Abstract In this paper, we propose a novel local feature extraction algorithm for the parallel vectors (PV) operator. Our method is based on Bézier clipping, which is a bracketing‐based root finding method that is commonly‐used in computer‐aided geometric design. Compared to Bézier bisection, our clipping method exhibits significantly faster convergence and reduces duplicate solutions. Compared to the general Newton‐Raphson algorithm, the success of our approach does not depend on the quality of an initial guess. The Bézier‐based formulation readily generalizes to polynomial higher‐order interpolants, such as for tricubic interpolation. With this, we derive the acceleration in the Sujudi‐Haimes vortex coreline criterion directly from the vector field interpolant. In addition, we describe the cross product of two polynomial vector fields in arbitrary polynomial degree, which can be used to approximate non‐polynomial interpolants. Further, we examine the effect of Newton refinement on the proposed Bézier clipping method. We evaluate our Bézier clipping method thoroughly on a range of different data sets.

Stochastic Monte Carlo solvers for partial differential equations (PDEs) recently gained popularity in computer graphics, finding applications in geometry processing, rendering, simulation, and visualization. At present, there exists no Monte Carlo solver for the rendering of biharmonic diffusion curves, an artist-friendly smooth vector graphics primitive. The fourth-order biharmonic equation of biharmonic diffusion curves can be split into two second-order PDEs, namely a Laplace and a Poisson equation. However, since biharmonic diffusion curves set Dirichlet and inhomogeneous Neumann conditions at the same time, these two second-order PDEs are tightly coupled and can hence not be solved directly. We propose to treat the rendering of biharmonic diffusion curves as an inverse problem, in which the Dirichlet data of the Laplace equation is unknown. We formulate a variational energy optimization, such that the user-defined boundary conditions are met. Thereby, the necessary gradients are estimated stochastically by solving two second-order problems with Dirichlet boundary conditions only.

Volumetric data arises in many scientific disciplines, describing the tissue in our body, the composition of the Earth, or the distribution of matter in the universe, to name a few. Such volume data is commonly visualized with direct methods such as ray marching, and with indirect methods such as isocontours. The combination of these approaches is fruitful to provide context and to enhance depth perception. A common challenge, however, is that three-dimensional data inherently leads to occlusions, for which visibility optimization techniques have been employed in the past. The joint visibility optimization of volumetric data (direct volume rendering) and surface geometry (indirect volume rendering) is challenging. With this paper, we provide an optimization approach in which explicit and implicit geometry, such as isocontours or closed context geometry, as well as the volume itself give way to reveal structures that have been identified as important by a user. We model the geometry implicitly as level set of a signed distance field, which is evolved under a normal flow to reduce the occlusion. The non-linear optimization involves gradient descent solvers, level set propagation, and multi-grid optimization. We compare our approach to previous visibility optimizations.

The detection and analysis of features in fluid flow are important tasks in fluid mechanics and flow visu alization. One recent class of methods to approach this problem is to first compute objective optimal reference frames, relative to which the input vector field becomes as steady as possible. However, existing methods either optimize locally over a fixed neighborhood, which might not match the extent of interesting features well, or perform global optimization, which is costly. We propose a novel objective method for the computation of optimal reference frames that automatically adapts to the flow field locally, without having to choose neighborhoods a priori. We enable adaptivity by formulating this problem as a moving least squares approximation, through which we determine a continuous field of reference frames. To incorporate fluid features into the computation of the reference frame field, we introduce the use of a scalar guidance field into the moving least squares approximation. The guidance field determines a curved manifold on which a regularly sampled input vector field becomes a set of irregularly spaced samples, which then forms the input to the moving least squares approximation. Although the guidance field can be any scalar field, by using a field that corresponds to flow features the resulting reference frame field will adapt accordingly. We show that using an FTLE field as the guidance field results in a reference frame field that adapts better to local features in the flow than prior wo rk. However, our moving least squares framework is formulated in a very general way, and therefore other types of guidance fields could be used in the future to adapt to local fluid features.

Abstract In data science, visual data exploration becomes increasingly more challenging due to the continued rapid increase of data dimensionality and data sizes. To manage complexity, two orthogonal approaches are commonly used in practice: First, data is frequently clustered in high‐dimensional space by fitting mixture models composed of normal distributions or Student t‐distributions. Second, dimensionality reduction is employed to embed high‐dimensional point clouds in a two‐ or three‐dimensional space. Those algorithms determine the spatial arrangement in low‐dimensional space without further user interaction. This leaves little room for a guided exploration and data analysis. In this paper, we propose a novel visualization system for the effective exploration and construction of potential subspaces onto which mixture models can be projected. The subspaces are spanned linearly via basis vectors, for which a vast number of basis vector combinations is theoretically imaginable. Our system guides the user step‐by‐step through the selection process by letting users choose one basis vector at a time. To guide the process, multiple choices are pre‐visualized at once on a multi‐faceted prism. In addition to the qualitative visualization of the distributions, multiple quantitative metrics are calculated by which subspaces can be compared and reordered, including variance, sparsity, and visibility. Further, a bookmarking tool lets users record and compare different basis vector combinations. The usability of the system is evaluated by data scientists and is tested on several high‐dimensional data sets.

In recent years, grid-free Monte Carlo methods have gained increasing popularity for solving fundamental partial differential equations. For a given point in the domain, the Walk-on-Spheres method solves a boundary integral equation by integrating recursively over the largest possible sphere. When the walks approach boundaries with Dirichlet conditions, the number of path vertices increases considerably, since the step size becomes smaller with decreasing distance to the boundary. In practice, the walks are terminated once they reach an epsilon-shell around the boundary. This, however, introduces bias, leading to a trade-off between accuracy and performance. Instead of using spheres, we propose to utilize geometric primitives that share more than one point with the boundary to increase the likelihood of immediately terminating. Along the boundary of those new geometric primitives a sampling probability is needed, which corresponds to the exit probability of a Brownian motion. This is known as a first passage problem. Utilizing that Laplace equations are invariant under conformal maps, we transform exit points from unit circles to the exit points of our geometric primitives, for which we describe a suitable placement strategy. With this, we obtain a novel approach to solve the Laplace equation in two dimensions, which does not require an epsilon-shell, significantly reduces the number of path vertices, and reduces inaccuracies near Dirichlet boundaries.

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.

Research on smooth vector graphics is separated into two independent research threads: one on interpolation-based gradient meshes and the other on diffusion-based curve formulations. With this paper, we propose a mathematical formulation that unifies gradient meshes and curve-based approaches as solution to a Poisson problem. To combine these two well-known representations, we first generate a non-overlapping intermediate patch representation that specifies for each patch a target Laplacian and boundary conditions. Unifying the treatment of boundary conditions adds further artistic degrees of freedoms to the existing formulations, such as Neumann conditions on diffusion curves. To synthesize a raster image for a given output resolution, we then rasterize boundary conditions and Laplacians for the respective patches and compute the final image as solution to a Poisson problem. We evaluate the method on various test scenes containing gradient meshes and curve-based primitives. Since our mathematical formulation works with established smooth vector graphics primitives on the front-end, it is compatible with existing content creation pipelines and with established editing tools. Rather than continuing two separate research paths, we hope that a unification of the formulations will lead to new rasterization and vectorization tools in the future that utilize the strengths of both approaches.

Abstract A long‐standing challenge in volume visualization is the effective communication of relevant spatial structures that might be hidden due to occlusions. Given a scalar field that indicates the importance of every point in the domain, previous work synthesized volume visualizations by weighted averaging of samples along view rays or by optimizing a spatially‐varying extinction field through an energy minimization. This energy minimization, however, did not directly measure the contribution of an individual sample to the final pixel color. In this paper, we measure the visibility of relevant structures directly by incorporating the transmittance into a non‐linear energy minimization. For the first time, we not only perform a transmittance‐based extinction optimization, we concurrently optimize the camera position to find ideal viewpoints. We derive the partial derivatives for the gradient‐based optimization symbolically, which makes the application of automatic differentiation methods unnecessary. The transmittance‐based formulation gives a direct visibility measure that is communicated to the user in order to make aware of potentially overlooked relevant structures. Our approach is compatible with any measure of importance and its versatility is demonstrated in multiple data sets.

Abstract Exploratory analysis of scalar fields on surface meshes presents significant challenges in identifying and visualizing important regions, particularly on the surface's backside. Previous visualization methods achieved only a limited visibility of significant features, i.e., regions with high or low scalar values, during interactive exploration. In response to this, we propose a novel technique, InverseVis, which leverages curved sphere tracing and uses the otherwise unused space to enhance visibility. Our approach combines direct and indirect rendering, allowing camera rays to wrap around the surface and reveal information from the backside. To achieve this, we formulate an energy term that guides the image synthesis in previously unused space, highlighting the most important regions of the backside. By quantifying the amount of visible important features, we optimize the camera position to maximize the visibility of the scalar field on both the front and backsides. InverseVis is benchmarked against state‐of‐the‐art methods and a derived technique, showcasing its effectiveness in revealing essential features and outperforming existing approaches.

Across many scientific disciplines, the pursuit of even higher grid resolutions leads to a severe scalability problem in scientific computing. Feature extraction is a commonly chosen approach to reduce the amount of information from dense fields down to geometric primitives that further enable a quantitative analysis. Examples of common features are isolines, extremal lines, or vortex corelines. Due to the rising complexity of the observed phenomena, or in the event of discretization issues with the data, a straightforward application of textbook feature definitions is unfortunately insufficient. Thus, feature extraction from spatial data often requires substantial pre- or post-processing to either clean up the results or to include additional domain knowledge about the feature in question. Such a separate pre- or post-processing of features not only leads to suboptimal and incomparable solutions, it also results in many specialized feature extraction algorithms arising in the different application domains. In this paper, we establish a mathematical language that not only encompasses commonly used feature definitions, it also provides a set of regularizers that can be applied across the bounds of individual application domains. By using the language of variational calculus, we treat features as variational minimizers, which can be combined and regularized as needed. Our formulation not only encompasses existing feature definitions as special case, it also opens the path to novel feature definitions. This work lays the foundations for many new research directions regarding formal definitions, data representations, and numerical extraction algorithms.

The field of smooth vector graphics explores the representation, creation, rasterization, and automatic generation of light-weight image representations, frequently used for scalable image content. Over the past decades, several conceptual approaches on the representation of images with smooth gradients have emerged that each led to separate research threads, including the popular gradient meshes and diffusion curves. As the computational models matured, the mathematical descriptions diverged and article started to focus more narrowly on subproblems, such as on the representation and creation of vector graphics, or the automatic vectorization from raster images. Most of the work concentrated on a specific mathematical model only. With this survey, we describe the established computational models in a consistent notation to spur further knowledge transfer, leveraging the recent advances in each field. We therefore categorize vector graphics article from the last decades based on their underlying mathematical representations as well as on their contribution to the vector graphics content creation pipeline, comprising representation, creation, rasterization, and automatic image vectorization. This survey is meant as an entry point for both artists and researchers. We conclude this survey with an outlook on promising research directions and challenges to overcome in the future.

As governed by personality trait theory, humans tackle problems differently depending on their long-term behavioral characteristics. Computational awareness of personality traits fuels affective computing research, which investigates how to reliably recognize and utilize personality traits. Applications are diverse, including therapy monitoring, learning assistance, and recommender systems. Data-driven approaches are a promising path forward towards personality-aware human-computer interactions. Thereby, central challenges are the non-disruptive data acquisition, the time frame over which data must be collected before predictions become accurate, and the feature-centered data reduction to train reliable and lightweight machine learning models. In this work, we address these challenges by presenting a fully-automatic feature extraction and machine learning pipeline that makes accurate personality trait predictions for the widely-used Five Factor Model from passively-collected, short-term smartphone typing data collected from 76 participants (68 university students) in the wild. Our model allows for personality trait assessments after one day of data collection, demonstrating that, despite being a long-term behavioral trend, personality traits can be inferred accurately from shorter time periods. We demonstrate that our system can accurately predict personality traits on two levels (low and high) with up to 74.5% accuracy and 0.72 AUC for a single day, and up to 84.5% accuracy and 0.79 AUC after subsequent refinement over 10 weeks.

The Gaussian mixture model (GMM) describes the distribution of random variables from several different populations. GMMs have widespread applications in probability theory, statistics, machine learning for unsupervised cluster analysis and topic modeling, as well as in deep learning pipelines. So far, few efforts have been made to explore the underlying point distribution in combination with the GMMs, in particular when the data becomes high-dimensional and when the GMMs are composed of many Gaussians. We present an analysis tool comprising various GPU-based visualization techniques to explore such complex GMMs. To facilitate the exploration of high-dimensional data, we provide a novel navigation system to analyze the underlying data. Instead of projecting the data to 2D, we utilize interactive 3D views to better support users in understanding the spatial arrangements of the Gaussian distributions. The interactive system is composed of two parts: (1) raycasting-based views that visualize cluster memberships, spatial arrangements, and support the discovery of new modes. (2) overview visualizations that enable the comparison of Gaussians with each other, as well as small multiples of different choices of basis vectors. Users are supported in their exploration with customization tools and smooth camera navigations. Our tool was developed and assessed by five domain experts, and its usefulness was evaluated with 23 participants. To demonstrate the effectiveness, we identify interesting features in several data sets.

Corporate meetings are a crucial part of business activities. While numerous academic papers investigated how to make the scheduling process of meetings faster or even automatic, little work has been done yet to facilitate the retrospective reasoning about how time is spent on meetings. Traditional calendar applications do not allow users to extract actionable statistics although it has been shown that reflection-oriented design can increase the users' understanding of their habits and can thereby encourage a shift towards better practices. In this paper, we present MineTime Insight, a tool made of multiple coordinated views for the exploration of personal calendar data, with the overarching goal of improving short and long-term scheduling decisions. Despite being focused on the working environment, our work builds upon recent results in the field of Personal Visual Analytics, as it targets users not necessarily expert in visualization and data analysis. We demonstrate the potential of MineTime Insight, when applied to the agenda of an executive manager. Finally, we discuss the results of an informal user study and a field study. Our results suggest that our visual representations are perceived as easy to understand and helpful towards a change in the scheduling habits.

In recent years, deep learning has opened countless research opportunities across many different disciplines. At present, visualization is mainly applied to explore and explain neural networks. Its counterpart-the application of deep learning to visualization problems-requires us to share data more openly in order to enable more scientists to engage in data-driven research. In this paper, we construct a large fluid flow data set and apply it to a deep learning problem in scientific visualization. Parameterized by the Reynolds number, the data set contains a wide spectrum of laminar and turbulent fluid flow regimes. The full data set was simulated on a high-performance compute cluster and contains 8000 time-dependent 2D vector fields, accumulating to more than 16 TB in size. Using our public fluid data set, we trained deep convolutional neural networks in order to set a benchmark for an improved post-hoc Lagrangian fluid flow analysis. In in-situ settings, flow maps are exported and interpolated in order to assess the transport characteristics of time-dependent fluids. Using deep learning, we improve the accuracy of flow map interpolations, allowing a more precise flow analysis at a reduced memory IO footprint.

Especially in the field of mechanical engineering, the market pressure for small and medium-sized enterprises increases because of faster developments and more complex designs. Nevertheless, standards and ergonomic safety regulations must be observed. In recent years, various applications were presented to help users understand and comply with inconvenient requirements. However, the time-consuming tools are primarily for ergonomics experts and often overwhelm engineers from small and medium-sized enterprises. We present an immersive concept that allows inexperienced users to quickly assess the ergonomic parameters of the visual field, represented by easy-to-understand visualizations. The solution is compared with standard market and scientific approaches.

An important goal of glyph based data visualization is to detect anomalies in data sets or to manage complex search tasks by guiding the users attention to regions of interest. Previous studies suggest that the user’s attention is tied to regions where different perspective projections meet. We describe a method for perspective distortion of glyphs within a virtual scene, while the rest of the scene is projected according to the common model of the computer graphics camera. The result is a multi-perspective image that directs the user’s attention. The perspective distortion is only applied if the user focuses on an irrelevant area of the data visualization. As soon as the gaze moves towards the relevant glyph, the perspective distortion is gradually removed. Therefore we evaluate eye tracking data in our prototypical implementation.

Abstract Dynamical systems are commonly used to describe the state of time‐dependent systems. In many engineering and control problems, the state space is high‐dimensional making it difficult to analyze and visualize the behavior of the system for varying input conditions. We present a novel dimensionality reduction technique that is tailored to high‐dimensional dynamical systems. In contrast to standard general purpose dimensionality reduction algorithms, we use energy minimization to preserve properties of the flow in the high‐dimensional space. Once the projection operator is optimized, further high‐dimensional trajectories are projected easily. Our 3D projection maintains a number of useful flow properties, such as critical points and flow maps, and is optimized to match geometric characteristics of the high‐dimensional input, as well as optional user constraints. We apply our method to trajectories traced in the phase spaces of second‐order dynamical systems, including finite‐sized objects in fluids, the circular restricted three‐body problem and a damped double pendulum. We compare the projections with standard visualization techniques, such as PCA, t‐SNE and UMAP, and visualize the dynamical systems with multiple coordinated views interactively, featuring a spatial embedding, projection to subspaces, our dimensionality reduction and a seed point exploration tool.

Potential vorticity is among the most important scalar quantities in atmospheric dynamics. For instance, potential vorticity plays a key role in particularly strong wind peaks in extratropical cyclones and it is able to explain the occurrence of frontal rain bands. Potential vorticity combines the key quantities of atmospheric dynamics, namely rotation and stratification. Under suitable wind conditions elongated banners of potential vorticity appear in the lee of mountains. Their role in atmospheric dynamics has recently raised considerable interest in the meteorological community for instance due to their influence in aviation wind hazards and maritime transport. In order to support meteorologists and climatologists in the analysis of these structures, we developed an extraction algorithm and a visual exploration framework consisting of multiple linked views. For the extraction we apply a predictor-corrector algorithm that follows streamlines and realigns them with extremal lines of potential vorticity. Using the agglomerative hierarchical clustering algorithm, we group banners from different sources based on their proximity. To visually analyze the time-dependent banner geometry, we provide interactive overviews and enable the query for detail on demand, including the analysis of different time steps, potentially correlated scalar quantities, and the wind vector field. In particular, we study the relationship between relative humidity and the banners for their potential in indicating the development of precipitation. Working with our method, the collaborating meteorologists gained a deeper understanding of the three-dimensional processes, which may spur follow-up research in the future.

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.

The topological analysis of unsteady vector fields remains to this day one of the largest challenges in flow visualization. We build up on recent work on vortex extraction to define a time-dependent vector field topology for 2D and 3D flows. In our work, we split the vector field into two components: a vector field in which the flow becomes steady, and the remaining ambient flow that describes the motion of topological elements (such as sinks, sources and saddles) and feature curves (vortex corelines and bifurcation lines). To this end, we expand on recent local optimization approaches by modeling spatially-varying deformations through displacement transformations from continuum mechanics. We compare and discuss the relationships with existing local and integration-based topology extraction methods, showing for instance that separatrices seeded from saddles in the optimal frame align with the integration-based streakline vector field topology. In contrast to the streakline-based approach, our method gives a complete picture of the topology for every time slice, including the steps near the temporal domain boundaries. With our work it now becomes possible to extract topological information even when only few time slices are available. We demonstrate the method in several analytical and numerically-simulated flows and discuss practical aspects, limitations and opportunities for future work.

Time-dependent fluid flows often contain numerous hyperbolic Lagrangian coherent structures, which act as transport barriers that guide the advection. The finite-time Lyapunov exponent is a commonly-used approximation to locate these repelling or attracting structures. Especially on large numerical simulations, the FTLE ridges can become arbitrarily sharp and very complex. Thus, the discrete sampling onto a grid for a subsequent direct volume rendering is likely to miss sharp ridges in the visualization. For this reason, an unbiased Monte Carlo-based rendering approach was recently proposed that treats the FTLE field as participating medium with single scattering. This method constructs a ground truth rendering without discretization, but it is prohibitively slow with render times in the order of days or weeks for a single image. In this paper, we accelerate the rendering process significantly, which allows us to compute video sequence of high-resolution FTLE animations in a much more reasonable time frame. For this, we follow two orthogonal approaches to improve on the rendering process: the volumetric light path integration in gradient domain and an acceleration of the transmittance estimation. We analyze the convergence and performance of the proposed method and demonstrate the approach by rendering complex FTLE fields in several 3D vector fields.

Over the past decades, scientific visualization became a fundamental aspect of modern scientific data analysis. Across all data-intensive research fields, ranging from structural biology to cosmology, data sizes increase rapidly. Dealing with the growing large-scale data is one of the top research challenges of this century. For the visual exploratory data analysis, interactivity, a view-dependent visibility optimization and frame coherence are indispensable. In this work, we extend the recent decoupled opacity optimization framework to enable a navigation without occlusion of important features through large geometric data. By expressing the accumulation of importance and optical depth in Fourier basis, the computation, evaluation and rendering of optimized transparent geometry become not only order-independent, but also operate within a fixed memory bound. We study the quality of our Fourier approximation in terms of accuracy, memory requirements and efficiency for both the opacity computation, as well as the order-independent compositing. We apply the method to different point, line and surface data sets originating from various research fields, including meteorology, health science, astrophysics and organic chemistry.

Mid-air interaction achieved through handtracking or controller in Virtual Reality (VR) is performed without an explicit spatial reference. One solution for creating such a reference is elastic feedback, which we realized via tension springs attached to a tracked VR-controller. Thereby, directional force is exerted and perceived by the user. A study was conducted to investigate the effects of elastic feedback on users in virtual travel tasks. Therefore, the elastic mode of the input device was compared to an isotonic mode, without any perceivable force. Also, two virtual travelling methods, a driving mode and a flight mode, were investigated in the study. The main goal of the study was to analyze, how elastic feedback influences user experience and presence perception. Statistical analysis with ANOVA of data from 24 subjects revealed significant main effects for the elastic mode on both, the User Experience Questionnaire (UEQ) and the Igroup Presence Questionnaire (IPQ). In addition, the subjects missed fewer target objects on average during the travel task. However, analysis of task completion times indicated no relevant differences. Although not all individual scales of UEQ and IPQ showed significant outcomes, the result indicates that immersive travel interfaces could benefit from the use of elastic feedback.

Abstract Robust feature extraction is an integral part of scientific visualization. In unsteady vector field analysis, researchers recently directed their attention towards the computation of near‐steady reference frames for vortex extraction, which is a numerically challenging endeavor. In this paper, we utilize a convolutional neural network to combine two steps of the visualization pipeline in an end‐to‐end manner: the filtering and the feature extraction. We use neural networks for the extraction of a steady reference frame for a given unsteady 2D vector field. By conditioning the neural network to noisy inputs and resampling artifacts, we obtain numerically stabler results than existing optimization‐based approaches. Supervised deep learning typically requires a large amount of training data. Thus, our second contribution is the creation of a vector field benchmark data set, which is generally useful for any local deep learning‐based feature extraction. Based on Vatistas velocity profile, we formulate a parametric vector field mixture model that we parameterize based on numerically‐computed example vector fields in near‐steady reference frames. Given the parametric model, we can efficiently synthesize thousands of vector fields that serve as input to our deep learning architecture. The proposed network is evaluated on an unseen numerical fluid flow simulation.

Abstract The art of representing images with triangles is known as image triangulation, which purposefully uses abstraction and simplification to guide the viewer's attention. The manual creation of image triangulations is tedious and thus several tools have been developed in the past that assist in the placement of vertices by means of image feature detection and subsequent Delaunay triangulation. In this paper, we formulate the image triangulation process as an optimization problem. We provide an interactive system that optimizes the vertex locations of an image triangulation to reduce the root mean squared approximation error. Along the way, the triangulation is incrementally refined by splitting triangles until certain refinement criteria are met. Thereby, the calculation of the energy gradients is expensive and thus we propose an efficient rasterization‐based GPU implementation. To ensure that artists have control over details, the system offers a number of direct and indirect editing tools that split, collapse and re‐triangulate selected parts of the image. For final display, we provide a set of rendering styles, including constant colours, linear gradients, tonal art maps and textures. Finally, we demonstrate temporal coherence for animations and compare our method with existing image triangulation tools.

Abstract Recurrent neural networks are prime candidates for learning evolutions in multi‐dimensional time series data. The performance of such a network is judged by the loss function, which is aggregated into a scalar value that decreases during training. Observing only this number hides the variation that occurs within the typically large training and testing data sets. Understanding these variations is of highest importance to adjust network hyper‐parameters, such as the number of neurons, number of layers or to adjust the training set to include more representative examples. In this paper, we design a comprehensive and interactive system that allows users to study the output of recurrent neural networks on both the complete training data and testing data. We follow a coarse‐to‐fine strategy, providing overviews of annual, monthly and daily patterns in the time series and directly support a comparison of different hyper‐parameter settings. We applied our method to a recurrent convolutional neural network that was trained and tested on 25 years of climate data to forecast meteorological attributes, such as temperature, pressure and wind velocity. We further visualize the quality of the forecasting models, when applied to various locations on the Earth and we examine the combination of several forecasting models.

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.

This paper presents a framework to explore multi-field data of aneurysms occurring at intracranial and cardiac arteries by using statistical graphics. The rupture of an aneurysm is often a fatal scenario, whereas during treatment serious complications for the patient can occur. Whether an aneurysm ruptures or whether a treatment is successful depends on the interaction of different morphological such as wall deformation and thickness, and hemodynamic attributes like wall shear stress and pressure. Therefore, medical researchers are very interested in better understanding these relationships. However, the required analysis is a time-consuming process, where suspicious wall regions are difficult to detect due to the time-dependent behavior of the data. Our proposed visualization framework enables medical researchers to efficiently assess aneurysm risk and treatment options. This comprises a powerful set of views including 2D and 3D depictions of the aneurysm morphology as well as statistical plots of different scalar fields. Brushing and linking aids the user to identify interesting wall regions and to understand the influence of different attributes on the aneurysm's state. Moreover, a visual comparison of pre- and post-treatment as well as different treatment options is provided. Our analysis techniques are designed in collaboration with domain experts, e.g., physicians, and we provide details about the evaluation.

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 In many scientific disciplines, the motion of finite‐sized objects in fluid flows plays an important role, such as in brownout engineering, sediment transport, oceanology or meteorology. These finite‐sized objects are called inertial particles and, in contrast to traditional tracer particles, their motion depends on their current position, their own particle velocity, the time and their size. Thus, the visualization of their motion becomes a high‐dimensional problem that entails computational and perceptual challenges. So far, no visualization explored and visualized the particle trajectories under variation of all seeding parameters. In this paper, we propose three coordinated views that visualize the different aspects of the high‐dimensional space in which the particles live. We visualize the evolution of particles over time, showing that particles travel different distances in the same time, depending on their size. The second view provides a clear illustration of the trajectories of different particle sizes and allows the user to easily identify differences due to particle size. Finally, we embed the trajectories in the space‐velocity domain and visualize their distance to an attracting manifold using ribbons. In all views, we support interactive linking and brushing, and provide abstraction through density volumes that are shown by direct volume rendering and isosurface slabs. Using our method, users gain deeper insights into the dynamics of inertial particles in 2D fluids, including size‐dependent separation, preferential clustering and attraction. We demonstrate the effectiveness of our method in multiple steady and unsteady 2D flows.

Traditionally, vector field visualization is concerned with 2D and 3D flows. Yet, many concepts can be extended to general dynamical systems, including the higher-dimensional problem of modeling the motion of finite-sized objects in fluids. In the steady case, the trajectories of these so-called inertial particles appear as tangent curves of a 4D or 6D vector field. These higher-dimensional flows are difficult to map to lower-dimensional spaces, which makes their visualization a challenging problem. We focus on vector field topology, which allows scientists to study asymptotic particle behavior. As recent work on the 2D case has shown, both extraction and classification of isolated critical points depend on the underlying particle model. In this paper, we aim for a model-independent classification technique, which we apply to two different particle models in not only 2D, but also 3D cases. We show that the classification can be done by performing an eigenanalysis of the spatial derivatives' velocity subspace of the higher-dimensional 4D or 6D flow. We construct glyphs that depict not only the types of critical points, but also encode the directional information given by the eigenvectors. We show that the eigenvalues and eigenvectors of the inertial phase space have sufficient symmetries and structure so that they can be depicted in 2D or 3D, instead of 4D or 6D.

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.

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.

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.

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.

The leading motive of the research demo is the utilization of the Augmented Virtuality technology and in particular the egocentric representation of real body parts in virtual and immersive environments. In this context, a prototypical application was developed, which superimposes the current view of the user's hands on the virtual scene in real-time. This is achieved in form of a captured video stream. Advantages compared to virtual avatars arise from the detailed and individual representation of the user's body and the saving of complex tracking hardware. Due to the integration of a toolbox, the visual appearance of the video overlay is highly customizable.

Abstract Even though much research was dedicated to the acceleration of consistent, progressive light transport simulations, the computation of fully converged images is still very time‐consuming. This is problematic, as for the practical use in production pipelines, the rapid editing of lighting effects is important. While previous approaches restart the simulation with every scene manipulation, we make use of the coherence between frames before and after a modification in order to accelerate convergence of the context that remained similar. This is especially beneficial if a scene is edited that has already been converging for a long time, because much of the previous result can be reused, e.g., sharp caustics cast or received by the unedited scene parts. In its essence, our method performs the scene modification stochastically by predicting and accounting for the difference image. In addition, we employ two heuristics to handle cases in which stochastic removal is likely to lead to strong noise. Typical scene interactions can be broken down into object adding and removal, material substitution, camera movement and light editing, which we all examine in a number of test scenes both qualitatively and quantitatively. As we focus on caustics, we chose stochastic progressive photon mapping as the underlying light transport algorithm. Further, we show preliminary results of bidirectional path tracing and vertex connection and merging.

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 Progressive light transport simulations aspire a physically‐based, consistent rendering to obtain visually appealing illumination effects, depth and realism. Thereby, the handling of large scenes is a difficult problem, as in typical scene subdivision approaches the parallel processing requires frequent synchronization due to the bouncing of light throughout the scene. In practice, however, only few object parts noticeably contribute to the radiance observable in the image, whereas large areas play only a minor role. In fact, a mesh simplification of the latter can go unnoticed by the human eye. This particular importance to the visible radiance in the image calls for an output‐sensitive mesh reduction that allows to render originally out‐of‐core scenes on a single machine without swapping of memory. Thus, in this paper, we present a preprocessing step that reduces the scene size under the constraint of radiance preservation with focus on high‐frequency effects such as caustics. For this, we perform a small number of preliminary light transport simulation iterations. Thereby, we identify mesh parts that contribute significantly to the visible radiance in the scene, and which we thus preserve during mesh reduction.

Abstract At present, stochastic progressive photon mapping (SPPM) is one of the most comprehensive methods for a consistent global illumination computation. Even though the number of photons is unlimited due to their progressive nature, the scene size is still bound by the available main memory. In this paper, we present the first consistent out‐of‐core SPPM algorithm. In order to cope with large scenes, we automatically subdivide the geometry and parallelly trace photons and eye rays in a portal‐based system, distributed across multiple machines in a commodity cluster. Moreover, modifications of the original SPPM method are introduced that keep both the utilization of tracer machines high and the network traffic low. Therefore, compared to a portal‐based single machine setup, our distributed approach achieves a significant speedup. We compare a GPU‐based with a CPU‐based implementation and demonstrate our system in multiple large test scenes of up to 90 million triangles.

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 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.

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.

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.

Three different neuromorphic vision sensors; a 2D smooth optical flow sensor, a 1D tracker sensor and a 1D motion sensor are modeled in a simulator. The sensors are modeled according to their spatio-temporal properties and the model is validated with experimental data obtained in previous work. The model parameters are fitted according to validation data, where a cost function is optimized with a gradient descent approach. The simulator used is the Open Dynamics Engine (ODE) and the experimental data from previous work was collected in a typical RoboCup scenario. A soccer playing robot is here used to provide reference information from a standard digital tracker system. By modeling the output from neuromorphic vision sensors, it is possible to reconstruct the experiment in the simulator. We will present simulated data from this particular experiment and compare it to the ground truth validation data. This work is important in order to reduce the time needed for real-world experiments for the set-up and optimization phase during the integration of such sensors into robotic systems. At the end we will also make an indication of the direction of future research.