constraint on eigenvalue sign, but also by Hessian degeneracies. Second, crease surfaces are not, in general, orientable. We describe an efficient algorithm for the extraction of crease surfaces which takes these insights into account and demonstrate that it produces more accurate results than previous approaches. Finally, we show that diffusion tensor magnetic resonance imaging (DT-MRI) stream surfaces, which were previously used for the analysis of planar regions in diffusion tensor MRI data, are mathematically ill-defined. As an example application of our method, creases in a measure of planarity are presented as a viable substitute.

Characteristic curves of vector fields include stream, path, and streak lines. Stream and path lines can be obtained by a simple vector field integration of an autonomous ODE system, i.e., they can be described as tangent curves of a vector field. This facilitates their mathematical analysis including the extraction of core lines around which stream or path lines exhibit swirling motion, or the computation of their curvature for every point in the domain without actually integrating them. Such a description of streak lines is not yet available, which excludes them from most of the feature extraction and analysis tools that have been developed in our community. In this paper, we develop the first description of streak lines as tangent curves of a derived vector field - the streak line vector field - and show how it can be computed from the spatial and temporal gradients of the flow map, i.e., a dense path line integration is required. We demonstrate the high accuracy of our approach by comparing it to solutions where the ground truth is analytically known and to solutions where the ground truth has been obtained using the classic streak line computation. Furthermore, we apply a number of feature extraction and analysis tools to the new streak line vector field including the extraction of cores of swirling streak lines and the computation of streak line curvature fields. These first applications foreshadow the large variety of possible future research directions based on our new mathematical description of streak lines.

In this paper we present techniques for the visualization of unsteady flows using streak surfaces, which allow for the first time an adaptive integration and rendering of such surfaces in real-time. The techniques consist of two main components, which are both realized on the GPU to exploit computational and bandwidth capacities for numerical particle integration and to minimize bandwidth requirements in the rendering of the surface. In the construction stage, an adaptive surface representation is generated. Surface refinement and coarsening strategies are based on local surface properties like distortion and curvature. We compare two different methods to generate a streak surface: a) by computing a patch-based surface representation that avoids any interdependence between patches, and b) by computing a particle-based surface representation including particle connectivity, and by updating this connectivity during particle refinement and coarsening. In the rendering stage, the surface is either rendered as a set of quadrilateral surface patches using high-quality point-based approaches, or a surface triangulation is built in turn from the given particle connectivity and the resulting triangle mesh is rendered. We perform a comparative study of the proposed techniques with respect to surface quality, visual quality and performance by visualizing streak surfaces in real flows using different rendering options.

Modern experimental and computational fluid mechanics are increasingly concerned with the structure nature of fluid motion. Recent research has highlighted the analysis of one transport structure which is called Lagrangian coherent structure. However, the quantity nature of the flow transport is still unclear. In this paper, we focus on the transport characteristics of physical quantities and propose an approach to visualize the finite-time transport structure of quantity advection. This is similar to an integral convolution over a scalar field along path-lines of a flow field. Applied to a well-chosen set of physical quantity fields this yields structures giving insights into the dynamical processes of the underlying flow. We demonstrate our approach on a number of test data sets.

Abstract Fiber tracking is a standard tool to estimate the course of major white matter tracts from diffusion tensor magnetic resonance imaging (DT‐MRI) data. In this work, we aim at supporting the visual analysis of classical streamlines from fiber tracking by integrating context from anatomical data, acquired by aT1‐weighted MRI measurement. To this end, we suggest a novel visualization metaphor, which is based on data‐driven deformation of geometry and has been inspired by a technique for anatomical fiber preparation known as Klingler dissection. We demonstrate that our method conveys the relation between streamlines and surrounding anatomical features more effectively than standard techniques like slice images and direct volume rendering. The method works automatically, but its GPU‐based implementation allows for additional, intuitive interaction.

Smoke rendering is a standard technique for flow visualization. Most approaches are based on a volumetric, particle based, or image based representation of the smoke. This paper introduces an alternative representation of smoke structures: as semi-transparent streak surfaces. In order to make streak surface integration fast enough for interactive applications, we avoid expensive adaptive retriangulations by coupling the opacity of the triangles to their shapes. This way, the surface shows a smoke-like look even in rather turbulent areas. Furthermore, we show modifications of the approach to mimic smoke nozzles, wool tufts, and time surfaces. The technique is applied to a number of test data sets.

Vector field based shape deformations (VFSD) have been introduced as an efficient method to deform shapes in a volume-preserving foldover-free manner. However, mainly simple implicitly defined shapes like spheres or cylinders have been explored as deformation tools by now. In contrast, boundary constraint modeling approaches enable the user to exactly define the support of the deformation on the surface. We present an approach to explicitly control VFSD: a scalar function together with two thresholds is placed directly on the shape to mark regions of full, zero, and blended deformation. The resulting deformation is volume-preserving and free of local self-intersections. In addition, the full deformation is steered by a 3D parametric curve and a parametric twisting function. This way our deformations appear to be a generalization of the boundary constraint modeling metaphor. We apply our approach in different scenarios. A parallelization of the computation on the GPU allows for editing high-resolution meshes at interactive speed.

While isosurfaces of anisotropy measures for data from diffusion tensor magnetic resonance imaging (DT-MRI) are known to depict major anatomical structures, the anisotropy metric reduces the rich tensor data to a simple scalar field. In this work, we suggest that the part of the data which has been ignored by the metric can be used to segment anisotropy isosurfaces into anatomically meaningful regions. For the implementation, we propose an edge-based watershed method that adapts and extends a method from curvature-based mesh segmentation [MW99]. Finally, we use the segmentation results to enhance visualization of the data.

Topological methods give concise and expressive visual representations of flow fields. The present work suggests a comparable method for the visualization of human brain diffusion MRI data. We explore existing techniques for the topological analysis of generic tensor fields, but find them inappropriate for diffusion MRI data. Thus, we propose a novel approach that considers the asymptotic behavior of a probabilistic fiber tracking method and define analogs of the basic concepts of flow topology, like critical points, basins, and faces, with interpretations in terms of brain anatomy. The resulting features are fuzzy, reflecting the uncertainty inherent in any connectivity estimate from diffusion imaging. We describe an algorithm to extract the new type of features, demonstrate its robustness under noise, and present results for two regions in a diffusion MRI dataset to illustrate that the method allows a meaningful visual analysis of probabilistic fiber tracking results.

In nature and in flow experiments particles form patterns of swirling motion in certain locations. Existing approaches identify these structures by considering the behavior of stream lines. However, in unsteady flows particle motion is described by path lines which generally gives different swirling patterns than stream lines. We introduce a novel mathematical characterization of swirling motion cores in unsteady flows by generalizing the approach of Sujudi/Haimes to path lines. The cores of swirling particle motion are lines sweeping over time, i.e., surfaces in the space-time domain. They occur at locations where three derived 4D vectors become coplanar. To extract them, we show how to re-formulate the problem using the Parallel Vectors operator. We apply our method to a number of unsteady flow fields.

This paper presents an approach to extracting a path line oriented topological segmentation for periodic 2D timedependent vector fields. Topological methods aiming in capturing the asymptotic behavior of path lines rarely exist because path lines are usually only defined over a fixed time-interval, making statements about their asymptotic behavior impossible. For the data class of periodic vector fields, this restriction does not apply any more. Our approach detects critical path lines as well as basins from which the path lines converge to the critical ones. We demonstrate our approach on a number of test data sets.

Abstract In this paper we extract and visualize the topological skeleton of two‐parameter‐dependent vector fields. This kind of vector data depends on two parameter dimensions, for instance physical time and a scale parameter. We show that two important classes of local bifurcations – fold and Hopf bifurcations – build line structures for which we present an approach to extract them. Furthermore we show that new kinds of structurally stable local bifurcations exist for this data, namely fold‐fold and Hopf‐fold bifurcations. We present a complete classification of them. We apply our topological extraction method to analyze a number of two‐parameter‐dependent vector fields with different physical interpretations of the two additional dimensions. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Line and Curve Generation I.3.3 [Computer Graphics]: Picture/Image Generation

We present an approach to define shape deformations by constructing and interactively modifying C 1 continuous time-dependent divergence-free vector fields. The deformation is obtained by a path line integration of the mesh vertices. This way, the deformation is volume-preserving, free of (local and global) self-intersections, feature preserving, smoothness preserving, and local. Different modeling metaphors support the approach which is able to modify the vector field on-the-fly according to the user input. The approach works at interactive frame rates for moderate mesh sizes, and the numerical integration preserves the volume with a high accuracy.

We present an approach to visualizing correlations in 3D multifield scalar data. The core of our approach is the computation of correlation fields, which are scalar fields containing the local correlations of subsets of the multiple fields. While the visualization of the correlation fields can be done using standard 3D volume visualization techniques, their huge number makes selection and handling a challenge. We introduce the Multifield-Graph to give an overview of which multiple fields correlate and to show the strength of their correlation. This information guides the selection of informative correlation fields for visualization. We use our approach to visually analyze a number of real and synthetic multifield datasets.

We introduce an approach to tracking vortex core lines in time-dependent 3D flow fields which are defined by the parallel vectors approach. They build surface structures in the 4D space-time domain. To extract them, we introduce two 4D vector fields which act as feature flow fields, i.e., their integration gives the vortex core structures. As part of this approach, we extract and classify local bifurcations of vortex core lines in space-time. Based on a 4D stream surface integration, we provide an algorithm to extract the complete vortex core structure. We apply our technique to a number of test data sets.

This paper presents an approach to extracting and classifying higher order critical points of 3D vector fields. To do so, we place a closed convex surface s around the area of interest. Then we show that the complete 3D classification of a critical point into areas of different flow behavior is equivalent to extracting the topological skeleton of an appropriate 2D vector field on s, if each critical point is equipped with an additional bit of information. Out of this skeleton, we create an icon which replaces the complete topological structure inside s for the visualization. We apply our method to find a simplified visual representation of clusters of critical points, leading to expressive visualizations of topologically complex 3D vector fields.

This paper describes approaches to topologically segmenting 2D time-dependent vector fields. For this class of vector fields, two important classes of lines exist: stream lines and path lines. Because of this, two segmentations are possible: either concerning the behavior of stream lines or of path lines. While topological features based on stream lines are well established, we introduce path line oriented topology as a new visualization approach in this paper. As a contribution to stream line oriented topology, we introduce new methods to detect global bifurcations like saddle connections and cyclic fold bifurcations as well as a method of tracking all isolated closed stream lines. To get the path line oriented topology, we segment the vector field into areas of attracting, repelling, and saddle-like behavior of the path lines. We compare both kinds of topologies and apply them to a number of test data sets.

In computer animation, human motion capture from video is a widely used technique to acquire motion parameters. The acquisition process typically requires an intrusion into the scene in the form of optical markers which are used to estimate the parameters of motion as well as the kinematic structure of the performer. Marker-free optical motion capture approaches exist, but due to their dependence on a specific type of a priori model they can hardly be used to track other subjects, e.g. animals. To bridge the gap between the generality of marker-based methods and the applicability of marker-free methods, we present a flexible non-intrusive approach that estimates both, a kinematic model and its parameters of motion from a sequence of voxel-volumes. The volume sequences are reconstructed from multi-view video data by means of a shape-from-silhouette technique. The described method is well-suited for but not limited to motion capture of human subjects.

We introduce a new technique for estimating the curvature tensor of a triangular mesh. The input of the algorithm is only a single triangle equipped with its (exact or estimated) vertex normals. This way we get a smooth junction of the curvature tensor inside each triangle of the mesh. We show that the error of the new method is comparable with the error of a cubic fitting approach if the incorporated normals are estimated. If the exact normals of the underlying surface are available at the vertices, the error drops significantly. We demonstrate the applicability of the new estimation at a rather complex data set.

Topological methods aim at the segmentation of a vector field into areas of different flow behavior. For 2D time-dependent vector fields, two such segmentations are possible: either concerning the behavior of stream lines, or of path lines. While stream line oriented topology is well established, we introduce path line oriented topology as a new visualization approach in this paper. As a contribution to stream line oriented topology we introduce new methods to detect global bifurcations like saddle connections and cyclic fold bifurcations. To get the path line oriented topology we segment the vector field into areas of attracting, repelling and saddle-like behavior of the path lines. We compare both kinds of topologies and apply them to a number of data sets.

For realistic animation of an artificial character a body model that represents the character's kinematic structure is required. Hierarchical skeleton models are widely used which represent bodies as chains of bones with interconnecting joints. In video motion capture, animation parameters are derived from the performance of a subject in the real world. For this acquisition procedure too, a kinematic body model is required. Typically, the generation of such a model for tracking and animation is, at best, a semi-automatic process. We present a novel approach that estimates a hierarchical skeleton model of an arbitrary moving subject from sequences of voxel data that were reconstructed from multi-view video footage. Our method does not require a-priori information about the body structure. We demonstrate its performance using synthetic and real data.

Abstract We present the first algorithm for constructing 3D vector fields based on their topological skeleton. The skeleton itself is modeled by interactively moving a number of control polygons. Then a piecewise linear vector field is automatically constructed which has the same topological skeleton as modeled before. This approach is based on a complete segmentation of the areas around critical points into sectors of different flow behavior. Based on this, we present the first approach to visualizing higher order critical points of 3D vector fields. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Line and Curve Generation I.3.7 [Computer Graphics]: Three‐Dimensional Graphics and Realism

Topological simplification techniques and topology preserving compression approaches for 2D vector fields have been developed quite independently of each other. In this paper we propose a combination of both approaches: a vector field should be compressed in such a way that its important topological features (both critical points and separatrices) are preserved while its unimportant features are allowed to collapse and disappear. To do so, a number of new solutions and modifications of pre-existing algorithms are presented. We apply the approach to a flow data set, which is both large and topologically complex, and achieve significant compression ratios there.

One of the reasons that topological methods have a limited popularity for the visualization of complex 3D flow fields is the fact that such topological structures contain a number of separating stream surfaces. Since these stream surfaces tend to hide each other as well as other topological features, for complex 3D topologies the visualizations become cluttered and hardly interpretable. This paper proposes to use particular stream lines called saddle connectors instead of separating stream surfaces and to depict single surfaces only on user demand. We discuss properties and computational issues of saddle connectors and apply these methods to complex flow data. We show that the use of saddle connectors makes topological skeletons available as a valuable visualization tool even for topologically complex 3D flow data.

Abstract In this paper we introduce a new compression technique for 2D vector fields which preserves the complete topology, i.e., the critical points and the connectivity of the separatrices. As the theoretical foundation of the algorithm, we show in a theorem that for local modifications of a vector field, it is possible to decide entirely by a local analysis whether or not the global topology is preserved. This result is applied in a compression algorithm which is based on a repeated local modification of the vector field ‐ namely a repeated edge collapse of the underlying piecewise linear domain. We apply the compression technique to a number of data sets with a complex topology and obtain significantly improved compression ratios in comparison to pre‐existing topology‐preserving techniques.

In this paper we study the exact contours of a piecewise trilinear scalar field. We show how to represent these contours exactly as trimmed surfaces of triangular rational cubic Bézier patches. As part of this, we introduce an extension of the marching cubes algorithm which gives a topologically exact triangular approximation of the contours for any case. Finally, we modify the exact contours to be globally G1 continuous without changing their topologies. We test the algorithm on both theoretical and practical data sets.

We introduce a scheme of control polygons to design topological skeletons for vector fields of arbitrary topology. Based on this we construct piecewise linear vector fields of exactly the topology specified by the control polygons. This way a controlled construction of vector fields of any topology is possible. Finally we apply this method for topology-preserving compression of vector fields consisting of a simple topology.

In this paper we present a new technique for visualizing multidimensional information. We describe objects of a higher dimensional information space as small closed free‐form‐surfaces in the visualization. The location, size and shape of these surfaces describe the original objects in information space uniquely. The underlying enhanced spring model is introduced. The technique is applied to two test data sets.