Visual Analysis of Trajectories in Multi‐Dimensional State Spaces

Multi‐dimensional data originate from many different sources and are relevant for many applications. One specific sub‐type of such data is continuous trajectory data in multi‐dimensional state spaces of complex systems. We adapt the concept of spatially continuous scatterplots and spatially continuous parallel coordinate plots to such trajectory data, leading to continuous‐time scatterplots and continuous‐time parallel coordinates. Together with a temporal heat map representation, we design coordinated views for visual analysis and interactive exploration. We demonstrate the usefulness of our visualization approach for three case studies that cover examples of complex dynamic systems: cyber‐physical systems consisting of heterogeneous sensors and actuators networks (the collection of time‐dependent sensor network data of an exemplary smart home environment), the dynamics of robot arm movement and motion characteristics of humanoids.     

Adaptable Radial Axes Plots for Improved Multivariate Data Visualization

Radial axes plots are multivariate visualization techniques that extend scatterplots in order to represent high‐dimensional data as points on an observable display. Well‐known methods include star coordinates or principal component biplots, which represent data attributes as vectors that define axes, and produce linear dimensionality reduction mappings. In this paper we propose a hybrid approach that bridges the gap between star coordinates and principal component biplots, which we denominate “adaptable radial axes plots”. It is based on solving convex optimization problems where users can: (a) update the axis vectors interactively, as in star coordinates, while producing mappings that enable to estimate attribute values optimally through labeled axes, similarly to principal component biplots; (b) use different norms in order to explore additional nonlinear mappings of the data; and (c) include weights and constraints in the optimization problems for sorting the data along one axis. The result is a flexible technique that complements, extends, and enhances current radial methods for data analysis.     

A Multi‐task Comparative Study on Scatter Plots and Parallel Coordinates Plots

Previous empirical studies for comparing parallel coordinates plots and scatter plots showed some uncertainty about their relative merits. Some of these studies focused on the task of value retrieval, where visualization usually has a limited advantage over reading data directly. In this paper, we report an empirical study that compares user performance, in terms of accuracy and response time, in the context of four different visualization tasks, namely value retrieval, clustering, outlier detection, and change detection. In order to evaluate the relative merits of the two types of plots with a common base line (i.e., reading data directly), we included three forms of stimuli, data tables, scatter plots, and parallel coordinate plots. Our results show that data tables are better suited for the value retrieval task, while parallel coordinates plots generally outperform the two other visual representations in three other tasks. Subjective feedbacks from the users are also consistent with the quantitative analyses. As visualization is commonly used for aiding multiple observational and analytical tasks, our results provided new evidence to support the prevailing enthusiasm for parallel coordinates plots in the field of visualization.     

Visual Coherence for Large‐Scale Line‐Plot Visualizations

Displaying a large number of lines within a limited amount of screen space is a task that is common to many different classes of visualization techniques such as time‐series visualizations, parallel coordinates, link‐node diagrams, and phase‐space diagrams. This paper addresses the challenging problems of cluttering and overdraw inherent to such visualizations. We generate a 2×2 tensor field during line rasterization that encodes the distribution of line orientations through each image pixel. Anisotropic diffusion of a noise texture is then used to generate a dense, coherent visualization of line orientation. In order to represent features of different scales, we employ a multi‐resolution representation of the tensor field. The resulting technique can easily be applied to a wide variety of line‐based visualizations. We demonstrate this for parallel coordinates, a time‐series visualization, and a phase‐space diagram. Furthermore, we demonstrate how to integrate a focus+context approach by incorporating a second tensor field. Our approach achieves interactive rendering performance for large data sets containing millions of data items, due to its image‐based nature and ease of implementation on GPUs. Simulation results from computational fluid dynamics are used to evaluate the performance and usefulness of the proposed method.     

Measuring Privacy and Utility in Privacy‐Preserving Visualization

In previous work, we proposed a technique for preserving the privacy of quasi‐identifiers in sensitive data when visualized using parallel coordinates. This paper builds on that work by introducing a number of metrics that can be used to assess both the level of privacy and the amount of utility that can be gained from the resulting visualizations. We also generalize our approach beyond parallel coordinates to scatter plots and other visualization techniques. Privacy preservation generally entails a trade‐off between privacy and utility: the more the data are protected, the less useful the visualization. Using a visually‐oriented approach, we can provide a higher amount of utility than directly applying data anonymization techniques used in data mining. To demonstrate this, we use the visual uncertainty framework for systematically defining metrics based on cluster artifacts and information theoretic principles. In a case study, we demonstrate the effectiveness of our technique as compared to standard data‐based clustering in the context of privacy‐preserving visualization.     

Smooth Transitions Between Parallel Coordinates and Scatter Plots via Polycurve Star Plots

This paper presents new techniques for seamlessly transitioning between parallel coordinate plots, star plots, and scatter plots. The star plot serves as a mediator visualization between parallel coordinate plots and scatter plots since it uses lines to represent data items as parallel coordinates do and can arrange axes orthogonally as used for scatter plots. The design of the transitions also motivated a new variant of the star plot, the polycurve star plot, that uses curved lines instead of straight ones and has advantages both in terms of space utilization and the detection of clusters. Furthermore, we developed a geometrically motivated method to embed scatter points from a scatter plot into star plots and parallel coordinate plots to track the transition of structural information such as clusters and correlations between the different plot types. The integration of our techniques into an interactive analysis tool for exploring multivariate data demonstrates the advantages and utility of our approach over a multi-view approach for scatter plots and parallel coordinate plots, which we confirmed in a user study and concrete usage scenarios.     

Dynamic Multi‐View Exploration of Shape Spaces

Statistical shape modeling is a widely used technique for the representation and analysis of the shapes and shape variations present in a population. A statistical shape model models the distribution in a high dimensional shape space, where each shape is represented by a single point. We present a design study on the intuitive exploration and visualization of shape spaces and shape models. Our approach focuses on the dual‐space nature of these spaces. The high‐dimensional shape space represents the population, whereas object space represents the shape of the 3D object associated with a point in shape space. A 3D object view provides local details for a single shape. The high dimensional points in shape space are visualized using a 2D scatter plot projection, the axes of which can be manipulated interactively. This results in a dynamic scatter plot, with the further extension that each point is visualized as a small version of the object shape that it represents. We further enhance the population‐object duality with a new type of view aimed at shape comparison. This new “shape evolution view” visualizes shape variability along a single trajectory in shape space, and serves as a link between the two spaces described above. Our three‐view exploration concept strongly emphasizes linked interaction between all spaces. Moving the cursor over the scatter plot or evolution views, shapes are dynamically interpolated and shown in the object view. Conversely, camera manipulation in the object view affects the object visualizations in the other views. We present a GPU‐accelerated implementation, and show the effectiveness of the three‐view approach using a number of real‐world cases. In these, we demonstrate how this multi‐view approach can be used to visually explore important aspects of a statistical shape model, including specificity, compactness and reconstruction error.     

Topology‐based Visualization of Transformation Pathways in Complex Chemical Systems

Studying transformation in a chemical system by considering its energy as a function of coordinates of the system's components provides insight and changes our understanding of this process. Currently, a lack of effective visualization techniques for high‐dimensional energy functions limits chemists to plot energy with respect to one or two coordinates at a time. In some complex systems, developing a comprehensive understanding requires new visualization techniques that show relationships between all coordinates at the same time. We propose a new visualization technique that combines concepts from topological analysis, multi‐dimensional scaling, and graph layout to enable the analysis of energy functions for a wide range of molecular structures. We demonstrate our technique by studying the energy function of a dimer of formic and acetic acids and a LTA zeolite structure, in which we consider diffusion of methane.     

Scattering Points in Parallel Coordinates

In this paper, we present a novel parallel coordinates design integrated with points (scattering points in parallel coordinates, SPPC), by taking advantage of both parallel coordinates and scatterplots. Different from most multiple views visualization frameworks involving parallel coordinates where each visualization type occupies an individual window, we convert two selected neighboring coordinate axes into a scatterplot directly. Multidimensional scaling is adopted to allow converting multiple axes into a single subplot. The transition between two visual types is designed in a seamless way. In our work, a series of interaction tools has been developed. Uniform brushing functionality is implemented to allow the user to perform data selection on both points and parallel coordinate polylines without explicitly switching tools. A GPU accelerated dimensional incremental multidimensional scaling (DIMDS) has been developed to significantly improve the system performance. Our case study shows that our scheme is more efficient than traditional multi-view methods in performing visual analysis tasks.     

Cycle Plot Revisited: Multivariate Outlier Detection Using a Distance‐Based Abstraction

The cycle plot is an established and effective visualization technique for identifying and comprehending patterns in periodic time series, like trends and seasonal cycles. It also allows to visually identify and contextualize extreme values and outliers from a different perspective. Unfortunately, it is limited to univariate data. For multivariate time series, patterns that exist across several dimensions are much harder or impossible to explore. We propose a modified cycle plot using a distance‐based abstraction (Mahalanobis distance) to reduce multiple dimensions to one overview dimension and retain a representation similar to the original. Utilizing this distance‐based cycle plot in an interactive exploration environment, we enhance the Visual Analytics capacity of cycle plots for multivariate outlier detection. To enable interactive exploration and interpretation of outliers, we employ coordinated multiple views that juxtapose a distance‐based cycle plot with Cleveland's original cycle plots of the underlying dimensions. With our approach it is possible to judge the outlyingness regarding the seasonal cycle in multivariate periodic time series.     

State‐of‐the‐Art in GPU‐Based Large‐Scale Volume Visualization

This survey gives an overview of the current state of the art in GPU techniques for interactive large‐scale volume visualization. Modern techniques in this field have brought about a sea change in how interactive visualization and analysis of giga‐, tera‐ and petabytes of volume data can be enabled on GPUs. In addition to combining the parallel processing power of GPUs with out‐of‐core methods and data streaming, a major enabler for interactivity is making both the computational and the visualization effort proportional to the amount and resolution of data that is actually visible on screen, i.e. ‘output‐sensitive’ algorithms and system designs. This leads to recent output‐sensitive approaches that are ‘ray‐guided’, ‘visualization‐driven’ or ‘display‐aware’. In this survey, we focus on these characteristics and propose a new categorization of GPU‐based large‐scale volume visualization techniques based on the notions of actual output‐resolution visibility and the current working set of volume bricks—the current subset of data that is minimally required to produce an output image of the desired display resolution. Furthermore, we discuss the differences and similarities of different rendering and data traversal strategies in volume rendering by putting them into a common context—the notion of address translation. For our purposes here, we view parallel (distributed) visualization using clusters as an orthogonal set of techniques that we do not discuss in detail but that can be used in conjunction with what we present in this survey.     

Visualization of Time‐Series Data in Parameter Space for Understanding Facial Dynamics

Over the past decade, computer scientists and psychologists have made great efforts to collect and analyze facial dynamics data that exhibit different expressions and emotions. Such data is commonly captured as videos and are transformed into feature‐based time‐series prior to any analysis. However, the analytical tasks, such as expression classification, have been hindered by the lack of understanding of the complex data space and the associated algorithm space. Conventional graph‐based time‐series visualization is also found inadequate to support such tasks. In this work, we adopt a visual analytics approach by visualizing the correlation between the algorithm space and our goal – classifying facial dynamics. We transform multiple feature‐based time‐series for each expression in measurement space to a multi‐dimensional representation in parameter space. This enables us to utilize parallel coordinates visualization to gain an understanding of the algorithm space, providing a fast and cost‐effective means to support the design of analytical algorithms.     

Visualizing High‐Dimensional Structures by Dimension Ordering and Filtering using Subspace Analysis

High‐dimensional data visualization is receiving increasing interest because of the growing abundance of high‐dimensional datasets. To understand such datasets, visualization of the structures present in the data, such as clusters, can be an invaluable tool. Structures may be present in the full high‐dimensional space, as well as in its subspaces. Two widely used methods to visualize high‐dimensional data are the scatter plot matrix (SPM) and the parallel coordinate plot (PCP). SPM allows a quick overview of the structures present in pairwise combinations of dimensions. On the other hand, PCP has the potential to visualize not only bi‐dimensional structures but also higher dimensional ones. A problem with SPM is that it suffers from crowding and clutter which makes interpretation hard. Approaches to reduce clutter are available in the literature, based on changing the order of the dimensions. However, usually this reordering has a high computational complexity. For effective visualization of high‐dimensional structures, also PCP requires a proper ordering of the dimensions. In this paper, we propose methods for reordering dimensions in PCP in such a way that high‐dimensional structures (if present) become easier to perceive. We also present a method for dimension reordering in SPM which yields results that are comparable to those of existing approaches, but at a much lower computational cost. Our approach is based on finding relevant subspaces for clustering using a quality criterion and cluster information. The quality computation and cluster detection are done in image space, using connected morphological operators. We demonstrate the potential of our approach for synthetic and astronomical datasets, and show that our method compares favorably with a number of existing approaches.     

Exploring High‐Dimensional Structure via Axis‐Aligned Decomposition of Linear Projections

Two‐dimensional embeddings remain the dominant approach to visualize high dimensional data. The choice of embeddings ranges from highly non‐linear ones, which can capture complex relationships but are difficult to interpret quantitatively, to axis‐aligned projections, which are easy to interpret but are limited to bivariate relationships. Linear project can be considered as a compromise between complexity and interpretability, as they allow explicit axes labels, yet provide significantly more degrees of freedom compared to axis‐aligned projections. Nevertheless, interpreting the axes directions, which are often linear combinations of many non‐trivial components, remains difficult. To address this problem we introduce a structure aware decomposition of (multiple) linear projections into sparse sets of axis‐aligned projections, which jointly capture all information of the original linear ones. In particular, we use tools from Dempster‐Shafer theory to formally define how relevant a given axis‐aligned project is to explain the neighborhood relations displayed in some linear projection. Furthermore, we introduce a new approach to discover a diverse set of high quality linear projections and show that in practice the information of k linear projections is often jointly encoded in ～ k axis‐aligned plots. We have integrated these ideas into an interactive visualization system that allows users to jointly browse both linear projections and their axis‐aligned representatives. Using a number of case studies we show how the resulting plots lead to more intuitive visualizations and new insights.     

Visual and quantitative analysis of great arteries’ blood flow jets in cardiac 4D PC‐MRI data

Flow in the great arteries (aorta, pulmonary artery) is normally laminar with a parabolic velocity profile. Eccentric flow jets are linked to various diseases like aneurysms. Cardiac 4D PC‐MRI data provide spatio‐temporally resolved blood flow information for the whole cardiac cycle. In this work, we establish a time‐dependent visualization and quantification of flow jets. For this purpose, equidistant measuring planes are automatically placed along the vessel's centerline. The flow jet position and region with highest velocities are extracted for every plane in each time step. This is done during pre‐processing and without user‐defined parameters. We visualize the main flow jet as geometric tube. High‐velocity areas are depicted as a net around this tube. Both geometries are time‐dependent and can be animated. Quantitative values are provided during cross‐sectional measuring plane‐based evaluation. Moreover, we offer a plot visualization as summary of flow jet characteristics for the selected plane. Our physiologically plausible results are in accordance with medical findings. Our clinical collaborators appreciate the possibility to view the flow jet in the whole vessel at once, which normally requires repeated pathline filtering due to varying velocities along the vessel course. The overview plots are considered as valuable for documentation purposes.     

Visualization Mosaics for Multivariate Visual Exploration

We present a new model for creating composite visualizations of multidimensional data sets using simple visual representations such as point charts, scatterplots and parallel coordinates as components. Each visual representation is contained in a tile, and the tiles are arranged in a mosaic of views using a space‐filling slice‐and‐dice layout. Tiles can be created, resized, split or merged using a versatile set of interaction techniques, and the visual representation of individual tiles can also be dynamically changed to another representation. Because each tile is self‐contained and independent, it can be implemented in any programming language, on any platform and using any visual representation. We also propose a formalism for expressing visualization mosaics. A Web‐based implementation called MosaicJS supporting multidimensional visual exploration showcases the versatility of the concept and illustrates how it can be used to integrate visualization components provided by different toolkits.     

Uncertainty‐Aware Exploration of Continuous Parameter Spaces Using Multivariate Prediction

Systems projecting a continuous n‐dimensional parameter space to a continuous m‐dimensional target space play an important role in science and engineering. If evaluating the system is expensive, however, an analysis is often limited to a small number of sample points. The main contribution of this paper is an interactive approach to enable a continuous analysis of a sampled parameter space with respect to multiple target values. We employ methods from statistical learning to predict results in real‐time at any user‐defined point and its neighborhood. In particular, we describe techniques to guide the user to potentially interesting parameter regions, and we visualize the inherent uncertainty of predictions in 2D scatterplots and parallel coordinates. An evaluation describes a real‐world scenario in the application context of car engine design and reports feedback of domain experts. The results indicate that our approach is suitable to accelerate a local sensitivity analysis of multiple target dimensions, and to determine a sufficient local sampling density for interesting parameter regions.     

平行坐标轴动态排列的地理空间多维数据可视分析

目的 平行坐标是经典的多维数据可视化方法,但在用于地理空间多维数据分析时,往往存在空间位置信息缺失和空间关联分析不确定等问题。对此,本文设计了一种有效关联平行坐标和地图的地理空间多维数据可视分析方法。方法 根据多维属性信息对地理空间位置进行聚类分析,引入Voronoi图和颜色明暗映射对地理空间各类区域进行显著标识,利用平行坐标呈现地理空间多维属性信息,引入互信息度量地理空间聚类与属性类别的相关性,动态地确定平行坐标轴排列顺序,进一步计算属性轴与地图之间数据线的绑定位置,对数据线的布局进行优化处理,降低地图与平行坐标系间数据线分布的紊乱程度。结果 有效集成上述可视化设计及数据分析方法,设计与实现一种基于平行坐标轴动态排列的地理空间多维数据可视化分析系统,提供便捷的用户交互模式,通过2组具有明显地理空间多维属性特征的数据进行测试,验证了本文可视分析方法的有效性和实用性。结论 本文提出的可视分析方法和工具可以帮助用户快速分析地理空间多维属性存在的空间分布特征及其关联模式,为地理空间多维数据的探索提供了有效手段。     

Finite‐Time Mass Separation for Comparative Visualizations of Inertial Particles

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