Linear embeddings support interactive visual exploration by mapping high-dimensional (nD) data into a two-dimensional space. Despite their popularity, selecting meaningful projection parameters remains a key challenge due to the infinite 2$n$-dimensional parameter space. Once an informative projection is found, users often seek similar ones that emphasize specific items differently while preserving global structure. For instance: Do clusters become outliers under slight changes? Can grouped items separate-or merge-through parameter adjustments? Which changes to the embedding parameters lead to such projections - and do they exist at all? Answering these questions efficiently is critical for effective visual search. Yet, current methods-such as projection tours or manual parameter tuning-are time-consuming and risk overlooking important views, including those of specific interest. We propose Composition Operators, a mathematical foundation for a novel set-of-point manipulation concept for linear embeddings-such as Star Coordinates-as an alternative approach to selecting informative embedding parameters in a more controllable manner with respect to the desired outcome. Users specify item-based constraints on the projection result; the corresponding 2n parameters are then derived automatically, eliminating the need to exhaustively search the entire parameter space to get a similar outcome. Neither the embedding space nor the set of parameters is altered - only the mechanism for navigating and selecting parameters is redefined. We provide closed-form solutions for this and demonstrate our interactive prototype on nD datasets from the UCI repository.

Abstract Radial axes plots are projection methods that represent high‐dimensional data samples as points on a two‐dimensional plane. These techniques define mappings through a set of axis vectors, each associated with a data variable, which users can manipulate interactively to create different plots and analyze data from multiple points of view. However, updating the direction and length of an axis vector is far from trivial. Users must consider the data analysis task, domain knowledge, the directions in which values should increase, the relative importance of each variable, or the correlations between variables, among other factors. Another issue is the difficulty to approximate high‐dimensional data values in the two‐dimensional visualizations, which can hamper searching for data with particular characteristics, analyzing the most common data values in clusters, inspecting outliers, etc. In this paper we present and analyze several optimization approaches for enhancing radial axes plots regarding their ability to represent high‐dimensional data values. The techniques can be used not only to approximate data values with greater accuracy, but also to guide users when updating axis vectors or extending visualizations with new variables, since they can reveal poor choices of axis vectors. The optimal axes can also be included in nonlinear plots. In particular, we show how they can be used within RadViz to assess the quality of a variable ordering. The in‐depth analysis carried out is useful for visualization designers developing radial axes techniques, or planning to incorporate axes into other visualization methods.

As of today, data analysis focuses primarily on the findings to be made inside the data and concentrates less on how those findings relate to the domain of investigation. Contemporary visualization as a field of research shows a strong tendency to adopt this data-centrism. Despite their decisive influence on the analysis result, qualitative aspects of the analysis process such as the structure, soundness, and complexity of the applied reasoning strategy are rarely discussed explicitly. We argue that if the purpose of visualization is the provision of domain insight rather than the depiction of data analysis results, a holistic perspective requires a qualitative component to to be added to the discussion of quantitative and human factors. To support this point, we demonstrate how considerations of qualitative factors in visual analysis can be applied to obtain explanations and possible solutions for a number of practical limitations inherent to the data-centric perspective on analysis. Based on this discussion of what we call qualitative visual analysis, we develop an inside-outside principle of nested levels of context that can serve as a conceptual basis for the development of visualization systems that optimally support the emergence of insight during analysis.

Abstract Analysts often use visualisation techniques like a scatterplot matrix (SPLOM) to explore multivariate datasets. The scatterplots of a SPLOM can help to identify and compare two‐dimensional global patterns. However, local patterns which might only exist within subsets of records are typically much harder to identify and may go unnoticed among larger sets of plots in a SPLOM. This paper explores the notion of local patterns and presents a novel approach to visually select, search for, and compare local patterns in a multivariate dataset. Model‐based and shape‐based pattern descriptors are used to automatically compare local regions in scatterplots to assist in the discovery of similar local patterns. Mechanisms are provided to assess the level of similarity between local patterns and to rank similar patterns effectively. Moreover, a relevance feedback module is used to suggest potentially relevant local patterns to the user. The approach has been implemented in an interactive tool and demonstrated with two real‐world datasets and use cases. It supports the discovery of potentially useful information such as clusters, functional dependencies between variables, and statistical relationships in subsets of data records and dimensions.

Star Coordinates are a popular projection technique in order to analyze and to disclose characteristic patterns of multidimensional data. Unfortunately, the shape, appearance, and distribution of such patterns are strongly affected by the given scaling of the data and can mislead the projection-based data analysis. In an extreme case, patterns might be more related to the choice of scaling than to the data themselves. Thus, we present the LloydRelaxer : a tool to minimize scaling-based effects in Star Coordinates. Our algorithm enforces a scaling configuration for which the data explains the observed patterns better than any scaling of them could do. It does so by an iterative minimizing and optimization process based on Voronoi diagrams and on the Lloyd relaxation within the projection space. We evaluate and test our approach by real benchmark multidimensional data of the UCI data repository.

This design study focuses on the analysis of a time sequence of categorical sequences. Such data is relevant for the geoscientific research field of landscape and climate development. It results from microscopic analysis of lake sediment cores. The goal is to gain hypotheses about landscape evolution and climate conditions in the past. To this end, geoscientists identify which categorical sequences are similar in the sense that they indicate similar conditions. Categorical sequences are similar if they have similar meaning (semantic similarity) and appear in similar time periods (temporal similarity). For data sets with many different categorical sequences, the task to identify similar sequences becomes a challenge. Our contribution is a tailored visual analysis concept that effectively supports the analytical process. Our visual interface comprises coupled visualizations of semantics and temporal context for the exploration and assessment of the similarity of categorical sequences. Integrated automatic methods reduce the analytical effort substantially. They (1) extract unique sequences in the data and (2) rank sequences by a similarity measure during the search for similar sequences. We evaluated our concept by demonstrations of our prototype to a larger audience and hands-on analysis sessions for two different lakes. According to geoscientists, our approach fills an important methodological gap in the application domain.

Abstract The UN Comtrade database is the world's largest repository of bilateral trade data. Their complexity poses a challenge to visualization systems, leading to issues such as scalability and visual clutter. Thus, we propose a radial layout‐based visual exploration system to enable the user to smoothly explore the change over time and to explore different commodity classes at once by using a novel edge bundling concept. We evaluated our system with the aid of a group of domain experts.

Abstract 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.

Abstract Data analysis often involves finding models that can explain patterns in data, and reduce possibly large data sets to more compact model‐based representations. In Statistics, many methods are available to compute model information. Among others, regression models are widely used to explain data. However, regression analysis typically searches for the best model based on the global distribution of data. On the other hand, a data set may be partitioned into subsets, each requiring individual models. While automatic data subsetting methods exist, these often require parameters or domain knowledge to work with. We propose a system for visual‐interactive regression analysis for scatter plot data, supporting both global and local regression modeling. We introduce a novel regression lens concept, allowing a user to interactively select a portion of data, on which regression analysis is run in interactive time. The lens gives encompassing visual feedback on the quality of candidate models as it is interactively navigated across the input data. While our regression lens can be used for fully interactive modeling, we also provide user guidance suggesting appropriate models and data subsets, by means of regression quality scores. We show, by means of use cases, that our regression lens is an effective tool for user‐driven regression modeling and supports model understanding.

Abstract One main task for domain experts in analysing their nD data is to detect and interpret class/cluster separations and outliers. In fact, an important question is, which features/dimensions separate classes best or allow a cluster‐based data classification. Common approaches rely on projections from nD to 2D, which comes with some challenges, such as: The space of projection contains an infinite number of items. How to find the right one? The projection approaches suffers from distortions and misleading effects. How to rely to the projected class/cluster separation? The projections involve the complete set of dimensions/features. How to identify irrelevant dimensions? Thus, to address these challenges, we introduce a visual analytics concept for the feature selection based on linear discriminative star coordinates (DSC), which generate optimal cluster separating views in a linear sense for both labeled and unlabeled data. This way the user is able to explore how each dimension contributes to clustering. To support to explore relations between clusters and data dimensions, we provide a set of cluster‐aware interactions allowing to smartly iterate through subspaces of both records and features in a guided manner. We demonstrate our features selection approach for optimal cluster/class separation analysis with a couple of experiments on real‐life benchmark high‐dimensional data sets.

Abstract To project high‐dimensional data to a 2D domain, there are two well‐established classes of approaches: RadViz and Star Coordinates. Both are well‐explored in terms of accuracy, completeness, distortions, and interaction issues. We present a generalization of both RadViz and Star Coordinates such that it unifies both approaches. We do so by considering the space of all projective projections. This gives additional degrees of freedom, which we use for three things: Firstly, we define a smooth transition between RadViz and Star Coordinates allowing the user to exploit the advantages of both approaches. Secondly, we define a data‐dependent magic lens to explore the data. Thirdly, we optimize the new degrees of freedom to minimize distortion. We apply our approach to a number of high‐dimensional benchmark datasets.

Finding good projections of n-dimensional datasets into a 2D visualization domain is one of the most important problems in Information Visualization. Users are interested in getting maximal insight into the data by exploring a minimal number of projections. However, if the number is too small or improper projections are used, then important data patterns might be overlooked. We propose a data-driven approach to find minimal sets of projections that uniquely show certain data patterns. For this we introduce a dissimilarity measure of data projections that discards affine transformations of projections and prevents repetitions of the same data patterns. Based on this, we provide complete data tours of at most n/2 projections. Furthermore, we propose optimal paths of projection matrices for an interactive data exploration. We illustrate our technique with a set of state-of-the-art real high-dimensional benchmark datasets.

Abstract The visual analysis of multivariate projections is a challenging task, because complex visual structures occur. This causes fatigue or misinterpretations, which distorts the analysis. In fact, the same projection can lead to different analysis results. We provide visual guidance pictograms to improve objectivity of the visual search. A visual guidance pictogram is an iconic visual density map encoding the visual structure of certain data properties. By using them to guide the analysis, structures in the projection can be better understood and mentally linked to properties in the data. We introduce a systematic scheme for designing such pictograms and provide a set of pictograms for standard visual tasks, such as correlation and distribution analysis, for standard projections like scatterplots, RadVis, and Star Coordinates. We conduct a study that compares the visual analysis of real data with and without the support of guidance pictograms. Our tests show that the training effort for a visual search can be decreased and the analysis bias can be reduced by supporting the user's visual search with guidance pictograms.

Abstract Direct Numerical Simulations of premixed combustion produce terabytes of raw data, which are prohibitively large to be stored, and have to be analyzed and visualized. A simultaneous and integrated treatment of data storage, data analysis and data visualization is required. For this, we introduce a sparse representation tailored to DNS data which can directly be used for both analysis and visualization. The method is based on the observation that most information is located in narrow‐band regions where the chemical reactions take place, but these regions are not well defined. An approach for the visual investigation of feature surfaces of the scalar fields involved in the simulation is shown as a possible application. We demonstrate our approach on multiple real datasets.

Understanding the hemodynamics of blood flow in vascular pathologies such as intracranial aneurysms is essential for both their diagnosis and treatment. Computational fluid dynamics (CFD) simulations of blood flow based on patient-individual data are performed to better understand aneurysm initiation and progression and more recently, for predicting treatment success. In virtual stenting, a flow-diverting mesh tube (stent) is modeled inside the reconstructed vasculature and integrated in the simulation. We focus on steady-state simulation and the resulting complex multiparameter data. The blood flow pattern captured therein is assumed to be related to the success of stenting. It is often visualized by a dense and cluttered set of streamlines.We present a fully automatic approach for reducing visual clutter and exposing characteristic flow structures by clustering streamlines and computing cluster representatives. While individual clustering techniques have been applied before to streamlines in 3D flow fields, we contribute a general quantitative and a domain-specific qualitative evaluation of three state-of-the-art techniques. We show that clustering based on streamline geometry as well as on domain-specific streamline attributes contributes to comparing and evaluating different virtual stenting strategies. With our work, we aim at supporting CFD engineers and interventional neuroradiologists.

Star coordinates is a popular projection technique from an nD data space to a 2D/3D visualization domain. It is defined by setting n coordinate axes in the visualization domain. Since it generally defines an affine projection, strong distortions can occur: an nD sphere can be mapped to an ellipse of arbitrary size and aspect ratio. We propose to restrict star coordinates to orthographic projections which map an nD sphere of radius r to a 2D circle of radius r. We achieve this by formulating conditions for the coordinate axes to define orthographic projections, and by running a repeated non-linear optimization in the background of every modification of the coordinate axes. This way, we define a number of orthographic interaction concepts as well as orthographic data tour sequences: a scatterplot tour, a principle component tour, and a grand tour. All concepts are illustrated and evaluated with synthetic and real data.

Abstract The scatterplot matrix (SPLOM) is a well‐established technique to visually explore high‐dimensional data sets. It is characterized by the number of scatterplots (plots) of which it consists of. Unfortunately, this number quadratically grows with the number of the data set’s dimensions. Thus, an SPLOM scales very poorly. Consequently, the usefulness of SPLOMs is restricted to a small number of dimensions. For this, several approaches already exist to explore such ‘small’ SPLOMs. Those approaches address the scalability problem just indirectly and without solving it. Therefore, we introduce a new greedy approach to manage ‘large’ SPLOMs with more than 100 dimensions. We establish a combined visualization and interaction scheme that produces intuitively interpretable SPLOMs by combining known quality measures, a pre‐process reordering and a perception‐based abstraction. With this scheme, the user can interactively find large amounts of relevant plots in large SPLOMs.

Abstract Two‐dimensional transfer functions are an effective and well‐accepted tool in volume classification. The design of them mostly depends on the user's experience and thus remains a challenge. Therefore, we present an approach in this paper to automate the transfer function design based on 2D density plots. By exploiting their smoothness, we adopted the Morse theory to automatically decompose the feature space into a set of valley cells. We design a simplification process based on cell separability to eliminate cells which are mainly caused by noise in the original volume data. Boundary persistence is first introduced to measure the separability between adjacent cells and to suitably merge them. Afterward, a reasonable classification result is achieved where each cell represents a potential feature in the volume data. This classification procedure is automatic and facilitates an arbitrary number and shape of features in the feature space. The opacity of each feature is determined by its persistence and size. To further incorporate the user's prior knowledge, a hierarchical feature representation is created by successive merging of the cells. With this representation, the user is allowed to merge or split features of interest and set opacity and color freely. Experiments on various volumetric data sets demonstrate the effectiveness and usefulness of our approach in transfer function generation.

Cerebral aneurysms are a pathological vessel dilatation that bear a high risk of rupture. For the understanding and evaluation of the risk of rupture, the analysis of hemodynamic information plays an important role. Besides quantitative hemodynamic information, also qualitative flow characteristics, e.g., the inflow jet and impingement zone are correlated with the risk of rupture. However, the assessment of these two characteristics is currently based on an interactive visual investigation of the flow field, obtained by computational fluid dynamics (CFD) or blood flow measurements. We present an automatic and robust detection as well as an expressive visualization of these characteristics. The detection can be used to support a comparison, e.g., of simulation results reflecting different treatment options. Our approach utilizes local streamline properties to formalize the inflow jet and impingement zone. We extract a characteristic seeding curve on the ostium, on which an inflow jet boundary contour is constructed. Based on this boundary contour we identify the impingement zone. Furthermore, we present several visualization techniques to depict both characteristics expressively. Thereby, we consider accuracy and robustness of the extracted characteristics, minimal visual clutter and occlusions. An evaluation with six domain experts confirms that our approach detects both hemodynamic characteristics reasonably.

Continuous Parallel Coordinates (CPC) are a contemporary visualization technique in order to combine several scalar fields, given over a common domain. They facilitate a continuous view for parallel coordinates by considering a smooth scalar field instead of a finite number of straight lines. We show that there are feature curves in CPC which appear to be the dominant structures of a CPC. We present methods to extract and classify them and demonstrate their usefulness to enhance the visualization of CPCs. In particular, we show that these feature curves are related to discontinuities in Continuous Scatterplots (CSP). We show this by exploiting a curve-curve duality between parallel and Cartesian coordinates, which is a generalization of the well-known point-line duality. Furthermore, we illustrate the theoretical considerations. Concluding, we discuss relations and aspects of the CPC's/CSP's features concerning the data analysis.

The concept of continuous scatterplot (CSP) is a modern visualization technique. The idea is to define a scalar density value based on the map between an n-dimensional spatial domain and an m-dimensional data domain, which describe the CSP space. Usually the data domain is two-dimensional to visually convey the underlying, density coded, data. In this paper we investigate kinds of map-based discontinuities, especially for the practical cases n = m = 2 and n = 3 | m = 2, and we depict relations between them and attributes of the resulting CSP itself. Additionally, we show that discontinuities build critical line structures, and we introduce algorithms to detect them. Further, we introduce a discontinuity-based visualization approach—called contribution map (CM)—which establishes a relationship between the CSP's data domain and the number of connected components in the spatial domain. We show that CMs enhance the CSP-based linking & brushing interaction. Finally, we apply our approaches to a number of synthetic as well as real data sets.