Contemporary materials science research is heavily conducted in silico, involving massive simulations of the atomic-scale evolution of materials. Cataloging basic patterns in the atomic displacements is key to understanding and predicting the evolution of physical properties. However, the combinatorial complexity of the space of possible transitions coupled with the overwhelming amount of data being produced by high-throughput simulations make such an analysis extremely challenging and time-consuming for domain experts. The development of visual analytics systems that facilitate the exploration of simulation data is an active field of research. While these systems excel in identifying temporal regions of interest, they treat each timestep of a simulation as an independent event without considering the behavior of the atomic displacements between timesteps. We address this gap by introducing LAMDA, a visual analytics system that allows domain experts to quickly and systematically explore state-to-state transitions. In LAMDA, transitions are hierarchically categorized, providing a basis for cataloging displacement behavior, as well as enabling the analysis of simulations at different resolutions, ranging from very broad qualitative classes of transitions to very narrow definitions of unit processes. LAMDA supports navigating the hierarchy of transitions, enabling scientists to visualize the commonalities between different transitions in each class in terms of invariant features characterizing local atomic environments, and LAMDA simplifies the analysis by capturing user inputs through annotations. We evaluate our system through a case study and report on findings from our domain experts.

Fiber surfaces and fiber lines, being the preimages of a bivariate function to a so-called control polygon in the function’s range, are the adaptation of isosurfaces and isolines to bivariate scalar fields. Previous works have proposed use cases for fiber surface and fiber line extraction using algorithmically generated control polygons with many line segments, but their authors have either not presented results or noted that the computation becomes prohibitive slow. We present an algorithm that speeds up fiber surface extraction by using dual bounding volume hierarchy (BVH) traversal, as well as a variation that combines the benefits of single and dual BVH traversal. We study the influence of the number of line segments in the control polygon on the performance of single and dual BVH traversal algorithms using data sets from various application domains and types of control polygons. We find that dual BVH traversal is several times faster in test cases where single BVH traversal is slow, facilitating the interactive exploration of fiber surfaces in many cases where existing methods are too slow.

Abstract Visual analytics (VA) is a paradigm for insight generation by using visual analysis techniques and automated reasoning by transforming data into hypotheses and visualization to extract new insights. The insights are fed back into the data to enhance it until the desired insight is found. Many applications use this principle to provide meaningful mechanisms to assist decision-makers in achieving their goals. This process can be affected by various uncertainties that can interfere with the user decision-making process. Currently, there are no methodical description and handling tool to include uncertainty in VA systematically. We provide a unified workflow to transform the classic VA cycle into an uncertainty-aware visual analytics (UAVA) cycle consisting of five steps. To prove its usability, three real-world applications represent examples of the UAVA cycle implementation and the described workflow.

Analyzing uncertainty in spatial data is a vital task in many domains, as for example with climate and weather simulation ensembles. Although many methods support the analysis of uncertain 2D data, such as uncertain isocontours or overlaying of statistical information on plots of the actual data, it is still a challenge to get a more detailed overview of 2D data together with its statistical properties. We present cumulative height fields, a visualization method for 2D scalar field ensembles using the marginal empirical distribution function and show preliminary results using volume rendering and slicing for the Max Planck Institute Grand Ensemble.

When learning a motor skill it is helpful to get corrective feedback from an instructor. This will support the learner to execute the movement correctly. With modern technology, it is possible to provide this feedback via mixed reality. In most cases, this involves visual cues to help the user understand the corrective feedback. We analyzed recent research approaches utilizing visual cues for feedback in mixed reality. The scope of this article is visual feedback for motor skill learning, which involves physical therapy, exercise, rehabilitation etc. While some of the surveyed literature discusses therapeutic effects of the training, this article focuses on visualization techniques. We categorized the literature from a visualization standpoint, including visual cues, technology and characteristics of the feedback. This provided insights into how visual feedback in mixed reality is applied in the literature and how different aspects of the feedback are related. The insights obtained can help to better adjust future feedback systems to the target group and their needs. This article also provides a deeper understanding of the characteristics of the visual cues in general and promotes future, more detailed research on this topic.

We develop an interactive approach for analyzing multi-field tensor data from simulations in close collaboration with domain scientists. Our approach is based on extensive application analysis and built around a multi-field clustering addressing multiple user-defined quantities which were required by the domain scientists. Established techniques like linked views complement the approach to support reasoning while offering an overview and detailed insight into the multi-field tensor data. Further, we include an evaluation containing a real-world use case and a user study with domain scientists to demonstrate the usefulness compared to existing tools.

Physics, medicine, earth sciences, mechanical engineering, geo-engineering, bio-engineering and many more application areas use tensorial data. For example, tensors are used in formulating the balance equations of charge, mass, momentum, or energy as well as the constitutive relations that complement them. Some of these tensors (i.e., stiffness tensor, strain gradient, photo-elastic tensor) are of order higher than two. Currently, there are nearly no visualization techniques for such data beyond glyphs. An important reason for this is the limit of currently used tensor decomposition techniques. In this article, we propose to use the deviatoric decomposition to draw lines describing tensors of arbitrary order in three dimensions. The deviatoric decomposition splits a three-dimensional tensor of any order with any type of index symmetry into totally symmetric, traceless tensors. These tensors, called deviators, can be described by a unique set of directions (called multipoles by J. C. Maxwell) and scalars. These multipoles allow the definition of multipole lines which can be computed in a similar fashion to tensor lines and allow a line-based visualization of three-dimensional tensors of any order. We give examples for the visualization of symmetric, second-order tensor fields as well as fourth-order tensor fields. To allow an interpretation of the multipole lines, we analyze the connection between the multipoles and the eigenvectors/eigenvalues in the second-order case. For the fourth-order stiffness tensor, we prove relations between multipoles and important physical quantities such as shear moduli as well as the eigenvectors of the second-order right Cauchy-Green tensor.

Electroactive polymers are frequently used in engineering applications due to their ability to change their shape and properties under the influence of an electric field. This process also works vice versa, such that mechanical deformation of the material induces an electric field in the EAP device. This specific behavior makes such materials highly attractive for the construction of actuators and sensors in various application areas. The electromechanical behaviour of electroactive polymers can be described by a third-order coupling tensor, which represents the sensitivity of mechanical stresses concerning the electric field, i.e., it establishes a relation between a second-order and a first-order tensor field. Due to this coupling tensor's complexity and the lack of meaningful visualization methods for third-order tensors in general, an interpretation of the tensor is rather difficult. Thus, the central engineering research question that this contribution deals with is a deeper understanding of electromechanical coupling by analyzing the third-order coupling tensor with the help of specific visualization methods. Starting with a deviatoric decomposition of the tensor, the multipoles of each deviator are visualized, which allows a first insight into this highly complex third-order tensor. In the present contribution, four examples, including electromechanical coupling, are simulated within a finite element framework and subsequently analyzed using the tensor visualization method.

Abstract In many applications, visual analytics (VA) has developed into a standard tool to ease data access and knowledge generation. VA describes a holistic cycle transforming data into hypothesis and visualization to generate insights that enhance the data. Unfortunately, many data sources used in the VA process are affected by uncertainty. In addition, the VA cycle itself can introduce uncertainty to the knowledge generation process but does not provide a mechanism to handle these sources of uncertainty. In this manuscript, we aim to provide an extended VA cycle that is capable of handling uncertainty by quantification, propagation, and visualization, defined as uncertainty-aware visual analytics (UAVA). Here, a recap of uncertainty definition and description is used as a starting point to insert novel components in the visual analytics cycle. These components assist in capturing uncertainty throughout the VA cycle. Further, different data types, hypothesis generation approaches, and uncertainty-aware visualization approaches are discussed that fit in the defined UAVA cycle. In addition, application scenarios that can be handled by such a cycle, examples, and a list of open challenges in the area of UAVA are provided.

In an era of quickly growing data set sizes, information reduction methods such as extracting or highlighting characteristic features become more and more important for data analysis. For single scalar fields, topological methods can fill this role by extracting and relating critical points. While such methods are regularly employed to study single scalar fields, it is less well studied how they can be extended to uncertain data, as produced, e.g., by ensemble simulations. Motivated by our previous work on visualization in climate research, we study new methods to characterize critical points in ensembles of 2D scalar fields. Previous work on this topic either assumed or required specific distributions, did not account for uncertainty introduced by approximating the underlying latent distributions by a finite number of fields, or did not allow to answer all our domain experts' questions. In this work, we use Bayesian inference to estimate the probability of critical points, either of the original ensemble or its bootstrapped mean. This does not make any assumptions on the underlying distribution and allows to estimate the sensitivity of the results to finite-sample approximations of the underlying distribution. We use color mapping to depict these probabilities and the stability of their estimation. The resulting images can, e.g., be used to estimate how precise the critical points of the mean-field are. We apply our method to synthetic data to validate its theoretical properties and compare it with other methods in this regard. We also apply our method to the data from our previous work, where it provides a more accurate answer to the domain experts' research questions.

Christopher Schröder, Kim Bürgl, Yves Annanias, Andreas Niekler, Lydia Müller, Daniel Wiegreffe, Christian Bender, Christoph Mengs, Gerik Scheuermann, Gerhard Heyer. Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers). 2021.

A driving factor of the winter weather in Western Europe is the North Atlantic Oscillation (NAO), manifested by fluctuations in the difference of sea level pressure between the Icelandic Low and the Azores High. Different methods have been developed that describe the strength of this oscillation, but they rely on certain assumptions, e.g., fixed positions of these two pressure systems. It is possible that climate change affects the mean location of both the Low and the High and thus the validity of these descriptive methods. This study is the first to visually analyze large ensemble climate change simulations (the MPI Grand Ensemble) to robustly assess shifts of the drivers of the NAO phenomenon using the uncertain northern hemispheric surface pressure fields. For this, we use a sliding window approach and compute empirical orthogonal functions (EOFs) for each window and ensemble member, then compare the uncertainty of local extrema in the results as well as their temporal evolution across different CO2scenarios. We find systematic northeastward shifts in the location of the pressure systems that correlate with the simulated warming. Applying visualization techniques for this analysis was not straightforward; we reflect and give some lessons learned for the field of visualization.

To assess the reliability of weather forecasts and climate simulations, common practice is to generate large ensembles of numerical simulations. Analyzing such data is challenging and requires pattern and feature detection. For single time-dependent scalar fields, empirical orthogonal functions (EOFs) are a proven means to identify the main variation. In this paper, we present an extension of that concept to time-dependent ensemble data. We applied our methods to two ensemble data sets from climate research in order to investigate the North Atlantic Oscillation (NAO) and East Atlantic (EA) pattern.

Abstract Medical imaging (image acquisition, image transformation, and image visualization) is a standard tool for clinicians in order to make diagnoses, plan surgeries, or educate students. Each of these steps is affected by uncertainty, which can highly influence the decision‐making process of clinicians. Visualization can help in understanding and communicating these uncertainties. In this manuscript, we aim to summarize the current state‐of‐the‐art in uncertainty‐aware visualization in medical imaging. Our report is based on the steps involved in medical imaging as well as its applications. Requirements are formulated to examine the considered approaches. In addition, this manuscript shows which approaches can be combined to form uncertainty‐aware medical imaging pipelines. Based on our analysis, we are able to point to open problems in uncertainty‐aware medical imaging.

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

Abstract Colormapping is one of the simplest and most widely used data visualization methods within and outside the visualization community. Uniformity, order, discriminative power, and smoothness of continuous colormaps are the most important criteria for evaluating and potentially improving colormaps. We present a local and a global automatic optimization algorithm in Euclidean color spaces for each of these design rules in this work. As a foundation for our optimization algorithms, we used the CCC‐Tool colormap specification (CMS); each algorithm has been implemented in this tool. In addition to synthetic examples that demonstrate each method's effect, we show the outcome of some of the methods applied to a typhoon simulation.

Many computer science disciplines (e.g., combinatorial optimization, natural language processing, and information retrieval) use standard or established test suites for evaluating algorithms. In visualization, similar approaches have been adopted in some areas (e.g., volume visualization), while user testimonies and empirical studies have been the dominant means of evaluation in most other areas, such as designing colormaps. In this paper, we propose to establish a test suite for evaluating the design of colormaps. With such a suite, the users can observe the effects when different continuous colormaps are applied to planar scalar fields that may exhibit various characteristic features, such as jumps, local extrema, ridge or valley lines, different distributions of scalar values, different gradients, different signal frequencies, different levels of noise, and so on. The suite also includes an expansible collection of real-world data sets including the most popular data for colormap testing in the visualization literature. The test suite has been integrated into a web-based application for creating continuous colormaps (https://ccctool.com/), facilitating close inter-operation between design and evaluation processes. This new facility complements traditional evaluation methods such as user testimonies and empirical studies.

Continuous colormaps are integral parts of many visualization techniques, such as heat-maps, surface plots, and flow visualization. Despite that the critiques of rainbow colormaps have been around and well-acknowledged for three decades, rainbow colormaps are still widely used today. One reason behind the resilience of rainbow colormaps is the lack of tools for users to create a continuous colormap that encodes semantics specific to the application concerned. In this paper, we present a web-based software system, CCC-Tool (short for Charting Continuous Colormaps) under the URL https://ccctool.com, for creating, editing, and analyzing such application-specific colormaps. We introduce the notion of "colormap specification (CMS)" that maintains the essential semantics required for defining a color mapping scheme. We provide users with a set of advanced utilities for constructing CMS's with various levels of complexity, examining their quality attributes using different plots, and exporting them to external application software. We present two case studies, demonstrating that the CCC-Tool can help domain scientists as well as visualization experts in designing semantically-rich colormaps.

Modern engineering uses optimization to design long-living and robust components. One part of this process is to find the optimal stress-aware design under given geometric constraints and loading conditions. Tensor visualization provides techniques to show and evaluate the stress distribution based on finite element method simulations. One such technique are hyperstreamlines. They allow us to explore the stress along a line following one principal stress direction while showing the other principal stress directions and their values. In this paper, we show shortcomings of this approach from an engineer’s point of view and propose a variant called tensor spines. It provides an improved perception of the relation between the principal stresses helping engineers to optimize their designs further.

Abstract Scientific visualization deals with increasingly complex data consisting of multiple fields. Typical disciplines generating multivariate data are fluid dynamics, structural mechanics, geology, bioengineering, and climate research. Quite often, scientists are interested in the relation between some of these variables. A popular visualization technique for a single scalar field is the extraction and rendering of isosurfaces. With this technique, the domain can be split into two parts, i.e. a volume with higher values and one with lower values than the selected isovalue. Fiber surfaces generalize this concept to two or three scalar variables up to now. This article extends the notion further to potentially any finite number of scalar fields. We generalize the fiber surface extraction algorithm of Raith et al. [RBN∗19] from 3 to d dimensions and demonstrate the technique using two examples from geology and climate research. The first application concerns a generic model of a nuclear waste repository and the second one an atmospheric simulation over central Europe. Both require complex simulations which involve multiple physical processes. In both cases, the new extended fiber surfaces helps us finding regions of interest like the nuclear waste repository or the power supply of a storm due to their characteristic properties.

Splat and antisplat events are a widely found phenomenon in three-dimensional turbulent flow fields. Splats are observed when fluid locally impinges on an impermeable surface transferring energy from the normal component to the tangential velocity components, while antisplats relate to the inverted situation. These events affect a variety of flow properties, such as the transfer of kinetic energy between velocity components and the transfer of heat, so that their investigation can provide new insight into these issues. Here, we propose the first Lagrangian method for the detection of splats and antisplats as features of an unsteady flow field. Our method utilizes the concept of strain tensors on flow-embedded flat surfaces to extract disjoint regions in which splat and antisplat events of arbitrary scale occur. We validate the method with artificial flow fields of increasing complexity. Subsequently, the method is used to analyze application data stemming from a direct numerical simulation of the turbulent flow over a backward facing step. Our results show that splat and antisplat events can be identified efficiently and reliably even in such a complex situation, demonstrating that the new method constitutes a well-suited tool for the analysis of turbulent flows.

Turbines are essential components of jet planes and power plants. Therefore, their efficiency and service life are of central engineering interest. In the case of jet planes or thermal power plants, the heating of the turbines due to the hot gas flow is critical. Besides effective cooling, it is a major goal of engineers to minimize heat transfer between gas flow and turbine by design. Since it is known that splat events have a substantial impact on the heat transfer between flow and immersed surfaces, we adapt a splat detection and visualization method to a turbine cascade simulation in this case study. Because splat events are small phenomena, we use a direct numerical simulation resolving the turbulence in the flow as the base of our analysis. The outcome shows promising insights into splat formation and its relation to vortex structures. This may lead to better turbine design in the future.

The use of clean and renewable energy and the abandoning of fossil energy have become goals of many national and international energy policies. But even when once accomplished, mankind has to take charge of the relics of the current energy supply system. For example, due to its harmful effects, nuclear waste has to be isolated from the biosphere safely and for sufficiently long times. The geological subsurface is considered as a promising option for the deposition of such by-or end products. In order to investigate the long-term evolution of a repository system, a multiphysics simulation was performed. It combines the structural mechanics of the host rock, the fluid dynamics of formation fluids, and the thermodynamics of all materials resulting in a highly multivariate data set. A visualization of such multiphysics data challenges the current methodology. In this article, we demonstrate how an analysis of a carefully selected subset of the variables in attribute space allows to visualize and interpret the simulation data. We apply a fiber surface extraction algorithm to explore the relationships between these variables. Studying the temporal evolution in attribute space, we found a regionally bulge that could be identified as an effect of the nuclear waste repository because it can be clearly separated from the natural geophysical state prior to waste disposal. Furthermore, we used the extracted fiber surface as a starting point to examine the distribution of other variables inside this area of the physical domain. We conclude this case study with lessons learned from the visualization as well as the geotechnical side.

Scientific visualization developed successful methods for scalar and vector fields. For tensor fields, however, effective, interactive visualizations are still missing despite progress over the last decades. We present a general approach for the generation of separating surfaces in symmetric, second-order, three-dimensional tensor fields. These surfaces are defined as fiber surfaces of the invariant space, i.e. as pre-images of surfaces in the range of a complete set of invariants. This approach leads to a generalization of the fiber surface algorithm by Klacansky et al. [16] to three dimensions in the range. This is due to the fact that the invariant space is three-dimensional for symmetric second-order tensors over a spatial domain. We present an algorithm for surface construction for simplicial grids in the domain and simplicial surfaces in the invariant space. We demonstrate our approach by applying it to stress fields from component design in mechanical engineering.

Abstract Sets of multiple scalar fields can be used to model many types of variation in data, such as uncertainty in measurements and simulations or time‐dependent behavior of scalar quantities. Many structural properties of such fields can be explained by dependencies between different points in the scalar field. Although these dependencies can be of arbitrary complexity, correlation, i.e., the linear dependency, already provides significant structural information. Existing methods for correlation analysis are usually limited to positive correlation, handle only local dependencies, or use combinatorial approximations to this continuous problem. We present a new approach for computing and visualizing correlated regions in sets of 2‐dimensional scalar fields. This paper describes the following three main contributions: (i) An algorithm for hierarchical correlation clustering resulting in a dendrogram, (ii) a generalization of topological landscapes for dendrogram visualization, and (iii) a new method for incorporating negative correlation values in the clustering and visualization. All steps are designed to preserve the special properties of correlation coefficients. The results are visualized in two linked views, one showing the cluster hierarchy as 2D landscape and the other providing a spatial context in the scalar field's domain. Different coloring and texturing schemes coupled with interactive selection support an exploratory data analysis.

A predominance map expresses the predominant data category for each geographical entity and colors are used to differentiate a small number of data categories. In tag maps, many data categories are present in the form of different tags, but related tag map approaches do not account for predominance, as tags are either displaced from their respective geographical locations or visual clutter occurs. We propose predominance tag maps, a layout algorithm that accounts for predominance for arbitrary aggregation granularities. The algorithm is able to utilize the font sizes of the tags as visual variable and it is further configurable to implement aggregation strategies beyond visualizing predominance. We introduce various measures to evaluate numerically the qualitative aspects of tag maps regarding local predominance, global features, and layout stability and we comparatively analyze our method to the tag map approach by Thom et al. [1] on the basis of real world data sets.

Abstract In 2005, Franco Moretti introduced Distant Reading to analyse entire literary text collections. This was a rather revolutionary idea compared to the traditional Close Reading, which focuses on the thorough interpretation of an individual work. Both reading techniques are the prior means of Visual Text Analysis. We present an overview of the research conducted since 2005 on supporting text analysis tasks with close and distant reading visualizations in the digital humanities. Therefore, we classify the observed papers according to a taxonomy of text analysis tasks, categorize applied close and distant reading techniques to support the investigation of these tasks and illustrate approaches that combine both reading techniques in order to provide a multi‐faceted view of the textual data. In addition, we take a look at the used text sources and at the typical data transformation steps required for the proposed visualizations. Finally, we summarize collaboration experiences when developing visualizations for close and distant reading, and we give an outlook on future challenges in that research area.

Abstract This paper presents the state of the art in the area of topology‐based visualization. It describes the process and results of an extensive annotation for generating a definition and terminology for the field. The terminology enabled a typology for topological models which is used to organize research results and the state of the art. Our report discusses relations among topological models and for each model describes research results for the computation, simplification, visualization, and application. The paper identifies themes common to subfields, current frontiers, and unexplored territory in this research area.

Abstract Simulations and measurements often result in scalar fields with uncertainty due to errors or output sensitivity estimates. Methods for analyzing topological features of such fields usually are not capable of handling all aspects of the data. They either are not deterministic due to using Monte Carlo approaches, approximate the data with confidence intervals, or miss out on incorporating important properties, such as correlation. In this paper, we focus on the analysis of critical points of Gaussian‐distributed scalar fields. We introduce methods to deterministically extract critical points, approximate their probability with high precision, and even capture relations between them resulting in an abstract graph representation. Unlike many other methods, we incorporate all information contained in the data including global correlation. Our work therefore is a first step towards a reliable and complete description of topological features of Gaussian‐distributed scalar fields.

Determining similar objects based upon the features of an object of interest is a common task for visual analytics systems. This process is called profiling, if the object of interest is a person with individual attributes. The profiling of musicians similar to a musician of interest with the aid of visual means became an interesting research question for musicologists working with the Bavarian Musicians Encyclopedia Online. This paper illustrates the development of a visual analytics profiling system that is used to address such research questions. Taking musicological knowledge into account, we outline various steps of our collaborative digital humanities project, priority (1) the definition of various measures to determine the similarity of musicians' attributes, and (2) the design of an interactive profiling system that supports musicologists in iteratively determining similar musicians. The utility of the profiling system is emphasized by various usage scenarios illustrating current research questions in musicology.

The papers in this special section were presenteda at the 2015 IEEE Pacific Visualization Symposium (PacificVis’15) that was held in Hangzhou from April 14 to 17, 2015.

We generalize the framework of moments and introduce a definition of invariants for three-dimensional vector fields. To do so, we use the method of moment normalization that has been shown to be useful in the two dimensions. Using invariant moments, we show how to search for patterns in these fields independent from their position, orientation and scale. From the first order vector moment tensor, we construct a complete and independent set of descriptors. We test the invariants in queries on synthetic and real world flow fields.

Combinatorial optimization problems and corresponding (meta-)heuristics have received much attention in the literature. Especially, the structural or topological analysis of search landscapes is important for evaluating the applicability and the performance of search operators for a given problem. However, this analysis is often tedious and usually the focus is on one specific problem and only a few operators. We present a visual analysis method that can be applied to a wide variety of problems and search operators. The method is based on steepest descent walks and shortest distances in the search landscape. The visualization shows the search landscape as seen by the search algorithm. It supports the topological analysis as well as the comparison of search landscapes. We showcase the method by applying it to two different search operators on the TSP, the QAP, and the SMTTP. Our results show how differences between search operators manifest in the search landscapes and how conclusions about the suitability of the search operator for different optimizations can be drawn.

Hierarchical fiber clustering is a promising way to analyze brain connectivity. A disadvantage of hierarchical fiber clustering is its difficult visualization. The simple presentation as a 2D tree is visually too complex because of the amount of several thousand leaves. We present a framework that allows the modification of the dendrogram visualization in a flexible way. The modified dendrogram visualization can convey additional information that grants an easier orientation within the hierarchical clustering. Besides the interaction with the dendrogram itself, it is also possible to make use of a 3D view and a clustering preview. To illustrate potential use cases, we present two usage examples that show the versatility of our framework.

The analysis of 2D flow data is often guided by the search for characteristic structures with semantic meaning. One way to approach this question is to identify structures of interest by a human observer, with the goal of finding similar structures in the same or other datasets. The major challenges related to this task are to specify the notion of similarity and define respective pattern descriptors. While the descriptors should be invariant to certain transformations, such as rotation and scaling, they should provide a similarity measure with respect to other transformations, such as deformations. In this paper, we propose to use moment invariants as pattern descriptors for flow fields. Moment invariants are one of the most popular techniques for the description of objects in the field of image recognition. They have recently also been applied to identify 2D vector patterns limited to the directional properties of flow fields. Moreover, we discuss which transformations should be considered for the application to flow analysis. In contrast to previous work, we follow the intuitive approach of moment normalization, which results in a complete and independent set of translation, rotation, and scaling invariant flow field descriptors. They also allow to distinguish flow features with different velocity profiles. We apply the moment invariants in a pattern recognition algorithm to a real world dataset and show that the theoretical results can be extended to discrete functions in a robust way.

The analysis of 2D flow data is often guided by the search for characteristic structures with semantic meaning. One way to approach this question is to identify structures of interest by a human observer. The challenge then, is to find similar structures in the same or other datasets on different scales and orientations. In this paper, we propose to use moment invariants as pattern descriptors for flow fields. Moment invariants are one of the most popular techniques for the description of objects in the field of image recognition. They have recently also been applied to identify 2D vector patterns limited to the directional properties of flow fields. In contrast to previous work, we follow the intuitive approach of moment normalization, which results in a complete and independent set of translation, rotation, and scaling invariant flow field descriptors. They also allow to distinguish flow features with different velocity profiles. We apply the moment invariants in a pattern recognition algorithm to a real world dataset and show that the theoretic results can be extended to discrete functions in a robust way.

This paper is the result of a close collaboration of mechanical engineers and visualization researchers. It showcases how interdisciplinary work can lead to new insight and progress in both fields. Our case is concerned with one step in the product development process. Its goal is the design of mechanical parts that are functional, meet required quality measures and can be manufactured with standard production methods. The collaboration started with unspecific goals and first experiments with the available data and visualization methods. During the course of the collaboration many concrete questions arose and in the end a hypothesis was developed which will be discussed and evaluated in this paper. We facilitate a case study to validate our hypothesis. For the case study we consider the design of a reinforcement structure of a brake lever, a plastic ribbing. Three new lever geometries are developed on basis of our hypothesis and are compared against each other and against a reference model. The validation comprises standard numerical and experimental tests. In our case, all new structures outperform the reference geometry. The results are very promising and suggest potential to impact the product development process also for more complex scenarios.

In the last years, illustrative methods have found their way into flow visualization since they communicate difficult information in a comprehensible way. This is of great benefit especially in domains where the audience does not necessarily have flow expertise. One such domain is the medical field where the development of 4D MR imaging (for in-vivo 3D blood flow measurement) lead to an increased demand for easy flow analysis techniques. The goal and the challenge is to transfer the data into simple visualizations supporting the physician with flow interpretation and decision making. In this work, we take one step towards this goal. We present an approach for the illustrative visualization of steady flow features occurring in 4D MRI data of heart and aorta. Like shown in manually created illustrations, we restrict our visualization to the main data characteristics and do not depict every flow detail. The input for our method are flow features extracted from a dataset's complete set of streamlines with the help of line predicates. We create an abstract depiction of these line bundles by selecting a set of bundle representatives reflecting the most important flow aspects. These lines are rendered as three-dimensional arrows that are fused in areas where they represent the same flow. Since vortices are another important flow information for a physician, we identify these regions in the 4D MRI data and display them as unobtrusive, tube-like structures. A hatching texture provides for a visual effect of rotational blood flow. By applying our illustration technique to diverse flow structures of several 4D MRI datasets, we demonstrate that the abstract visualization is useful to gain an easier insight into the data. Feedback of medical experts confirmed the usefulness and revealed limitations of our work. The images are restricted to the essential flow features and, therefore, clearer and less cluttered. Our method has great potential and offers many possible applications, e.g., in comparative visualization and also beyond the medical domain.

Abstract How can the notion of topological structures for single scalar fields be extended to multifields? In this paper we propose a definition for such structures using the concepts of Pareto optimality and Pareto dominance. Given a set of piecewise‐linear, scalar functions over a common simplical complex of any dimension, our method finds regions of “consensus” among single fields’ critical points and their connectivity relations. We show that our concepts are useful to data analysis on real‐world examples originating from fluid‐flow simulations; in two cases where the consensus of multiple scalar vortex predictors is of interest and in another case where one predictor is studied under different simulation parameters. We also compare the properties of our approach with current alternatives.

Abstract Particle swarm optimization (PSO) is a metaheuristic that has been applied successfully to many continuous and combinatorial optimization problems, e.g., in the fields of economics, engineering, and natural sciences. In PSO, a swarm of particles moves within a search space in order to find an optimal solution. Unfortunately, it is hard to understand in detail why and how changes in the design of PSO algorithms affect the optimization behavior. Visualizing the particle states could provide substantially better insight into PSO algorithms. Though in case of combinatorial optimization problems, it often raises the problem of illustrating the states within the discrete search space that cannot be embedded spatially. We propose a visualization approach to depict the optimization problem topologically using a landscape metaphor. This visualization is augmented by an illustration of the time‐dependent states of the particles. Thus, the user of dPSO‐Vis is able to analyze the swarm's behavior within the search space. In principle, our method can be used for any optimization algorithm where a swarm of individuals searches within a discrete search space. Our approach is verified with a case study for the PSO algorithm HelixPSO that predicts the secondary structure of RNA molecules.

Four-dimensional MRI is an in vivo flow imaging modality that is expected to significantly enhance the understanding of cardiovascular diseases. Among other fields, 4D MRI provides valuable data for the research of cardiac blood flow and with that the development, diagnosis, and treatment of various cardiac pathologies. However, to gain insights from larger research studies or to apply 4D MRI in the clinical routine later on, analysis techniques become necessary that allow to robustly identify important flow characteristics without demanding too much time and expert knowledge. Heart muscle contractions and the particular complexity of the flow in the heart imply further challenges when analyzing cardiac blood flow. Working toward the goal of simplifying the analysis of 4D MRI heart data, we present a visual analysis method using line predicates. With line predicates precalculated integral lines are sorted into bundles with similar flow properties, such as velocity, vorticity, or flow paths. The user can combine the line predicates flexibly and by that carve out interesting flow features helping to gain overview. We applied our analysis technique to 4D MRI data of healthy and pathological hearts and present several flow aspects that could not be shown with current methods. Three 4D MRI experts gave feedback and confirmed the additional benefit of our method for their understanding of cardiac blood flow.

Rendering large numbers of dense line bundles in three dimensions is a common need for many visualization techniques, including streamlines and fiber tractography. Unfortunately, depiction of spatial relations inside these line bundles is often difficult but critical for understanding the represented structures. Many approaches evolved for solving this problem by providing special illumination models or tube-like renderings. Although these methods improve spatial perception of individual lines or related sets of lines, they do not solve the problem for complex spatial relations between dense bundles of lines. In this paper, we present a novel approach that improves spatial and structural perception of line renderings by providing a novel ambient occlusion approach suited for line rendering in real time.

Analyzing high-dimensional point clouds is a classical challenge in visual analytics. Traditional techniques, such as projections or axis-based techniques, suffer from projection artifacts, occlusion, and visual complexity. We propose to split data analysis into two parts to address these shortcomings. First, a structural overview phase abstracts data by its density distribution. This phase performs topological analysis to support accurate and nonoverlapping presentation of the high-dimensional cluster structure as a topological landscape profile. Utilizing a landscape metaphor, it presents clusters and their nesting as hills whose height, width, and shape reflect cluster coherence, size, and stability, respectively. A second local analysis phase utilizes this global structural knowledge to select individual clusters or point sets for further, localized data analysis. Focusing on structural entities significantly reduces visual clutter in established geometric visualizations and permits a clearer, more thorough data analysis. This analysis complements the global topological perspective and enables the user to study subspaces or geometric properties, such as shape.

4D MRI is an in vivo flow imaging modality which has the potential to significantly enhance diagnostics and therapy of cardiovascular diseases. However, current analysis methods demand too much time and expert knowledge in order to apply 4D MRI in the clinics or larger clinical studies. One missing piece are methods allowing to gain a quick overview of the flow data's main properties. We present a line predicate approach that sorts precalculated integral lines, which capture the complete flow dynamics, into bundles with similar properties. We introduce several streamline and pathline predicates that allow to structure the flow according to various features useful for blood flow analysis, such as, e.g., velocity distribution, vortices, and flow paths. The user can combine these predicates flexibly and by that create flow structures that help to gain overview and carve out special features of the current dataset. We show the usefulness of our approach by means of a detailed discussion of 4D MRI datasets of healthy and pathological aortas.

Existing methods for analyzing separation of streamlines are often restricted to a finite time or a local area. In our paper we introduce a new method that complements them by allowing an infinite-time-evaluation of steady planar vector fields. Our algorithm unifies combinatorial and probabilistic methods and introduces the concept of separation in time-discrete Markov-Chains. We compute particle distributions instead of the streamlines of single particles. We encode the flow into a map and then into a transition matrix for each time direction. Finally, we compare the results of our grid-independent algorithm to the popular Finite-Time-Lyapunov-Exponents and discuss the discrepancies.

In many fields of science or engineering, we are confronted with uncertain data. For that reason, the visualization of uncertainty received a lot of attention, especially in recent years. In the majority of cases, Gaussian distributions are used to describe uncertain behavior, because they are able to model many phenomena encountered in science. Therefore, in most applications uncertain data is (or is assumed to be) Gaussian distributed. If such uncertain data is given on fixed positions, the question of interpolation arises for many visualization approaches. In this paper, we analyze the effects of the usual linear interpolation schemes for visualization of Gaussian distributed data. In addition, we demonstrate that methods known in geostatistics and machine learning have favorable properties for visualization purposes in this case.

We developed a simulation tool for investigating the evolution of early metabolism, allowing us to speculate on the formation of metabolic pathways from catalyzed chemical reactions and on the development of their characteristic properties. Our model consists of a protocellular entity with a simple RNA-based genetic system and an evolving metabolism of catalytically active ribozymes that manipulate a rich underlying chemistry. Ensuring an almost open-ended and fairly realistic simulation is crucial for understanding the first steps in metabolic evolution. We show here how our simulation tool can be helpful in arguing for or against hypotheses on the evolution of metabolic pathways. We demonstrate that seemingly mutually exclusive hypotheses may well be compatible when we take into account that different processes dominate different phases in the evolution of a metabolic system. Our results suggest that forward evolution shapes metabolic network in the very early steps of evolution. In later and more complex stages, enzyme recruitment supersedes forward evolution, keeping a core set of pathways from the early phase.

The contour tree compactly describes scalar field topology. From the viewpoint of graph drawing, it is a tree with attributes at vertices and optionally on edges. Standard tree drawing algorithms emphasize structural properties of the tree and neglect the attributes. Applying known techniques to convey this information proves hard and sometimes even impossible. We present several adaptions of popular graph drawing approaches to the problem of contour tree drawing and evaluate them. We identify five esthetic criteria for drawing contour trees and present a novel algorithm for drawing contour trees in the plane that satisfies four of these criteria. Our implementation is fast and effective for contour tree sizes usually used in interactive systems (around 100 branches) and also produces readable pictures for larger trees, as is shown for an 800 branch example.

We present a novel method to visualize multidimensional point clouds. While conventional visualization techniques, like scatterplot matrices or parallel coordinates, have issues with either overplotting of entities or handling many dimensions, we abstract the data using topological methods before presenting it. We assume the input points to be samples of a random variable with a high-dimensional probability distribution which we approximate using kernel density estimates on a suitably reconstructed mesh. From the resulting scalar field we extract the join tree and present it as a topological landscape, a visualization metaphor that utilizes the human capability of understanding natural terrains. In this landscape, dense clusters of points show up as hills. The nesting of hills indicates the nesting of clusters. We augment the landscape with the data points to allow selection and inspection of single points and point sets. We also present optimizations to make our algorithm applicable to large data sets and to allow interactive adaption of our visualization to the kernel window width used in the density estimation.

In this paper, we present a novel three-stage process to visualize the structure of point clouds in arbitrary dimensions. To get insight into the structure and complexity of a data set, we would most preferably just look into it, e.g. by plotting its corresponding point cloud. Unfortunately, for orthogonal scatter plots, this only works up to three dimensions, and other visualizations, like parallel coordinates or scatterplot matrices, also have problems handling many dimensions and visual overlap of data entities. The presented solution tackles the problem of visualizing point clouds indirectly by visualizing the topology of their density distribution. The benefit of this approach is that this topology can be computed in arbitrary dimensions. Similar to examining scatter plots, this gives the important information like the number, size and nesting structure of accumulated regions. We view our approach as an alternative to cluster visualization. To create the visualization, we first estimate the density function using a novel high-dimensional interpolation scheme. Second, we compute that function's topology by means of the join tree, generate a corresponding 3-D terrain using the topological landscape metaphor introduced by Weber et al. (2007), and finally augment that landscape by placing the original data points at suitable locations.

Abstract Automatic detection of relevant structures in scientific data sets is still one of the big challenges in visualization. Techniques based on information theory have shown to be a promising direction to automatically highlight interesting subsets of a time‐dependent data set. The methods that have been proposed so far, however, were restricted to the Eulerian view. In the Eulerian description of motion, a position fixed in space is observed over time. In fluid dynamics, however, not only the site‐specific analysis of the flow is of interest, but also the temporal evolution of particles that are advected through the domain by the flow. This second description of motion is called the Lagrangian perspective. To support these two different frames of reference widely used in CFD research, we extend the notion of local statistical complexity (LSC) to make them applicable to Lagrangian and Eulerian flow descriptions. Thus, coherent structures can be identified by highlighting positions that either feature unusual temporal dynamics at a fixed position or that hold a particle that experiences such dynamics while passing through the position. A new area of application is opened by LagrangianLSC, which can be applied to short pathlines running through each position in the data set, as well as to individual pathlines computed for longer time intervals. Coloring the pathline according to the local complexity helps to detect extraordinary dynamics while the particle passes through the domain. The two techniques are explained and compared using different fluid flow examples.

Abstract We present an algorithm that allows stream surfaces to recognize and adapt to vector field topology. Standard stream surface algorithms either refine the surface uncontrolled near critical points which slows down the computation considerably and may lead to a poor surface approximation. Alternatively, the concerned region is omitted from the stream surface by severing it into two parts thus generating an incomplete stream surface. Our algorithm utilizes topological information to provide a fast, accurate, and complete triangulation of the stream surface near critical points. The required topological information is calculated in a preprocessing step. We compare our algorithm against the standard approach both visually and in performance.

Stream surfaces are an intuitive approach to represent 3D vector fields. In many cases, however, they are challenging objects to visualize and to understand, due to a high degree of self-occlusion. Despite the need for adequate rendering methods, little work has been done so far in this important research area. In this paper, we present an illustrative rendering strategy for stream surfaces. In our approach, we apply various rendering techniques, which are inspired by the traditional flow illustrations drawn by Dallmann and Abraham \& Shaw in the early 1980s. Among these techniques are contour lines and halftoning to show the overall surface shape. Flow direction as well as singularities on the stream surface are depicted by illustrative surface streamlines. ;To go beyond reproducing static text book images, we provide several interaction features, such as movable cuts and slabs allowing an interactive exploration of the flow and insights into subjacent structures, e.g., the inner windings of vortex breakdown bubbles. These methods take only the parameterized stream surface as input, require no further preprocessing, and can be freely combined by the user. We explain the design, GPU-implementation, and combination of the different illustrative rendering and interaction methods and demonstrate the potential of our approach by applying it to stream surfaces from various flow simulations. ;

Graphs are a versatile structure and abstraction for binary relationships between objects. To gain insight into such relationships, their corresponding graph can be visualized. In the past, many classes of graphs have been defined, e.g. trees, planar graphs, directed acyclic graphs, and visualization algorithms were proposed for these classes. Although many graphs may only be classified as "general" graphs, they can contain substructures that belong to a certain class. Archambault proposed the TopoLayout framework: rather than draw any arbitrary graph using one method, split the graph into components that are homogeneous with respect to one graph class and then draw each component with an algorithm best suited for this class. Graph products constitute a class that arises frequently in graph theory, but for which no visualization algorithm has been proposed until now. In this paper, we present an algorithm for drawing graph products and the aesthetic criterion graph product's drawings are subject to. We show that the popular High-Dimensional Embedder approach applied to cartesian products already respects this aestetic criterion, but has disadvantages. We also present how our method is integrated as a new component into the TopoLayout framework. Our implementation is used for further research of graph products in a biological context.

Abstract We introduce a novel technique for the construction of smooth stream surfaces of 4th order precision. While common stream surface techniques use linear interpolation for generating seed points for new streamlines in the refinement phase, we use Hermite interpolation. The derivatives needed for Hermite interpolation are obtained by integration along the streamlines. This yields stream surfaces of4th order precision. Additionally, we analyse the accuracy ofthe well known Hultquist approach and our new algorithm and proof that Hultquist's method is exact for linear vector fields. We compare both methods using the well known distance based and a novel error based refinement strategy. Our resulting surface is C1‐continuous, enabling improved rendering among other benefits.

Due to its nonlinear nature, the climate system shows quite high natural variability on different time scales, including multiyear oscillations such as the El Niño Southern Oscillation phenomenon. Beside a shift of the mean states and of extreme values of climate variables, climate change may also change the frequency or the spatial patterns of these natural climate variations. Wavelet analysis is a well established tool to investigate variability in the frequency domain. However, due to the size and complexity of the analysis results, only few time series are commonly analyzed concurrently. In this paper we will explore different techniques to visually assist the user in the analysis of variability and variability changes to allow for a holistic analysis of a global climate model data set consisting of several variables and extending over 250 years. Our new framework and data from the IPCC AR4 simulations with the coupled climate model ECHAM5/MPI-OM are used to explore the temporal evolution of El Niño due to climate change.

Abstract The notions of Finite‐Time Lyapunov Exponent (FTLE) and Lagrangian Coherent Structures provide a strong framework for the analysis and visualization of complex technical flows. Their definition is simple and intuitive, and they are built on a deep theoretical foundation. We apply these concepts to enable the analysis of flows in the immediate vicinity of the boundaries of flow‐embedded objects by limiting the Lagrangian analysis to surfaces closely neighboring these boundaries. To this purpose, we present an approach to approximate FTLE fields over such surfaces. Furthermore, we achieve an effective depiction of boundary‐related flow structures such as separation and attachment over object boundaries and specific insight into the surrounding flow using several specifically chosen visualization techniques. We document the applicability of our methods by presenting a number of application examples.

Abstract Current unsteady multi‐field simulation data‐sets consist of millions of data‐points. To efficiently reduce this enormous amount of information, local statistical complexity was recently introduced as a method that identifies distinctive structures using concepts from information theory. Due to high computational costs this method was so far limited to 2D data. In this paper we propose a new strategy for the computation that is substantially faster and allows for a more precise analysis. The bottleneck of the original method is the division of spatio‐temporal configurations in the field (light‐cones) into different classes of behavior. The new algorithm uses a density‐driven Voronoi tessellation for this task that more accurately captures the distribution of configurations in the sparsely sampled high‐dimensional space. The efficient computation is achieved using structures and algorithms from graph theory. The ability of the method to detect distinctive regions in 3D is illustrated using flow and weather simulations.

The visualization and exploration of multivariate data is still a challenging task. Methods either try to visualize all variables simultaneously at each position using glyph-based approaches or use linked views for the interaction between attribute space and physical domain such as brushing of scatterplots. Most visualizations of the attribute space are either difficult to understand or suffer from visual clutter. We propose a transformation of the high-dimensional data in attribute space to 2D that results in a point cloud, called attribute cloud, such that points with similar multivariate attributes are located close to each other. The transformation is based on ideas from multivariate density estimation and manifold learning. The resulting attribute cloud is an easy to understand visualization of multivariate data in two dimensions. We explain several techniques to incorporate additional information into the attribute cloud, that help the user get a better understanding of multivariate data. Using different examples from fluid dynamics and climate simulation, we show how brushing can be used to explore the attribute cloud and find interesting structures in physical space.

Understanding fluid flow data, especially vortices, is still a challenging task. Sophisticated visualization tools help to gain insight. In this paper, we present a novel approach for the interactive comparison of scalar fields using isosurfaces, and its application to fluid flow datasets. Features in two scalar fields are defined by largest contour segmentation after topological simplification. These features are matched using a volumetric similarity measure based on spatial overlap of individual features. The relationships defined by this similarity measure are ranked and presented in a thumbnail gallery of feature pairs and a graph representation showing all relationships between individual contours. Additionally, linked views of the contour trees are provided to ease navigation. The main render view shows the selected features overlapping each other. Thus, by displaying individual features and their relationships in a structured fashion, we enable exploratory visualization of correlations between similar structures in two scalar fields. We demonstrate the utility of our approach by applying it to a number of complex fluid flow datasets, where the emphasis is put on the comparison of vortex related scalar quantities.

The huge amount of different automatic clustering methods emphasizes one thing: there is no optimal clustering method for all possible cases. In certain application domains, like genomics and natural language processing, it is not even clear if any of the already known clustering methods suffice. In such cases, an automatic clustering method is often followed by manual refinement. The refined version may then be used as either an illustration, a reference, or even an input for a rule based or other machine learning algorithm as a new clustering method. In this paper, we describe a novel interaction technique to manual cluster refinement using the metaphor of soap bubbles, represented by special implicit surfaces (blobs). For instance, entities can simply be moved inside and outside of these blobs. A modified force-directed layout process automatically arranges entities equidistant on the screen. The modifications include a reduction to the expected amount of computation per iteration down to O(|V| log |V|+|E|) in order to achieve a high response time for use in an interactive system. We also spend a considerable amount of effort making the display of blobs fast enough for an interactive system.

Modern unsteady (multi-)field visualizations require an effective reduction of the data to be displayed. From a huge amount of information the most informative parts have to be extracted. Instead of the fuzzy application dependent notion of feature, a new approach based on information theoretic concepts is introduced in this paper to detect important regions. This is accomplished by extending the concept of local statistical complexity from finite state cellular automata to discretized (multi-)fields. Thus, informative parts of the data can be highlighted in an application-independent, purely mathematical sense. The new measure can be applied to unsteady multifields on regular grids in any application domain. The ability to detect and visualize important parts is demonstrated using diffusion, flow, and weather simulations.

We present, extend, and apply a method to extract the contribution of a subregion of a data set to the global flow. To isolate this contribution, we decompose the flow in the subregion into a potential flow that is induced by the original flow on the boundary and a localized flow. The localized flow is obtained by subtracting the potential flow from the original flow. Since the potential flow is free of both divergence and rotation, the localized flow retains the original features and captures the region-specific flow that contains the local contribution of the considered subdomain to the global flow. In the remainder of the paper, we describe an implementation on unstructured grids in both two and three dimensions for steady and unsteady flow fields. We discuss the application of some widely used feature extraction methods on the localized flow and describe applications like reverse-flow detection using the potential flow. Finally, we show that our algorithm is robust and scalable by applying it to various flow data sets and giving performance figures.

We present a method to extract and visualize vortices that originate from bounding walls of three-dimensional time-dependent flows. These vortices can be detected using their footprint on the boundary, which consists of critical points in the wall shear stress vector field. In order to follow these critical points and detect their transformations, affected regions of the surface are parameterized. Thus, an existing singularity tracking algorithm devised for planar settings can be applied. The trajectories of the singularities are used as a basis for seeding particles. This leads to a new type of streak line visualization, in which particles are released from a moving source. These generalized streak lines visualize the particles that are ejected from the wall. We demonstrate the usefulness of our method on several transient fluid flow datasets from computational fluid dynamics simulations.

Analysis of the spatial structure of proteins including folding processes is a challenge for modern bioinformatics. Due to limited experimental access to folding processes, computer simulations are a standard approach. Since realistic continuous (all-atom) simulations are far too expensive, lattice based protein folding simulations are a common coarse-graining. In this paper, we present a visualization tool for lattice based protein folding simulations. The system is based on Shneiderman’s mantra "Overview first, zoom and filter, details on demand" and uses a collection of information visualization techniques including multiple views, focus+context and table lenses which have been tailored towards our data. We demonstrate the potential of information visualization techniques for providing insight into such simulations.

Due to the amount of data nowadays, automatic detection, classification and visualization of features is necessary for a thorough inspection of flow data sets. Pattern matching using vector valued templates has already been applied successfully for the detection of features. In this paper, the approach is extended to automatically compute feature based segmentations of flow data sets. Different problems of the segmentation like the influence of thresholds, overlapping features, and classification errors are discussed. Visualizations of the segmentation display important structures of the flow and highlight the interesting features. The segmentation algorithm presented in this paper is applicable to 2D and 3D vector fields as well as to time-dependent data.

Dynamical models that explain the formation of spatial structures of RNA molecules have reached a complexity that requires novel visualization methods that help to analyze the validity of these models. Here, we focus on the visualization of so-called folding landscapes of a growing RNA molecule. Folding landscapes describe the energy of a molecule as a function of its spatial configuration; thus they are huge and high dimensional. Their most salient features, however, are encapsulated by their so-called barrier tree that reflects the local minima and their connecting saddle points. For each length of the growing RNA chain there exists a folding landscape. We visualize the sequence of folding landscapes by an animation of the corresponding barrier trees. To generate the animation, we adapt the foresight layout with tolerance algorithm for general dynamic graph layout problems. Since it is very general, we give a detailed description of each phase: constructing a supergraph for the trees, layout of that supergraph using a modified DoT algorithm, and presentation techniques for the final animation.

Predicates are functions that return Boolean values. They are an essential tool in computer science. A close look at flow feature definitions reveals that they can be seen as point predicates that tell if a specific feature exists at a certain point. Besides the information about features, scientists and engineers like to know the overall behavior of all streamlines in the flow, typically in the connection with the important features in their application domain. We call this a structure definition for the flow. A successful example for a structure definition is flow topology. In this paper, we present streamline predicates as functions that tell the user about the connection between streamlines and features selected by the user. This means answers to questions like: Which streamlines flow through a given vortex, separation bubble, or shock wave? It can be shown that streamline predicates may refine flow topology so that it also reveals questions about vortices in 3D.

In this paper, three-component particle image velocimetry (3-C PIV) measurements within the wake of a helicopter rotor from the HART II test are analyzed. These PIV-images are quite a challenge as the noise due to the measurement method and the inherent turbulence of the flow can not be distinguished. Furthermore, features are often hidden by a mean flow, which is influenced by vortices and therefore not easy to determine. The authors present some image processing methods adapted to these vector fields for the computation of position, size, and direction of the vortices in this data. These methods are quite robust in terms of noise and independent of any mean flow and therefore appropriate for this analysis. The results of the analysis allow a more descriptive and intuitive visualization of the vortices.

In this paper we present an approach to the analysis of the contribution of a small subregion in a dataset to the global flow. To this purpose, we subtract the potential flow that is induced by the boundary of the sub-domain from the original flow. Since the potential flow is free of both divergence and rotation, the localized flow field retains the original features. In contrast to similar approaches, by making explicit use of the boundary flow of the subregion, we manage to isolate the region-specific flow that contains exactly the local contribution of the considered subdomain to the global flow. In the remainder of the paper, we describe an implementation on unstructured grids in both two and three dimensions. We discuss the application of several widely used feature extraction methods on the localized flow, with an emphasis on topological schemes.

Tensors occur in many areas of science and engineering. Especially, they are used to describe charge, mass and energy transport (i.e. electrical conductivity tensor, diffusion tensor, thermal conduction tensor resp.) If the locale transport pattern is complicated, usual second order tensor representation is not sufficient. So far, there are no appropriate visualization methods for this case. We point out similarities of symmetric higher order tensors and spherical harmonics. A spherical harmonic representation is used to improve tensor glyphs. This paper unites the definition of streamlines and tensor lines and generalizes tensor lines to those applications where second order tensors representations fail. The algorithm is tested on the tractography problem in diffusion tensor magnetic resonance imaging (DT-MRI) and improved for this special application.

It is a challenging task to visualize the behavior of time-dependent 3D vector fields. Most of the time an overview of unsteady fields is provided via animations, but, unfortunately, animations provide only transient impressions of momentary flow. In this paper we present two approaches to visualize time varying fields with fixed geometry. Path lines and streak lines represent such a steady visualization of unsteady vector fields, but because of occlusion and visual clutter it is useless to draw them all over the spatial domain. A selection is needed. We show how bundles of streak lines and path lines, running at different times through one point in space, like through an eyelet, yield an insightful visualization of flow structure ("eyelet lines"). To provide a more intuitive and appealing visualization we also explain how to construct a surface from these lines. As second approach, we use a simple measurement of local changes of a field over time to determine regions with strong changes. We visualize these regions with isosurfaces to give an overview of the activity in the dataset. Finally we use the regions as a guide for placing eyelets.

Image processing and computer vision have robust methods for feature extraction and the computation of derivatives of scalar fields. Furthermore, interpolation and the effects of applying a filter can be analyzed in detail and can be advantages when applying these methods to vector fields to obtain a solid theoretical basis for feature extraction. We recently introduced the Clifford convolution, which is an extension of the classical convolution on scalar fields and provides a unified notation for the convolution of scalar and vector fields. It has attractive geometric properties that allow pattern matching on vector fields. In image processing, the convolution and the Fourier transform operators are closely related by the convolution theorem and, in this paper, we extend the Fourier transform to include general elements of Clifford Algebra, called multivectors, including scalars and vectors. The resulting convolution and derivative theorems are extensions of those for convolution and the Fourier transform on scalar fields. The Clifford Fourier transform allows a frequency analysis of vector fields and the behavior of vector-valued filters. In frequency space, vectors are transformed into general multivectors of the Clifford Algebra. Many basic vector-valued patterns, such as source, sink, saddle points, and potential vortices, can be described by a few multivectors in frequency space.

We present an approach for monitoring the positions of vector field singularities and related structural changes in time-dependent datasets. The concept of singularity index is discussed and extended from the well-understood planar case to the more intricate three-dimensional setting. Assuming a tetrahedral grid with linear interpolation in space and time, vector field singularities obey rules imposed by fundamental invariants (Poincare index), which we use as a basis for an efficient tracking algorithm. We apply the presented algorithm to CFD datasets to illustrate its purpose. We examine structures that exhibit topological variations with time and describe some of the insight gained with our method. Examples are given that show a correlation in the evolution of physical quantities that play a role in vortex breakdown.

Vortex breakdowns and flow recirculation are essential phenomena in aeronautics where they appear as a limiting factor in the design of modern aircrafts. Because of the inherent intricacy of these features, standard flow visualization techniques typically yield cluttered depictions. The paper addresses the challenges raised by the visual exploration and validation of two CFD simulations involving vortex breakdown. To permit accurate and insightful visualization we propose a new approach that unfolds the geometry of the breakdown region by letting a plane travel through the structure along a curve. We track the continuous evolution of the associated projected vector field using the theoretical framework of parametric topology. To improve the understanding of the spatial relationship between the resulting curves and lines we use direct volume rendering and multidimensional transfer functions for the display of flow-derived scalar quantities. This enriches the visualization and provides an intuitive context for the extracted topological information. Our results offer clear, synthetic depictions that permit new insight into the structural properties of vortex breakdowns.

We present a new method for topological segmentation in steady three-dimensional vector fields. Depending on desired properties, the algorithm replaces the original vector field by a derived segmented data set, which is utilized to produce separating surfaces in the vector field. We define the concept of a segmented data set, develop methods that produce the segmented data by sampling the vector field with streamlines, and describe algorithms that generate the separating surfaces. This method is applied to generate local separatrices in the field, defined by a movable boundary region placed in the field. The resulting partitions can be visualized using standard techniques for a visualization of a vector field at a higher level of abstraction.

The goal of this paper is to define a convolution operation which transfers image processing and pattern matching to vector fields from flow visualization. For this, a multiplication of vectors is necessary. Clifford algebra provides such a multiplication of vectors. We define a Clifford convolution on vector fields with uniform grids. The Clifford convolution works with multivector filter masks. Scalar and vector masks can be easily converted to multivector fields. So, filter masks from image processing on scalar fields can be applied as well as vector and scalar masks. Furthermore, a method for pattern matching with Clifford convolution on vector fields is described. The method is independent of the direction of the structures. This provides an automatic approach to feature detection. The features can be visualized using any known method like glyphs, isosurfaces or streamlines. The features are defined by filter masks instead of analytical properties and thus the approach is more intuitive.

Isosurfaces are commonly used to visualize scalar fields. Critical isovalues indicate isosurface topology changes: the creation of new surface components, merging of surface components or the formation of holes in a surface component. Therefore, they highlight interesting isosurface behavior and are helpful in exploration of large trivariate data sets. We present a method that detects critical isovalues in a scalar field defined by piecewise trilinear interpolation over a rectilinear grid and describe how to use them when examining volume data. We further review varieties of the marching cubes (MC) algorithm, with the intention of preserving topology of the trilinear interpolant when extracting an isosurface. We combine and extend two approaches in such a way that it is possible to extract meaningful isosurfaces even when a critical value is chosen as the isovalue.

This paper presents a new algorithm for the calculation of stream surfaces for tetrahedral grids. It propagates the surface through the tetrahedra, one at a time, calculating the intersections with the tetrahedral faces. The method allows us to incorporate topological information from the cells, e.g. critical points. The calculations are based on barycentric coordinates, since this simplifies the theory and the algorithm. The stream surfaces are ruled surfaces inside each cell, and their construction starts with line segments on the faces. Our method supports the analysis of velocity fields resulting from computational fluid dynamics (CFD) simulations.

Vector fields can present complex structural behavior, especially in turbulent computational fluid dynamics. The topological analysis of these data sets reduces the information, but one is usually still left with too many details for interpretation. In this paper, we present a simplification approach that removes pairs of critical points from the data set, based on relevance measures. In contrast to earlier methods, no grid changes are necessary, since the whole method uses small local changes of the vector values defining the vector field. An interpretation in terms of bifurcations underlines the continuous, natural flavor of the algorithm.

Topological methods produce simple and meaningful depictions of symmetric, second order two-dimensional tensor fields. Extending previous work dealing with vector fields, we propose here a scheme for the visualization of time-dependent tensor fields. Basic notions of unsteady tensor topology are discussed. Topological changes - known as bifurcations - are precisely detected and identified by our method which permits an accurate tracking of degenerate points and related structures.

The analysis and visualization of flows is a central problem in visualization. Topology-based methods have gained increasing interest in recent years. This article describes a method for the detection of closed streamlines in flows. It is based on a special treatment of cases where a streamline reenters a cell to prevent infinite cycling during streamline calculation. The algorithm checks for possible exits of a loop of crossed edges and detects structurally stable closed streamlines. These global features are not detected by conventional topology and feature detection algorithms.

Topology analysis of plane, turbulent vector fields results in visual clutter caused by critical points indicating vortices of finer and finer scales. A simplification can be achieved by merging critical points within a prescribed radius into higher order critical points. After building clusters containing the singularities to merge, the method generates a piecewise linear representation of the vector field in each cluster containing only one (higher order) singularity. Any visualization method can be applied to the result after this process. Using different maximal distances for the critical points to be merged results in a hierarchy of simplified vector fields that can be used for analysis on different scales.

We present a method for visualizing three dimensional vector fields which are defined on a two dimensional manifold only. These vector fields do exist in real application, as we show by an example of an optical measuring instrument which can gauge the displacement at the surface of a mechanical part. The general idea is to compute LIC textures in the manifold's tangent space and to deform the manifold according to the normal information. The resulting LIC texture is mapped onto the deformed manifold and is rendered as a three dimensional scene. Due to the light's reflection on the deformed manifold, one can interactively explore the result of the deformation.

An application of C/sup 1/ scalar interpolation for 2D vector field topology visualization is presented. Powell-Sabin and Nielson interpolants are considered which both make use of Nielson's Minimum Norm Network for the precomputation of the derivatives in our implementation. A comparison of their results to the commonly used linear interpolant underlines their significant improvement of singularity location and topological skeleton depiction. Evaluation is based upon the processing of polynomial vector fields with known topology containing higher order singularities.

The article deals with dependencies of a piecewise linear vector field and the triangulation of the domain. It shows that the topology of the field may depend on the triangulation and gives a suitable choice to obtain a simple topology by changes of the triangulation. The main point is the appearance of pairs of critical points with positive and negative Poincare index. Many of these occurrences can be avoided by changing the grid. This is proved in the article. An algorithm is presented which uses these results to extract simpler topological skeletons than usual methods. Finally there are several examples comparing these algorithms to demonstrate the effects of the data dependent triangulation.

We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorithms for vector field topology are based on piecewise linear or bilinear approximation, and that these methods destroy the local topology if nonlinear behavior is present. Our algorithm looks for such situations, chooses an appropriate polynomial approximation in these areas, and, finally, visualizes the topology. This overcomes the problem, and the algorithm is still very fast because we are using linear approximation outside these small but important areas. The paper contains a detailed description of the algorithm and a basic introduction to Clifford algebra.

Presents an algorithm for the visualization of vector field topology based on Clifford algebra. It allows the detection of higher-order singularities. This is accomplished by first analysing the possible critical points and then choosing a suitable polynomial approximation, because conventional methods based on piecewise linear or bilinear approximation do not allow higher-order critical points and destroy the topology in such cases. The algorithm is still very fast, because of using linear approximation outside the areas with several critical points.