Monitoring chemical processes is essential to prevent catastrophic failures, optimize costs and profits, and ensure the safety of employees and the environment. A key component of modern monitoring systems is the automated detection of anomalies in sensor data over time, called time series, enabling partial automation of plant operation and adding additional layers of supervision to crucial components. The development of anomaly detection methods in this domain is challenging, since real chemical process data are usually proprietary, and simulated data are generally not a sufficient replacement. In this paper, we present NoBOOM, the first collection of datasets for anomaly detection in real-world chemical process data, including labeled data from a running process at our industry partner BASF SE — one of the world’s leading chemical companies — and several chemical processes run in laboratory‑scale and pilot‑scale plants. While we are not able to share every detail about the industrial process, for the laboratory‑ and pilot‑scale plants, we provide comprehensive information on plant configuration, process operation, and, in particular, anomaly events, enabling a differentiated analysis of anomaly detection methods. To demonstrate the complexity of the benchmark, we analyze the data with regard to common issues of time-series anomaly detection (TSAD) benchmarks, including potential triviality and bias.

In this paper, we present a novel heuristic algorithm for the stable but NP-complete deformation-based edit distance on merge trees. Our key contribution is the introduction of a user-controlled look-ahead parameter that allows to trade off accuracy and computational cost. We achieve a fixed parameter tractable running time that is polynomial in the size of the input but exponential in the look-ahead value. This extension unlocks the potential of the deformation-based edit distance in handling saddle swaps, while maintaining feasible computation times. Experimental results demonstrate the computational efficiency and effectiveness of this approach in handling specific perturbations.

Abstract Machine learning algorithms are widely applied to create powerful prediction models. With increasingly complex models, humans' ability to understand the decision function (that maps from a high‐dimensional input space) is quickly exceeded. To explain a model's decisions, black‐box methods have been proposed that provide either non‐linear maps of the global topology of the decision boundary, or samples that allow approximating it locally. The former loses information about distances in input space, while the latter only provides statements about given samples, but lacks a focus on the underlying model for precise ‘What‐If'‐reasoning. In this paper, we integrate both approaches and propose an interactive exploration method using local linear maps of the decision space. We create the maps on high‐dimensional hyperplanes—2D‐slices of the high‐dimensional parameter space—based on statistical and personal feature mutability and guided by feature importance. We complement the proposed workflow with established model inspection techniques to provide orientation and guidance. We demonstrate our approach on real‐world datasets and illustrate that it allows identification of instance‐based decision boundary structures and can answer multi‐dimensional ‘What‐If'‐questions, thereby identifying counterfactual scenarios visually.

Abstract Edit distances between merge trees of scalar fields have many applications in scientific visualization, such as ensemble analysis, feature tracking or symmetry detection. In this paper, we propose branch mappings, a novel approach to the construction of edit mappings for merge trees. Classic edit mappings match nodes or edges of two trees onto each other, and therefore have to either rely on branch decompositions of both trees or have to use auxiliary node properties to determine a matching. In contrast, branch mappings employ branch properties instead of node similarity information, and are independent of predetermined branch decompositions. Especially for topological features, which are typically based on branch properties, this allows a more intuitive distance measure which is also less susceptible to instabilities from small‐scale perturbations. For trees with 𝒪(n) nodes, we describe an 𝒪(n4) algorithm for computing optimal branch mappings, which is faster than the only other branch decomposition‐independent method in the literature by more than a linear factor. Furthermore, we compare the results of our method on synthetic and real‐world examples to demonstrate its practicality and utility.

We present Knowledge Rocks, an implementation strategy and guideline for augmenting visualization systems to knowledge-assisted visualization systems, as defined by the KAVA model. Visualization systems become more and more sophisticated. Hence, it is increasingly important to support users with an integrated knowledge base in making constructive choices and drawing the right conclusions. We support the effective reactivation of visualization software resources by augmenting them with knowledge-assistance. To provide a general and yet supportive implementation strategy, we propose an implementation process that bases on an application-agnostic architecture. This architecture is derived from existing knowledge-assisted visualization systems and the KAVA model. Its centerpiece is an ontology that is able to automatically analyze and classify input data, linked to a database to store classified instances. We discuss design decisions and advantages of the KR framework and illustrate its broad area of application in diverse integration possibilities of this architecture into an existing visualization system. In addition, we provide a detailed case study by augmenting an it-security system with knowledge-assistance facilities.

Embeddings of high-dimensional data are widely used to explore data, to verify analysis results, and to communicate information. Their explanation, in particular with respect to the input attributes, is often difficult. With linear projects like PCA the axes can still be annotated meaningfully. With non-linear projections this is no longer possible and alternative strategies such as attribute-based color coding are required. In this paper, we review existing augmentation techniques and discuss their limitations. We present the Non-Linear Embeddings Surveyor (NoLiES) that combines a novel augmentation strategy for projected data (rangesets) with interactive analysis in a small multiples setting. Rangesets use a set-based visualization approach for binned attribute values that enable the user to quickly observe structure and detect outliers. We detail the link between algebraic topology and rangesets and demonstrate the utility of NoLiES in case studies with various challenges (complex attribute value distribution, many attributes, many data points) and a real-world application to understand latent features of matrix completion in thermodynamics.

Abstract In order to discuss the kinds of reasoning a visualization supports and the conclusions that can be drawn within the analysis context, a theoretical framework is needed that enables a formal treatment of the reasoning process. Such a model needs to encompass three stages of the visualization pipeline: encoding, decoding and interpretation. The encoding details how data are transformed into a visualization and what can be seen in the visualization. The decoding explains how humans construct graphical contexts inside the depicted visualization and how they interpret them assigning meaning to displayed structures according to a formal reasoning strategy. In the presented model, we adapt and combine theories for the different steps into a unified formal framework such that the analysis process is modelled as an assignment of meaning to displayed structures according to a formal reasoning strategy. Additionally, we propose the ConceptGraph, a combined graph‐based representation of the finite‐state transducers resulting from the three stages, that can be used to formalize and understand the reasoning process. We apply the new model to several visualization types and investigate reasoning strategies for various tasks.

Operation technology networks, i.e. hard- and software used for monitoring and controlling physical/industrial processes, have been considered immune to cyber attacks for a long time. A recent increase of attacks in these networks proves this assumption wrong. Several technical constraints lead to approaches to detect attacks on industrial processes using available sensor data. This setting differs fundamentally from anomaly detection in IT-network traffic and requires new visualization approaches adapted to the common periodical behavior in OT-network data. We present a tailored visualization system that utilizes inherent features of measurements from industrial processes to full capacity to provide insight into the data and support triage analysis by laymen and experts. The novel combination of spiral plots with results from anomaly detection was implemented in an interactive system. The capabilities of our system are demonstrated using sensor and actuator data from a real-world water treatment process with introduced attacks. Exemplary analysis strategies are presented. Finally, we evaluate effectiveness and usability of our system and perform an expert evaluation.

This work describes an approach for the interactive visual analysis of large-scale simulations, where numerous superlevel set components and their evolution are of primary interest. The approach first derives, at simulation runtime, a specialized Cinema database that consists of images of component groups, and topological abstractions. This database is processed by a novel graph operation-based nested tracking graph algorithm (GO-NTG) that dynamically computes NTGs for component groups based on size, overlap, persistence, and level thresholds. The resulting NTGs are in turn used in a feature-centered visual analytics framework to query specific database elements and update feature parameters, facilitating flexible post hoc analysis.

Abstract In order to gain probabilistic results, ensemble simulation techniques are increasingly applied in the weather and climate sciences (as well as in various other scientific disciplines). In many cases, however, only mean results or other abstracted quantities such as percentiles are used for further analyses and dissemination of the data. In this work, we aim at a more detailed visualization of the temporal development of the whole ensemble that takes the variability of all single members into account. We propose a visual analytics tool that allows an effective analysis process based on a hierarchical clustering of the time‐dependent scalar fields. The system includes a flow chart that shows the ensemble members' cluster affiliation over time, reflecting the whole cluster hierarchy. The latter one can be dynamically explored using a visualization derived from a dendrogram. As an aid in linking the different views, we have developed an adaptive coloring scheme that takes into account cluster similarity and the containment relationships. Finally, standard visualizations of the involved field data (cluster means, ground truth data, etc.) are also incorporated. We include results of our work on real‐world datasets to showcase the utility of our approach.

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

Ensemble simulations are used in climate research to account for natural variability. For medium-term decadal predictions, each simulation run is initialized with real observations from a different day resulting in a set of possible climatic futures. Understanding the variability and the predictive power in this wealth of data is still a challenging task. In this paper, we introduce a visual analytics system to explore variability within ensembles of decadal climate predictions. We propose a new interactive visualization technique (clustering timeline) based on the Sankey diagram, which conveys a concise summary of data similarity and its changes over time. We augment the system with two additional visualizations, filled contour maps and heatmaps, to provide analysts with additional information relating the new diagram to raw data and automatic clustering results. The usefulness of the technique is demonstrated by case studies and user interviews.

Complex networks require effective tools and visualizations for their analysis and comparison. Clique communities have been recognized as a powerful concept for describing cohesive structures in networks. We propose an approach that extends the computation of clique communities by considering persistent homology, a topological paradigm originally introduced to characterize and compare the global structure of shapes. Our persistence-based algorithm is able to detect clique communities and to keep track of their evolution according to different edge weight thresholds. We use this information to define comparison metrics and a new centrality measure, both reflecting the relevance of the clique communities inherent to the network. Moreover, we propose an interactive visualization tool based on nested graphs that is capable of compactly representing the evolving relationships between communities for different thresholds and clique degrees. We demonstrate the effectiveness of our approach on various network types.

Abstract Tracking graphs are a well established tool in topological analysis to visualize the evolution of components and their properties over time, i.e., when components appear, disappear, merge, and split. However, tracking graphs are limited to a single level threshold and the graphs may vary substantially even under small changes to the threshold. To examine the evolution of features for varying levels, users have to compare multiple tracking graphs without a direct visual link between them. We propose a novel, interactive, nested graph visualization based on the fact that the tracked superlevel set components for different levels are related to each other through their nesting hierarchy. This approach allows us to set multiple tracking graphs in context to each other and enables users to effectively follow the evolution of components for different levels simultaneously. We demonstrate the effectiveness of our approach on datasets from finite pointset methods, computational fluid dynamics, and cosmology simulations.

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 Clustering algorithms support exploratory data analysis by grouping inputs that share similar features. Especially the clustering of unlabelled data is said to be a fiendishly difficult problem, because users not only have to choose a suitable clustering algorithm but also a suitable number of clusters. The known issues of existing clustering validity measures comprise instabilities in the presence of noise and restrictive assumptions about cluster shapes. In addition, they cannot evaluate individual clusters locally. We present a new measure for assessing and comparing different clusterings both on a global and on a local level. Our measure is based on the topological method of persistent homology, which is stable and unbiased towards cluster shapes. Based on our measure, we also describe a new visualization that displays similarities between different clusterings (using a global graph view) and supports their comparison on the individual cluster level (using a local glyph view). We demonstrate how our visualization helps detect different—but equally valid—clusterings of data sets from multiple application domains.

Animal development is marked by the repeated reorganization of cells and cell populations, which ultimately determine form and shape of the growing organism. One of the central questions in developmental biology is to understand precisely how cells reorganize, as well as how and to what extent this reorganization is coordinated. While modern microscopes can record video data for every cell during animal development in 3D+t, analyzing these videos remains a major challenge: reconstruction of comprehensive cell tracks turned out to be very demanding especially with decreasing data quality and increasing cell densities. In this paper, we present an analysis pipeline for coordinated cellular motions in developing embryos based on the optical flow of a series of 3D images. We use numerical integration to reconstruct cellular long-term motions in the optical flow of the video, we take care of data validation, and we derive a LIC-based, dense flow visualization for the resulting pathlines. This approach allows us to handle low video quality such as noisy data or poorly separated cells, and it allows the biologists to get a comprehensive understanding of their data by capturing dynamic growth processes in stills. We validate our methods using three videos of growing fruit fly embryos.

Abstract High‐dimensional data sets are a prevalent occurrence in many application domains. This data is commonly visualized using dimensionality reduction (DR) methods. DR methods provide e.g. a two‐dimensional embedding of the abstract data that retains relevant high‐dimensional characteristics such as local distances between data points. Since the amount of DR algorithms from which users may choose is steadily increasing, assessing their quality becomes more and more important. We present a novel technique to quantify and compare the quality of DR algorithms that is based on persistent homology. An inherent beneficial property of persistent homology is its robustness against noise which makes it well suited for real world data. Our pipeline informs about the best DR technique for a given data set and chosen metric (e.g. preservation of local distances) and provides knowledge about the local quality of an embedding, thereby helping users understand the shortcomings of the selected DR method. The utility of our method is demonstrated using application data from multiple domains and a variety of commonly used DR methods.

Abstract Topological and geometrical methods constitute common tools for the analysis of high‐dimensional scientific data sets. Geometrical methods such as projection algorithms focus on preserving distances in the data set. Topological methods such as contour trees, by contrast, focus on preserving structural and connectivity information. By combining both types of methods, we want to benefit from their individual advantages. To this end, we describe an algorithm that uses persistent homology to analyse the topology of a data set. Persistent homology identifies high‐dimensional holes in data sets, describing them as simplicial chains. We localize these chains using geometrical information of the data set, which we obtain from geodesic distances on a neighbourhood graph. The localized chains describe the structure of point clouds. We represent them using an interactive graph, in which each node describes a single chain and its geometrical properties. This graph yields a more intuitive understanding of multivariate point clouds and simplifies comparisons of time‐varying data. Our method focuses on detecting and analysing inhomogeneous regions, i.e. holes, in a data set because these regions characterize data in a different manner, thereby leading to new insights. We demonstrate the potential of our method on data sets from particle physics, political science and meteorology.

Currently, user centered transfer function design begins with the user interacting with a one or two-dimensional histogram of the volumetric attribute space. The attribute space is visualized as a function of the number of voxels, allowing the user to explore the data in terms of the attribute size/magnitude. However, such visualizations provide the user with no information on the relationship between various attribute spaces (e.g., density, temperature, pressure, x, y, z) within the multivariate data. In this work, we propose a modification to the attribute space visualization in which the user is no longer presented with the magnitude of the attribute; instead, the user is presented with an information metric detailing the relationship between attributes of the multivariate volumetric data. In this way, the user can guide their exploration based on the relationship between the attribute magnitude and user selected attribute information as opposed to being constrained by only visualizing the magnitude of the attribute. We refer to this modification to the traditional histogram widget as an abstract attribute space representation. Our system utilizes common one and two-dimensional histogram widgets where the bins of the abstract attribute space now correspond to an attribute relationship in terms of the mean, standard deviation, entropy, or skewness. In this manner, we exploit the relationships and correlations present in the underlying data with respect to the dimension(s) under examination. These relationships are often times key to insight and allow us to guide attribute discovery as opposed to automatic extraction schemes which try to calculate and extract distinct attributes a priori. In this way, our system aids in the knowledge discovery of the interaction of properties within volumetric data.

Abstract In recent years, a collection of new techniques which deal with video as input data, emerged in computer graphics and visualization. In this survey, we report the state of the art in video‐based graphics and video visualization. We provide a review of techniques for making photo‐realistic or artistic computer‐generated imagery from videos, as well as methods for creating summary and/or abstract visual representations to reveal important features and events in videos. We provide a new taxonomy to categorize the concepts and techniques in this newly emerged body of knowledge. To support this review, we also give a concise overview of the major advances in automated video analysis, as some techniques in this field (e.g. feature extraction, detection, tracking and so on) have been featured in video‐based modelling and rendering pipelines for graphics and visualization.

The extraction of significant structures in arbitrary high-dimensional data sets is a challenging task. Moreover, classifying data points as noise in order to reduce a data set bears special relevance for many application domains. Standard methods such as clustering serve to reduce problem complexity by providing the user with classes of similar entities. However, they usually do not highlight relations between different entities and require a stopping criterion, e.g. the number of clusters to be detected. In this paper, we present a visualization pipeline based on recent advancements in algebraic topology. More precisely, we employ methods from persistent homology that enable topological data analysis on high-dimensional data sets. Our pipeline inherently copes with noisy data and data sets of arbitrary dimensions. It extracts central structures of a data set in a hierarchical manner by using a persistence-based filtering algorithm that is theoretically well-founded. We furthermore introduce persistence rings, a novel visualization technique for a class of topological features-the persistence intervals-of large data sets. Persistence rings provide a unique topological signature of a data set, which helps in recognizing similarities. In addition, we provide interactive visualization techniques that assist the user in evaluating the parameter space of our method in order to extract relevant structures. We describe and evaluate our analysis pipeline by means of two very distinct classes of data sets: First, a class of synthetic data sets containing topological objects is employed to highlight the interaction capabilities of our method. Second, in order to affirm the utility of our technique, we analyse a class of high-dimensional real-world data sets arising from current research in cultural heritage.

Abstract We present a novel approach for the evaluation of 2D flow visualizations based on the visual reconstructability of the input vector fields. According to this metric, a visualization has high quality if the underlying data can be reliably reconstructed from the image. This approach provides visualization creators with a cost‐effective means to assess the quality of visualization results objectively. We present a vision‐based reconstruction system for the three most commonly‐used visual representations of vector fields, namely streamlines, arrow glyphs, and line integral convolution. To demonstrate the use of visual reconstructability as a quality metric, we consider a selection of vector fields obtained from numerical simulations, containing typical flow features. We apply the three types of visualization to each dataset, and compare the visualization results based on their visual reconstructability of the original vector field.

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 Sound is an integral part of most movies and videos. In many situations, viewers of a video are unable to hear the sound track, for example, when watching it in a fast forward mode, viewing it by hearing‐impaired viewers or when the plot is given as a storyboard. In this paper, we present an automated visualization solution to such problems. The system first detects the common components (such as music, speech, rain, explosions, and so on) from a sound track, then maps them to a collection of programmable visual metaphors, and generates a composite visualization. This form of sound visualization, which is referred to as SoundRiver, can be also used to augment various forms of video abstraction and annotated key frames and to enhance graphical user interfaces for video handling software. The SoundRiver conveys more semantic information to the viewer than traditional graphical representations of sound illustration, such as phonoautographs, spectrograms or artistic audiovisual animations.

Abstract Salience detection is a principle mechanism to facilitate visual attention. A good visualization guides the observer's attention to the relevant aspects of the representation. Hence, the distribution of salience over a visualization image is an essential measure of the quality of the visualization. We describe a method for computing such a metric for a visualization image in the context of a given dataset. We show how this technique can be used to analyze a visualization's salience, improve an existing visualization, and choose the best representation from a set of alternatives. The usefulness of this proposed metric is illustrated using examples from information visualization, volume visualization and flow visualization.

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.

Pixel-based visualization is a popular method of conveying large amounts of numerical data graphically. Application scenarios include business and finance, bioinformatics and remote sensing. In this work, we examined how the usability of such visual representations varied across different tasks and block resolutions. The main stimuli consisted of temporal pixel-based visualization with a white-red color map, simulating monthly temperature variation over a six-year period. In the first study, we included 5 separate tasks to exert different perceptual loads. We found that performance varied considerably as a function of task, ranging from 75% correct in low-load tasks to below 40% in high-load tasks. There was a small but consistent effect of resolution, with the uniform patch improving performance by around 6% relative to higher block resolution. In the second user study, we focused on a high-load task for evaluating month-to-month changes across different regions of the temperature range. We tested both CIE L*u*v* and RGB color spaces. We found that the nature of the change-evaluation errors related directly to the distance between the compared regions in the mapped color space. We were able to reduce such errors by using multiple color bands for the same data range. In a final study, we examined more fully the influence of block resolution on performance, and found block resolution had a limited impact on the effectiveness of pixel-based visualization.

In this paper, we examine whether or not information theory can be one of the theoretic frameworks for visualization. We formulate concepts and measurements for qualifying visual information. We illustrate these concepts with examples that manifest the intrinsic and implicit use of information theory in many existing visualization techniques. We outline the broad correlation between visualization and the major applications of information theory, while pointing out the difference in emphasis and some technical gaps. Our study provides compelling evidence that information theory can explain a significant number of phenomena or events in visualization, while no example has been found which is fundamentally in conflict with information theory. We also notice that the emphasis of some traditional applications of information theory, such as data compression or data communication, may not always suit visualization, as the former typically focuses on the efficient throughput of a communication channel, whilst the latter focuses on the effectiveness in aiding the perceptual and cognitive process for data understanding and knowledge discovery. These findings suggest that further theoretic developments are necessary for adopting and adapting information theory for visualization.

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 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.

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 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.