Dynamic Mode Decomposition (DMD) is a data-driven and model-free decomposition technique. It is suitable for revealing spatio-temporal features of both numerically and experimentally acquired data. Conceptually, DMD performs a low-dimensional spectral decomposition of the data into the following components: the modes, called DMD modes, encode the spatial contribution of the decomposition, whereas the DMD amplitudes specify their impact. Each associated eigenvalue, referred to as DMD eigenvalue, characterizes the frequency and growth rate of the DMD mode. In this paper, we demonstrate how the components of DMD can be utilized to obtain temporal and spatial information from time-dependent flow fields. We begin with the theoretical background of DMD and its application to unsteady flow. Next, we examine the conventional process with DMD mathematically and put it in relationship to the discrete Fourier transform. Our analysis shows that the current use of DMD components has several drawbacks. To resolve these problems we adjust the components and provide new and meaningful insights into the decomposition: we show that our improved components capture the spatio-temporal patterns of the flow better. Moreover, we remove redundancies in the decomposition and clarify the interplay between components, allowing users to understand the impact of components. These new representations, which respect the spatio-temporal character of DMD, enable two clustering methods that segment the flow into physically relevant sections and can therefore be used for the selection of DMD components. With a number of typical examples, we demonstrate that the combination of these techniques allows new insights with DMD for unsteady flow.

We present time-aligned edge plots: time- and edge-scalable representations of dynamic graphs. Vertices are mapped to two vertical parallel axes. The left axis depicts the source vertices, whereas the right one depicts the destination vertices. The time axis is horizontally embedded in-between the two axes, resulting in a two-dimensional graph layout. Edges are added by drawing straight lines connecting the corresponding source and destination vertices through time, while the pixels along the lines are used to encode the time-varying information. In this way, the depiction of edges at the individual timepoints is reduced to only a few pixels, resulting in a less cluttered representation of dynamic graphs, while the alignment of edges over time reveals the temporal patterns in the data and preserves the users' mental map. We evaluate our approach by comparing it theoretically and empirically against the state-of-the-art using dynamic graphs of varying complexities.

We present an approach for interactively analyzing large dynamic graphs consisting of several thousand time steps with a particular focus on temporal aspects. we employ a static representation of the time-varying graph based on the concept of space-time cubes, i.e., we create a volumetric representation of the graph by stacking the adjacency matrices of each of its time steps. To achieve an efficient analysis of complex data, we discuss three classes of analytics methods of particular importance in this context: data views, aggregation and filtering, and comparison. For these classes, we present a GPU-based implementation of respective analysis methods that enable the interactive analysis of large graphs. We demonstrate the utility as well as the scalability of our approach by presenting application examples for analyzing different time-varying data sets.

Abstract The visualization of dynamic graphs demands visually encoding at least three major data dimensions: vertices, edges, and time steps. Many of the state‐of‐the‐art techniques can show an overview of vertices and edges but lack a data‐scalable visual representation of the time aspect. In this paper, we address the problem of displaying dynamic graphs with a thousand or more time steps. Our proposed interleaved parallel edge splatting technique uses a time‐to‐space mapping and shows the complete dynamic graph in a static visualization. It provides an overview of all data dimensions, allowing for visually detecting time‐varying data patterns; hence, it serves as a starting point for further data exploration. By applying clustering and ordering techniques on the vertices, edge splatting on the links, and a dense time‐to‐space mapping, our approach becomes visually scalable in all three dynamic graph data dimensions. We illustrate the usefulness of our technique by applying it to call graphs and US domestic flight data with several hundred vertices, several thousand edges, and more than a thousand time steps.

The analysis of eye tracking data often requires the annotation of areas of interest (AOIs) to derive semantic interpretations of human viewing behavior during experiments. This annotation is typically the most time-consuming step of the analysis process. Especially for data from wearable eye tracking glasses, every independently recorded video has to be annotated individually and corresponding AOIs between videos have to be identified. We provide a novel visual analytics approach to ease this annotation process by image-based, automatic clustering of eye tracking data integrated in an interactive labeling and analysis system. The annotation and analysis are tightly coupled by multiple linked views that allow for a direct interpretation of the labeled data in the context of the recorded video stimuli. The components of our analytics environment were developed with a user-centered design approach in close cooperation with an eye tracking expert. We demonstrate our approach with eye tracking data from a real experiment and compare it to an analysis of the data by manual annotation of dynamic AOIs. Furthermore, we conducted an expert user study with 6 external eye tracking researchers to collect feedback and identify analysis strategies they used while working with our application.

Visual search can be time-consuming, especially if the scene contains a large number of possibly relevant objects. An instance of this problem is present when using geographic or schematic maps with many different elements representing cities, streets, sights, and the like. Unless the map is well-known to the reader, the full map or at least large parts of it must be scanned to find the elements of interest. In this paper, we present a controlled eye-tracking study (30 participants) to compare four variants of map annotation with labels: within-image annotations, grid reference annotation, directional annotation, and miniature annotation. Within-image annotation places labels directly within the map without any further search support. Grid reference annotation corresponds to the traditional approach known from atlases. Directional annotation utilizes a label in combination with an arrow pointing in the direction of the label within the map. Miniature annotation shows a miniature grid to guide the reader to the area of the map in which the label is located. The study results show that within-image annotation is outperformed by all other annotation approaches. Best task completion times are achieved with miniature annotation. The analysis of eye-movement data reveals that participants applied significantly different visual task solution strategies for the different visual annotations.

We facilitate the comparative visual analysis of eye tracking data from multiple participants with a visualization that represents the temporal changes of viewing behavior. Common approaches to visually analyze eye tracking data either occlude or ignore the underlying visual stimulus, impairing the interpretation of displayed measures. We introduce fixation-image charts: a new technique to display the temporal changes of fixations in the context of the stimulus without visual overlap between participants. Fixation durations, the distance and direction of saccades between consecutive fixations, as well as the stimulus context can be interpreted in one visual representation. Our technique is not limited to static stimuli, but can be applied to dynamic stimuli as well. Using fixation metrics and the visual similarity of stimulus regions, we complement our visualization technique with an interactive filter concept that allows for the identification of interesting fixation sequences without the time-consuming annotation of areas of interest. We demonstrate how our technique can be applied to different types of stimuli to perform a range of analysis tasks. Furthermore, we discuss advantages and shortcomings derived from a preliminary user study.

Collecting and processing provenance, i.e., information describing the production process of some end product, is important in various applications, e.g., to assess quality, to ensure reproducibility, or to reinforce trust in the end product. In the past, different types of provenance meta-data have been proposed, each with a different scope. The first part of the proposed tutorial provides an overview and comparison of these different types of provenance. To put provenance to good use, it is essential to be able to interact with and present provenance data in a user-friendly way. Often, users interested in provenance are not necessarily experts in databases or query languages, as they are typically domain experts of the product and production process for which provenance is collected (biologists, journalists, etc.). Furthermore, in some scenarios, it is difficult to use solely queries for analyzing and exploring provenance data. The second part of this tutorial therefore focuses on enabling users to leverage provenance through adapted visualizations. To this end, we will present some fundamental concepts of visualization before we discuss possible visualizations for provenance.

We present a new visualization approach for displaying eye tracking data from multiple participants. We aim to show the spatio-temporal data of the gaze points in the context of the underlying image or video stimulus without occlusion. Our technique, denoted as gaze stripes, does not require the explicit definition of areas of interest but directly uses the image data around the gaze points, similar to thumbnails for images. A gaze stripe consists of a sequence of such gaze point images, oriented along a horizontal timeline. By displaying multiple aligned gaze stripes, it is possible to analyze and compare the viewing behavior of the participants over time. Since the analysis is carried out directly on the image data, expensive post-processing or manual annotation are not required. Therefore, not only patterns and outliers in the participants' scanpaths can be detected, but the context of the stimulus is available as well. Furthermore, our approach is especially well suited for dynamic stimuli due to the non-aggregated temporal mapping. Complementary views, i.e., markers, notes, screenshots, histograms, and results from automatic clustering, can be added to the visualization to display analysis results. We illustrate the usefulness of our technique on static and dynamic stimuli. Furthermore, we discuss the limitations and scalability of our approach in comparison to established visualization techniques.

Module workflows are used to generate custom applications with modular software frameworks. They describe data flow between the modular components and their execution under certain parameter configurations. In many cases, module workflows are modeled in a graphical way by the user. To come up with the final result or to explore multiple solutions, they often undergo many iterations of adaptation. Furthermore, existing workflows may be reused for new applications. We visualize the evolution of module workflows with a focus-and-context approach and visualization techniques for time-dependent data. Our approach provides insight into user behavior and the characteristics of the underlying systems. As our examples show, this can help identify usability issues and indicate options to improve the effectiveness of the system. We demonstrate our approach for module workflows in Vis Trails, a modular visualization system that allows building custom visualizations by combining different modules for processing and visualizing data.

The visual exploration of spatio-temporal eye movement data is challenging, especially if we are interested in the movement patterns of a large number of study participants. For example, if popular visualization techniques like heat maps or gaze plots are used, we may lose the temporal information or get lost in visual clutter. To address these issues, we propose an approach for filtering saccadic eye movement data called hierarchical filter wheels, which employs a radial representation of saccade information. It supports the analysis of sequences of saccades by filtering them with respect to direction and length. The focus of our approach is a fast initial analysis of data from eye tracking studies without the need of defining areas of interest (AOIs) or other preprocessing of the data. The hierarchical filters are interactively generated on users' demand by creating a hierarchy of multiple filter wheels each filtering one element of the sequence. We use a bubble tree layout to represent the generated filter hierarchy. The node positions in our layout directly represent the spatial properties of the filter criteria allowing an intuitive incremental generation and understanding of filter hierarchies. We illustrate the approach by applying it to eye movement data formerly recorded in an eye tracking study investigating the readability of different node-link tree diagrams. We further demonstrate how the hierarchical filter wheels can be used in combination with gaze plots.

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

We present a visual representation for dynamic, weighted graphs based on the concept of adjacency lists. Two orthogonal axes are used: one for all nodes of the displayed graph, the other for the corresponding links. Colors and labels are employed to identify the nodes. The usage of color allows us to scale the visualization to single pixel level for large graphs. In contrast to other techniques, we employ an asymmetric mapping that results in an aligned and compact representation of links. Our approach is independent of the specific properties of the graph to be visualized, but certain graphs and tasks benefit from the asymmetry. As we show in our results, the strength of our technique is the visualization of dynamic graphs. In particular, sparse graphs benefit from the compact representation. Furthermore, our approach uses visual encoding by size to represent weights and therefore allows easy quantification and comparison. We evaluate our approach in a quantitative user study that confirms the suitability for dynamic and weighted graphs. Finally, we demonstrate our approach for two examples of dynamic graphs.

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

A new type of glyph is introduced to visualize unsteady flow with static images, allowing easier analysis of time-dependent phenomena compared to animated visualization. Adopting the visual metaphor of radar displays, this glyph represents flow directions by angles and time by radius in spherical coordinates. Dense seeding of flow radar glyphs on the flow domain naturally lends itself to multi-scale visualization: zoomed-out views show aggregated overviews, zooming-in enables detailed analysis of spatial and temporal characteristics. Uncertainty visualization is supported by extending the glyph to display possible ranges of flow directions. The paper focuses on 2D flow, but includes a discussion of 3D flow as well. Examples from CFD and the field of stochastic hydrogeology show that it is easy to discriminate regions of different spatiotemporal flow behavior and regions of different uncertainty variations in space and time. The examples also demonstrate that parameter studies can be analyzed because the glyph design facilitates comparative visualization. Finally, different variants of interactive GPU-accelerated implementations are discussed.

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

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