Abstract Through training and gathered experience, domain experts attain a mental model of the uncertainties inherent in the visual analytics processes for their respective domain. For an accurate data analysis and trustworthiness of the analysis results, it is essential to include this knowledge and consider this model of uncertainty during the analytical process. For multi‐dimensional data analysis, Parallel Coordinates are a widely used approach due to their linear scalability with the number of dimensions and bijective (i.e., loss‐less) data transformation. However, selections in Parallel Coordinates are typically achieved by a binary brushing operation on the axes, which does not allow the users to map their mental model of uncertainties to their selection. We, therefore, propose Probabilistic Parallel Coordinates as a natural extension of the classical Parallel Coordinates approach that integrates probabilistic brushing on the axes. It supports the interactive modeling of a probability distribution for each parallel coordinate. The selections on multiple axes are combined accordingly. An efficient rendering on a compute shader facilitates interactive frame rates. We evaluated our open‐source tool with practitioners and compared it to classical Parallel Coordinates on multiple regression and uncertain selection tasks in user studies.

Partitionings (or segmentations) divide a given domain into disjoint connected regions whose union forms again the entire domain. Multi-dimensional partitionings occur, for example, when analyzing parameter spaces of simulation models, where each segment of the partitioning represents a region of similar model behavior. Having computed a partitioning, one is commonly interested in understanding how large the segments are and which segments lie next to each other. While visual representations of 2D domain partitionings that reveal sizes and neighborhoods are straightforward, this is no longer the case when considering multi-dimensional domains of three or more dimensions. We propose an algorithm for computing 2D embeddings of multi-dimensional partitionings. The embedding shall have the following properties: It shall maintain the topology of the partitioning and optimize the area sizes and joint boundary lengths of the embedded segments to match the respective sizes and lengths in the multi-dimensional domain. We demonstrate the effectiveness of our approach by applying it to different use cases, including the visual exploration of 3D spatial domain segmentations and multi-dimensional parameter space partitionings of simulation ensembles. We numerically evaluate our algorithm with respect to how well sizes and lengths are preserved depending on the dimensionality of the domain and the number of segments.

Sensitivity analyses of simulation ensembles determine how simulation parameters influence the simulation's outcome. Commonly, one global numerical sensitivity value is computed per simulation parameter. However, when considering 3D spatial simulations, the analysis of localized sensitivities in different spatial regions is of importance in many applications. For analyzing the spatial variation of parameter sensitivity, one needs to compute a spatial sensitivity scalar field per simulation parameter. Given $n$n simulation parameters, we obtain multi-field data consisting of $n$n scalar fields when considering all simulation parameters. We propose an interactive visual analytics solution to analyze the multi-field sensitivity data. It supports the investigation of how strongly and in what way individual parameters influence the simulation outcome, in which spatial regions this is happening, and what the interplay of the simulation parameters is. Its central component is an overview visualization of all sensitivity fields that avoids 3D occlusions by linearizing the data using an adapted scheme of data-driven space-filling curves. The spatial sensitivity values are visualized in a combination of a Horizon Graph and a line chart. We validate our approach by applying it to synthetic and real-world ensemble data.

When analyzing heterogeneous data comprising numerical and categorical attributes, it is common to treat the different data types separately or transform the categorical attributes to numerical ones. The transformation has the advantage of facilitating an integrated multi-variate analysis of all attributes. We propose a novel technique for transforming categorical data into interpretable numerical feature vectors using Large Language Models (LLMs). The LLMs are used to identify the categorical attributes' main characteristics and assign numerical values to these characteristics, thus generating a multi-dimensional feature vector. The transformation can be computed fully automatically, but due to the interpretability of the characteristics, it can also be adjusted intuitively by an end user. We provide a respective interactive tool that aims to validate and possibly improve the AI-generated outputs. Having transformed a categorical attribute, we propose novel methods for ordering and color-coding the categories based on the similarities of the feature vectors.

Scatterplots provide a visual representation of bivariate data (or 2D embeddings of multivariate data) that allows for effective analyses of data dependencies, clusters, trends, and outliers. Unfortunately, classical scatterplots suffer from scalability issues, since growing data sizes eventually lead to overplotting and visual clutter on a screen with a fixed resolution, which hinders the data analysis process. We propose an algorithm that compensates for irregular sample distributions by a smooth transformation of the scatterplot's visual domain. Our algorithm evaluates the scatterplot's density distribution to compute a regularization mapping based on integral images of the rasterized density function. The mapping preserves the samples' neighborhood relations. Few regularization iterations suffice to achieve a nearly uniform sample distribution that efficiently uses the available screen space. We further propose approaches to visually convey the transformation that was applied to the scatterplot and compare them in a user study. We present a novel parallel algorithm for fast GPU-based integral-image computation, which allows for integrating our de-cluttering approach into interactive visual data analysis systems.

A high number of samples often leads to occlusion in scatter-plots, which hinders data perception and analysis. De-cluttering approaches based on spatial transformation reduce visual clutter by remapping samples using the entire available scatterplot domain. Such regularized scatterplots may still be used for data analysis tasks, if the spatial transformation is smooth and preserves the original neighborhood relations of samples. Recently, Rave et al. [21] proposed an efficient regularization method based on integral images. We propose a generalization of their regularization scheme using sector-based transformations with the aim of increasing sample uniformity of the resulting scatterplot. We document the improvement of our approach using various uniformity measures.

Abstract Latent feature spaces of deep neural networks are frequently used to effectively capture semantic characteristics of a given dataset. In the context of spatio‐temporal ensemble data, the latent space represents a similarity space without the need of an explicit definition of a field similarity measure. Commonly, these networks are trained for specific data within a targeted application. We instead propose a general training strategy in conjunction with a deep neural network architecture, which is readily applicable to any spatio‐temporal ensemble data without re‐training. The latent‐space visualization allows for a comprehensive visual analysis of patterns and temporal evolution within the ensemble. With the use of SimilarityNet, we are able to perform similarity analyses on large‐scale spatio‐temporal ensembles in less than a second on commodity consumer hardware. We qualitatively compare our results to visualizations with established field similarity measures to document the interpretability of our latent space visualizations and show that they are feasible for an in‐depth basic understanding of the underlying temporal evolution of a given ensemble.

Abstract The application of parallel axes for the interactive visual analysis of multidimensional data is a widely used concept. While multidimensional data sets are commonly heterogeneous in nature, i.e. data items contain both numerical and categorical (including ordinal) attribute values, the use of parallel axes often assumes either numerical or categorical attributes. While Parallel Coordinates and their large variety of extensions focus on numerical data, Parallel Sets and related methods focus on categorical attributes. While both concepts allow for displaying heterogeneous data, no clear strategies have been defined for representing categories in Parallel Coordinates or discretization of continuous ranges in Parallel Sets. In practice, type conversion as a pre‐processing step can be used as well as coordinated views of numerical and categorical data visualizations. We evaluate traditional and state‐of‐the‐art approaches with respect to the interplay of categorical and numerical dimensions for querying probability‐based events. We also compare against a heterogeneous Parallel Coordinates/Parallel Set approach with a novel interface between categorical and numerical axes . We show that approaches for mapping categorical data to numerical axis representations can lead to lower accuracy in answering probability‐based questions and higher response times than hybrid approaches in multiple‐event scenarios.

Abstract Given a time‐varying scalar field, the analysis of correlations between different spatial regions, i.e., the linear dependence of time series within these regions, provides insights into the structural properties of the data. In this context, regions are connected components of the spatial domain with high time series correlations. The detection and analysis of such regions is often performed globally, which requires pairwise correlation computations that are quadratic in the number of spatial data samples. Thus, operations based on all pairwise correlations are computationally demanding, especially when dealing with ensembles that model the uncertainty in the spatio‐temporal phenomena using multiple simulation runs. We propose a two‐step procedure: In a first step, we map the spatial samples to a 3D embedding based on a pairwise correlation matrix computed from the ensemble of time series. The 3D embedding allows for a one‐to‐one mapping to a 3D color space such that the outcome can be visually investigated by rendering the colors for all samples in the spatial domain. In a second step, we generate a hierarchical image segmentation based on the color images. From then on, we can visually analyze correlations of regions at all levels in the hierarchy within an interactive setting, which includes the uncertainty‐aware analysis of the region's time series correlation and respective time lags.

The development of new pseudo-random number generators (PRNGs) has steadily increased over the years. Commonly, PRNGs' randomness is "measured" by using statistical pass/fail suite tests, but the question remains, which PRNG is the best when compared to others. Existing randomness tests lack means for comparisons between PRNGs, since they are not quantitatively analysing. It is, therefore, an important task to analyze the quality of randomness for each PRNG, or, in general, comparing the randomness property among PRNGs. In this paper, we propose a novel visual approach to analyze PRNGs randomness allowing for a ranking comparison concerning the PRNGs' quality. Our analysis approach is applied to ensembles of time series which are outcomes of different PRNG runs. The ensembles are generated by using a single PRNG method with different parameter settings or by using different PRNG methods. We propose a similarity metric for PRNG time series for randomness and apply it within an interactive visual approach for analyzing similarities of PRNG time series and relating them to an optimal result of perfect randomness. The interactive analysis leads to an unsupervised classification, from which respective conclusions about the impact of the PRNGs' parameters or rankings of PRNGs on randomness are derived. We report new findings using our approach in a study of randomness for state-of-the-art numerical PRNGs such as LCG, PCG, SplitMix, Mersenne Twister, and RANDU as well as chaos-based PRNG families such as K-Logistic map and K-Tent map with varying parameter K.

Data tours follow an exploratory multi-dimensional data visualization concept that provides animations of projections of the multidimensional data to a 2D visual space. To create an animation, a sequence of key projections is provided and morphings between each pair of consecutive key projections are computed, which then can be stitched together to form the data tour. The morphings should be smooth so that a user can easily follow the transformations, and their computations shall be fast to allow for their integration into an interactive visual exploration process. Moreover, if the key projections are chosen to satisfy additional conditions, it is desirable that these conditions are maintained during morphing. Shape preservation is such a desirable condition, as it avoids shape distortions that may otherwise be caused by a projection. We develop a novel efficient morphing algorithms for computing shape-preserving data tours, i.e., data tours constructed for a sequence of shape-preserving linear projections. We propose a stepping strategy for the morphing to avoid discontinuities in the evolution of the projections, where we represent the linear projections using a star-coordinates system. Our algorithms are less computationally involved, produce smoother morphings, and require less user-defined parameter settings than existing state-of-the-art approaches.

Abstract Mixed data sets containing numerical and categorical attributes are nowadays ubiquitous. Converting them to one attribute type may lead to a loss of information. We present an approach for handling numerical and categorical attributes in a holistic view. For data sets with many attributes, dimensionality reduction (DR) methods can help to generate visual representations involving all attributes. While automatic DR for mixed data sets is possible using weighted combinations, the impact of each attribute on the resulting projection is difficult to measure. Interactive support allows the user to understand the impact of data dimensions in the formation of patterns. Star Coordinates is a well‐known interactive linear DR technique for multi‐dimensional numerical data sets. We propose to extend Star Coordinates and its initial configuration schemes to mixed data sets. In conjunction with analysing numerical attributes, our extension allows for exploring the impact of categorical dimensions and individual categories on the structure of the entire data set. The main challenge when interacting with Star Coordinates is typically to find a good configuration of the attribute axes. We propose a guided mixed data analysis based on maximizing projection quality measures by the use of recommended transformations, named hints, in order to find a proper configuration of the attribute axes.

Principal curves are a long-standing and well-known method for summarizing large scatterplots. They are defined as self-consistent curves (or curve sets in the more general case) that locally pass through the middle of the scatterplot data. However, computing principal curves that capture well complex scatterplot topologies and are robust to noise is hard and/or slow for large scatterplots. We present a fast and robust approach for computing principal graphs (a generalization of principal curves for more complex topologies) inspired by the similarity to medial descriptors (curves locally centered in a shape). Compared to state-of-the-art methods for computing principal graphs, we outperform these in terms of computational scalability and robustness to noise and resolution. We also demonstrate the advantages of our method over other scatterplot summarization approaches.

In immersive virtual environments (IVE), users' visual and auditory perception is replaced by computer-generated stimuli. Thus, knowing the positions of real objects is crucial for physical safety. While some solutions exist, e. g., using virtual replicas or visible cues indicating the interaction space boundaries, these are limiting the IVE design or depend on the hardware setup. Moreover, most solutions cannot handle lost tracking, erroneous tracker calibration, or moving obstacles. However, these are common scenarios especially for the increasingly popular home virtual reality settings. In this paper, we present a stand-alone hardware device designed to alert IVE users for potential collisions with real-world objects. It uses distance sensors mounted on a head-mounted display (HMD) and vibro-tactile actuators inserted into the HMD's face cushion. We implemented different types of sensor-actuator mappings with the goal to find a mapping function that is minimally obtrusive in normal use, but efficiently alerting in risk situations.

In this paper, we present Haptic Prop, a semi-passive, pico-powered, tangible prop, which is able to provide programmable friction for interaction with a tabletop setup, such as interactive workbenches or fish-tank VR. We explore the interaction space, its basic components, and constraints. Haptic Prop can be used to provide haptic feedback to the user at different levels and in different directions. We have conducted a preliminary user study evaluating the users' acceptance for the device and their ability to detect the programmed level of friction for rotation and linear movements. While currently still preliminary, the results demonstrate the utility of our device and outline some promising directions for future work.

Abstract The purpose of multi‐run simulations is often to capture the variability of the output with respect to different initial settings. Comparative analysis of multi‐run spatio‐temporal simulation data requires us to investigate the differences in the dynamics of the simulations' changes over time. To capture the changes and differences, aggregated statistical information may often be insufficient, and it is desirable to capture the local differences between spatial data fields at different times and between different runs. To calculate the pairwise similarity between data fields, we generalize the concept of isosurface similarity from individual surfaces to entire fields and propose efficient computation strategies. The described approach can be applied considering a single scalar field for all simulation runs or can be generalized to a similarity measure capturing all data fields of a multi‐field data set simultaneously. Given the field similarity, we use multi‐dimensional scaling approaches to visualize the similarity in two‐dimensional or three‐dimensional projected views as well as plotting one‐dimensional similarity projections over time. Each simulation run is depicted as a polyline within the similarity maps. The overall visual analysis concept can be applied using our proposed field similarity or any other existing measure for field similarity. We evaluate our measure in comparison to popular existing measures for different configurations and discuss their advantages and limitations. We apply them to generate similarity maps for real‐world data sets within the overall concept for comparative visualization of multi‐run spatio‐temporal data and discuss the results.

Dimensionality reduction is commonly applied to multidimensional data to reduce the complexity of their analysis. In visual analysis systems, projections embed multidimensional data into 2D or 3D spaces for graphical representation. To facilitate a robust and accurate analysis, essential characteristics of the multidimensional data shall be preserved when projecting. Orthographic star coordinates is a state-of-the-art linear projection method that avoids distortion of multidimensional clusters by restricting interactive exploration to orthographic projections. However, existing numerical methods for computing orthographic star coordinates have a number of limitations when putting them into practice. We overcome these limitations by proposing the novel concept of shapepreserving star coordinates where shape preservation is assured using a superset of orthographic projections. Our scheme is explicit, exact, simple, fast, parameter-free, and stable. To maintain a valid shape-preserving star-coordinates configuration during user interaction with one of the star-coordinates axes, we derive an algorithm that only requires us to modify the configuration of one additional compensatory axis. Different design goals can be targeted by using different strategies for selecting the compensatory axis. We propose and discuss four strategies including a strategy that approximates orthographic star coordinates very well and a data-driven strategy. We further present shape-preserving morphing strategies between two shape-preserving configurations, which can be adapted for the generation of data tours. We apply our concept to multiple data analysis scenarios to document its applicability and validate its desired properties.

Scatterplot matrices (SPLOMs) are widely used for exploring multidimensional data. Scatterplot diagnostics (scagnostics) approaches measure characteristics of scatterplots to automatically find potentially interesting plots, thereby making SPLOMs more scalable with the dimension count. While statistical measures such as regression lines can capture orientation, and graph-theoretic scagnostics measures can capture shape, there is no scatterplot characterization measure that uses both descriptors. Based on well-known results in shape analysis, we propose a scagnostics approach that captures both scatterplot shape and orientation using skeletons (or medial axes). Our representation can handle complex spatial distributions, helps discovery of principal trends in a multiscale way, scales visually well with the number of samples, is robust to noise, and is automatic and fast to compute. We define skeleton-based similarity metrics for the visual exploration and analysis of SPLOMs. We perform a user study to measure the human perception of scatterplot similarity and compare the outcome to our results as well as to graph-based scagnostics and other visual quality metrics. Our skeleton-based metrics outperform previously defined measures both in terms of closeness to perceptually-based similarity and computation time efficiency.

Abstract To understand how topology shapes the dynamics in excitable networks is one of the fundamental problems in network science when applied to computational systems biology and neuroscience. Recent advances in the field discovered the influential role of two macroscopic topological structures, namely hubs and modules. We propose a visual analytics approach that allows for a systematic exploration of the role of those macroscopic topological structures on the dynamics in excitable networks. Dynamical patterns are discovered using the dynamical features of excitation ratio and co‐activation. Our approach is based on the interactive analysis of the correlation of topological and dynamical features using coordinated views. We designed suitable visual encodings for both the topological and the dynamical features. A degree map and an adjacency matrix visualization allow for the interaction with hubs and modules, respectively. A barycentric‐coordinates layout and a multi‐dimensional scaling approach allow for the analysis of excitation ratio and co‐activation, respectively. We demonstrate how the interplay of the visual encodings allows us to quickly reconstruct recent findings in the field within an interactive analysis and even discovered new patterns. We apply our approach to network models of commonly investigated topologies as well as to the structural networks representing the connectomes of different species. We evaluate our approach with domain experts in terms of its intuitiveness, expressiveness, and usefulness.

Multi-run simulations are widely used to investigate how simulated processes evolve depending on varying initial conditions. Frequently, such simulations model the change of spatial phenomena over time. Isocontours have proven to be effective for the visual representation and analysis of 2D and 3D spatial scalar fields. We propose a novel visualization approach for multi-run simulation data based on isocontours. By introducing a distance function for isocontours, we generate a distance matrix used for a multidimensional scaling projection. Multiple simulation runs are represented by polylines in the projected view displaying change over time. We propose a fast calculation of isocontour differences based on a quasi-Monte Carlo approach. For interactive visual analysis, we support filtering and selection mechanisms on the multi-run plot and on linked views to physical space visualizations. Our approach can be effectively used for the visual representation of ensembles, for pattern and outlier detection, for the investigation of the influence of simulation parameters, and for a detailed analysis of the features detected. The proposed method is applicable to data of any spatial dimensionality and any spatial representation (gridded or unstructured). We validate our approach by performing a user study on synthetic data and applying it to different types of multi-run spatio-temporal simulation data.

Similarity-based layouts generated by multidimensional projections or other dimension reduction techniques are commonly used to visualize high-dimensional data. Many projection techniques have been recently proposed addressing different objectives and application domains. Nonetheless, very little is known about the effectiveness of the generated layouts from a user's perspective, how distinct layouts from the same data compare regarding the typical visualization tasks they support, or how domain-specific issues affect the outcome of the techniques. Learning more about projection usage is an important step towards both consolidating their role in high-dimensional data analysis and taking informed decisions when choosing techniques. This work provides a contribution towards this goal. We describe the results of an investigation on the performance of layouts generated by projection techniques as perceived by their users. We conducted a controlled user study to test against the following hypotheses: (1) projection performance is task-dependent; (2) certain projections perform better on certain types of tasks; (3) projection performance depends on the nature of the data; and (4) subjects prefer projections with good segregation capability. We generated layouts of high-dimensional data with five techniques representative of different projection approaches. As application domains we investigated image and document data. We identified eight typical tasks, three of them related to segregation capability of the projection, three related to projection precision, and two related to incurred visual cluttering. Answers to questions were compared for correctness against `ground truth' computed directly from the data. We also looked at subject confidence and task completion times. Statistical analysis of the collected data resulted in Hypotheses 1 and 3 being confirmed, Hypothesis 2 being confirmed partially and Hypotheses 4 could not be confirmed. We discuss our findings in comparison with some numerical measures of projection layout quality. Our results offer interesting insight on the use of projection layouts in data visualization tasks and provide a departing point for further systematic investigations.

Multidimensional data impose a challenge for visual analyses. Commonly, dimensionality reduction techniques are used to project the multidimensional data into a 2D visual space. Poco et al. [9] showed that projection into a 3D visual space can increase the performance of common visual analysis tasks due to a higher projection precision. They also backed up their findings with a user study. However, when conducting the user study they displayed the 3D visual space on a 2D screen, which may impede the correct perception of the third dimension. In this paper, we present a study that investigates the effect of stereoscopic environments when used for the visual analysis of multidimensional data after projection into a 3D visual space. We conducted a controlled user study to compare correctness, timing, and confidence in segregation and precision tasks when performed in stereoscopic immersive environments and on a nonstereoscopic 2D screen. In terms of the stereoscopic immersive environments, we operated on and compared results obtained with two setup: a single screen and a six-sided highly immersive system, in both of which interaction was performed with a 3D input device. We investigated whether the stereoscopic immersive environments have an effect on user performance depending on the visual encodings. We used both 3D scatter plots and cluster visualizations in the form of enclosing surfaces or hulls for the visual analysis tasks.

Visual attention and eye movement are important subjects of investigation with a wide area of application scenarios. The analysis of eye tracking data comprises examinations of differences between subjects and groups of subjects and identification of patterns and outliers within and between groups. To allow for such complex queries in an intuitive way, we present an interactive visual analysis tool using coordinated views. The views display areas of interest, statistical graphics of fixation times and time series and comprise novel visual encodings for spatiotemporal analysis and subject similarities. Interaction mechanisms allow for a refined analysis that takes into account all aspects of the data. We apply our approach to eye tracking data from psychological experiments and conduct a case study to demonstrate its effectiveness and intuitiveness.

Abstract For the visual analysis of multidimensional data, dimension reduction methods are commonly used to project to a lower‐dimensional visual space. In the context of multifields, i.e., volume data with a multidimensional attribute space, the spatial arrangement of the samples in the volumetric domain can be exploited to generate a Continuous Representation of the Projected Attribute Space (CoRPAS). Here, the sample locations in the volumetric domain may be arranged in a structured or unstructured way and may or may not be connected by a grid or a mesh. We propose an approach to generate CoRPAS for any sample arrangement using an isotropic density function. An interactive visual exploration system with three coordinated views of volume visualization, CoRPAS, and an interaction widget based on star coordinates is presented. The star‐coordinates widget provides an intuitive means for the user to change the projection matrix. The coordinated views allow for feature selection in form of brushing and linking. The approach is applied to both synthetic data and data resulting from numerical simulations of physical phenomena. In particular, simulations based on Smoothed Particle Hydrodynamics are addressed, where the simulation kernel can be used to produce a CoRPAS that is consistent with the simulation. We also show how a logarithmic scaling of attribute values in CoRPAS is supported, which is of high practical relevance.

Path generations are a necessary integral part of any automated machining approach using 3-axis robots. Given an input path in form of a piecewise linear curve, we automatically generate an optimized path that lies within a given error bound or tolerance band of the input path. The optimization is targeted at minimizing the processing time of the machining process. As sharp turns require the robot to slow down, we want to minimize the local curvature at each point of the curve. Our approach is an efficient offline algorithm that consists of several processing steps. Ina preprocessing step, we analyze the input path and split it into small groups. The groups are categorized and can be handled independently and locally. We apply a local sleeve concept for complicated groups and a local Bezierapproximation for simple groups. In a post processing step the groups are combined to form a G2-continuous path. Our approach achieves high-quality results that are comparable to the sleeves approach while being significantly more efficient (speed-up of one order of magnitude) when applied to real-world problems.

Abstract Visualization of high‐dimensional data requires a mapping to a visual space. Whenever the goal is to preserve similarity relations a frequent strategy is to use 2D projections, which afford intuitive interactive exploration, e.g., by users locating and selecting groups and gradually drilling down to individual objects. In this paper, we propose a framework for projecting high‐dimensional data to 3D visual spaces, based on a generalization of the Least‐Square Projection (LSP). We compare projections to 2D and 3D visual spaces both quantitatively and through a user study considering certain exploration tasks. The quantitative analysis confirms that 3D projections outperform 2D projections in terms of precision. The user study indicates that certain tasks can be more reliably and confidently answered with 3D projections. Nonetheless, as 3D projections are displayed on 2D screens, interaction is more difficult. Therefore, we incorporate suitable interaction functionalities into a framework that supports 3D transformations, predefined optimal 2D views, coordinated 2D and 3D views, and hierarchical 3D cluster definition and exploration. For visually encoding data clusters in a 3D setup, we employ color coding of projected data points as well as four types of surface renderings. A second user study evaluates the suitability of these visual encodings. Several examples illustrate the framework's applicability for both visual exploration of multidimensional abstract (non‐spatial) data as well as the feature space of multi‐variate spatial data.

BACKGROUND: Quantification of different types of cells is often needed for analysis of histological images. In our project, we compute the relative number of proliferating hepatocytes for the evaluation of the regeneration process after partial hepatectomy in normal rat livers. RESULTS: Our presented automatic approach for hepatocyte (HC) quantification is suitable for the analysis of an entire digitized histological section given in form of a series of images. It is the main part of an automatic hepatocyte quantification tool that allows for the computation of the ratio between the number of proliferating HC-nuclei and the total number of all HC-nuclei for a series of images in one processing run. The processing pipeline allows us to obtain desired and valuable results for a wide range of images with different properties without additional parameter adjustment. Comparing the obtained segmentation results with a manually retrieved segmentation mask which is considered to be the ground truth, we achieve results with sensitivity above 90% and false positive fraction below 15%. CONCLUSIONS: The proposed automatic procedure gives results with high sensitivity and low false positive fraction and can be applied to process entire stained sections.

Abstract Signed distance functions (SDF) to explicit or implicit surface representations are intensively used in various computer graphics and visualization algorithms. Among others, they are applied to optimize collision detection, are used to reconstruct data fields or surfaces, and, in particular, are an obligatory ingredient for most level set methods. Level set methods are common in scientific visualization to extract surfaces from scalar or vector fields. Usual approaches for the construction of an SDF to a surface are either based on iterative solutions of a special partial differential equation or on marching algorithms involving a polygonization of the surface. We propose a novel method for a non‐iterative approximation of an SDF and its derivatives in a vicinity of a manifold. We use a second‐order algebraic fitting scheme to ensure high accuracy of the approximation. The manifold is defined (explicitly or implicitly) as an isosurface of a given volumetric scalar field. The field may be given at a set of irregular and unstructured samples. Stability and reliability of the SDF generation is achieved by a proper scaling of weights for the Moving Least Squares approximation, accurate choice of neighbors, and appropriate handling of degenerate cases. We obtain the solution in an explicit form, such that no iterative solving is necessary, which makes our approach fast.

BACKGROUND: Protein-protein interaction (PPI) is fundamental to many biological processes. In the course of evolution, biological networks such as protein-protein interaction networks have developed. Biological networks of different species can be aligned by finding instances (e.g. proteins) with the same common ancestor in the evolutionary process, so-called orthologs. For a better understanding of the evolution of biological networks, such aligned networks have to be explored. Visualization can play a key role in making the various relationships transparent. RESULTS: We present a novel visualization system for aligned biological networks in 3D space that naturally embeds existing 2D layouts. In addition to displaying the intra-network connectivities, we also provide insight into how the individual networks relate to each other by placing aligned entities on top of each other in separate layers. We optimize the layout of the entire alignment graph in a global fashion that takes into account inter- as well as intra-network relationships. The layout algorithm includes a step of merging aligned networks into one graph, laying out the graph with respect to application-specific requirements, splitting the merged graph again into individual networks, and displaying the network alignment in layers. In addition to representing the data in a static way, we also provide different interaction techniques to explore the data with respect to application-specific tasks. CONCLUSION: Our system provides an intuitive global understanding of aligned PPI networks and it allows the investigation of key biological questions. We evaluate our system by applying it to real-world examples documenting how our system can be used to investigate the data with respect to these key questions. Our tool VANLO (Visualization of Aligned Networks with Layout Optimization) can be accessed at http://www.math-inf.uni-greifswald.de/VANLO.

Abstract Visual analytics of multidimensional multivariate data is a challenging task because of the difficulty in understanding metrics in attribute spaces with more than three dimensions. Frequently, the analysis goal is not to look into individual records but to understand the distribution of the records at large and to find clusters of records with similar attribute values. A large number of (typically hierarchical) clustering algorithms have been developed to group individual records to clusters of statistical significance. However, only few visualization techniques exist for further exploring and understanding the clustering results. We propose visualization and interaction methods for analyzing individual clusters as well as cluster distribution within and across levels in the cluster hierarchy. We also provide a clustering method that operates on density rather than individual records. To not restrict our search for clusters, we compute density in the given multidimensional multivariate space. Clusters are formed by areas of high density. We present an approach that automatically computes a hierarchical tree of high density clusters. To visually represent the cluster hierarchy, we present a 2D radial layout that supports an intuitive understanding of the distribution structure of the multidimensional multivariate data set. Individual clusters can be explored interactively using parallel coordinates when being selected in the cluster tree. Furthermore, we integrate circular parallel coordinates into the radial hierarchical cluster tree layout, which allows for the analysis of the overall cluster distribution. This visual representation supports the comprehension of the relations between clusters and the original attributes. The combination of the 2D radial layout and the circular parallel coordinates is used to overcome the overplotting problem of parallel coordinates when looking into data sets with many records. We apply an automatic coloring scheme based on the 2D radial layout of the hierarchical cluster tree encoding hue, saturation, and value of the HSV color space. The colors support linking the 2D radial layout to other views such as the standard parallel coordinates or, in case data is obtained from multidimensional spatial data, the distribution in object space.

Abstract Point clusters occur in both spatial and non‐spatial data. In the former context they may represent segmented particle data, in the latter context they may represent clusters in scatterplots. In order to visualize such point clusters, enclosing surfaces lead to much better comprehension than pure point renderings. We propose a flexible system for the generation of enclosing surfaces for 3D point clusters. We developed a GPU‐based 3D discrete Voronoi diagram computation that supports all surface extractions. Our system provides three different types of enclosing surfaces. By generating a discrete distance field to the point cluster and extracting an isosurface from the field, an enclosing surface with any distance to the point cluster can be generated. As a second type of enclosing surfaces, a hull of the point cluster is extracted. The generation of the hull uses a projection of the discrete Voronoi diagram of the point cluster to an isosurface to generate a polygonal surface. Generated hulls of non‐convex clusters are also non‐convex. The third type of enclosing surfaces can be created by computing a distance field to the hull and extracting an isosurface from the distance field. This method exhibits reduced bumpiness and can extract surfaces arbitrarily close to the point cluster without losing connectedness. We apply our methods to the visualization of multidimensional spatial and non‐spatial data. Multidimensional clusters are extracted and projected into a 3D visual space, where the point clusters are visualized. The respective clusters can also be visualized in object space when dealing with multidimensional particle data.

Data sets resulting from physical simulations typically contain a multitude of physical variables. It is, therefore, desirable that visualization methods take into account the entire multi-field volume data rather than concentrating on one variable. We present a visualization approach based on surface extraction from multi-field particle volume data. The surfaces segment the data with respect to the underlying multi-variate function. Decisions on segmentation properties are based on the analysis of the multi-dimensional feature space. The feature space exploration is performed by an automated multi-dimensional hierarchical clustering method, whose resulting density clusters are shown in the form of density level sets in a 3D star coordinate layout. In the star coordinate layout, the user can select clusters of interest. A selected cluster in feature space corresponds to a segmenting surface in object space. Based on the segmentation property induced by the cluster membership, we extract a surface from the volume data. Our driving applications are Smoothed Particle Hydrodynamics (SPH) simulations, where each particle carries multiple properties. The data sets are given in the form of unstructured point-based volume data. We directly extract our surfaces from such data without prior resampling or grid generation. The surface extraction computes individual points on the surface, which is supported by an efficient neighborhood computation. The extracted surface points are rendered using point-based rendering operations. Our approach combines methods in scientific visualization for object-space operations with methods in information visualization for feature-space operations.

Smooth surface extraction using partial differential equations (PDEs) is a well-known and widely used technique for visualizing volume data. Existing approaches operate on gridded data and mainly on regular structured grids. When considering unstructured point-based volume data where sample points do not form regular patterns nor are they connected in any form, one would typically resample the data over a grid prior to applying the known PDE-based methods. We propose an approach that directly extracts smooth surfaces from unstructured point-based volume data without prior resampling or mesh generation. When operating on unstructured data one needs to quickly derive neighborhood information. The respective information is retrieved by partitioning the 3D domain into cells using a kd-tree and operating on its cells. We exploit neighborhood information to estimate gradients and mean curvature at every sample point using a four-dimensional least-squares fitting approach. Gradients and mean curvature are required for applying the chosen PDE-based method that combines hyperbolic advection to an isovalue of a given scalar field and mean curvature flow. Since we are using an explicit time-integration scheme, time steps and neighbor locations are bounded to ensure convergence of the process. To avoid small global time steps, we use asynchronous local integration. We extract the surface by successively fitting a smooth auxiliary function to the data set. This auxiliary function is initialized as a signed distance function. For each sample and for every time step we compute the respective gradient, the mean curvature, and a stable time step. With these informations the auxiliary function is manipulated using an explicit Euler time integration. The process successively continues with the next sample point in time. If the norm of the auxiliary function gradient in a sample exceeds a given threshold at some time, the auxiliary function is reinitialized to a signed distance function. After convergence of the evolution, the resulting smooth surface is obtained by extracting the zero isosurface from the auxiliary function using direct isosurface extraction from unstructured point-based volume data and rendering the extracted surface using point-based rendering methods.

In modern biology, major efforts are undertaken to understand diseases, aging, evolution, and many other aspects of life. Biological networks play a key role in this research. Examples of such networks are protein-protein interaction (PPI) networks, gene regulatory networks, and metabolic pathways. Intuitive and comprehensible visualizations can significantly support the understanding and the analysis of such biological networks. Therefore, much research is conducted in the areas of bio-informatics and information visualization dealing with the problem of visualizing networks. In the course of evolution biological networks changed gradually. Therefore, the conserved core parts of the networks of different species can be aligned by matching the corresponding instances in the species, such as orthologous proteins or genes. Analyzing these alignments is of high relevance, as they convey significant information on protein function and organismal phenotype. For analysis purposes, the alignments have to be made transparent. Visualization methods are needed to display alignments in conjunction with the individual network structures. This is no standard network visualization problem, as we have to deal with more than one network and inter-network relationships. Several approaches to visualize aligned networks exist. In this survey we present and discuss these different approaches and report on their advantages and drawbacks. We draw conclusions on the applicability of the various approaches.

Vector field visualization approaches can broadly be categorized into approaches that directly visualize local or integrated flow and approaches that analyze the topological structure and visualize extracted features. Our goal was to come up with a method that falls into the first category, yet reveals structural information. We have developed a dense flow visualization method that shows the overall flow behavior while accentuating structural information without performing a topological analysis. Our method is based on a geometry-based flow integration step and a texture-based visual exploration step. The flow integration step generates a density field, which is written into a texture. The density field is generated by tracing particles under the influence of the underlying vector field. When using a quasi-random seeding strategy for initialization, the resulting density is high in attracting regions and low in repelling regions. Density is measured by the number of particles per region accumulated over time. We generate one density field using forward and one using backward propagation. The density fields are explored using texture-based rendering techniques. We generate the two output images separately and blend the results, which allows us to distinguish between inflow and outflow regions. We obtained dense flow visualizations that display the overall flow behavior, emphasize critical and separating regions, and indicate flow direction in the neighborhood of these regions. We have test our method for isolated first-order singularities and real data sets.

Isosurface extraction is a standard visualization method for scalar volume data and has been subject to research for decades. Nevertheless, to our knowledge, no isosurface extraction method exists that directly extracts surfaces from scattered volume data without 3D mesh generation or reconstruction over a structured grid. We propose a method based on spatial domain partitioning using a kd-tree and an indexing scheme for efficient neighbor search. Our approach consists of a geometry extraction and a rendering step. The geometry extraction step computes points on the isosurface by linearly interpolating between neighboring pairs of samples. The neighbor information is retrieved by partitioning the 3D domain into cells using a kd-tree. The cells are merely described by their index and bitwise index operations allow for a fast determination of potential neighbors. We use an angle criterion to select appropriate neighbors from the small set of candidates. The output of the geometry step is a point cloud representation of the isosurface. The final rendering step uses point-based rendering techniques to visualize the point cloud. Our direct isosurface extraction algorithm for scattered volume data produces results of quality close to the results from standard isosurface extraction algorithms for gridded volume data (like marching cubes). In comparison to 3D mesh generation algorithms (like Delaunay tetrahedrization), our algorithm is about one order of magnitude faster for the examples used in this paper.

We present a visual exploration system supporting protein analysis when using gel-free data acquisition methods. The data to be analyzed is obtained by coupling liquid chromatography (LC) with mass spectrometry (MS). LC-MS data have the properties of being nonequidistantly distributed in the time dimension (measured by LC) and being scattered in the mass-to-charge ratio dimension (measured by MS). We describe a hierarchical data representation and visualization method for large LC-MS data. Based on this visualization, we have developed a tool that supports various data analysis steps. Our visual tool provides a global understanding of the data, intuitive detection and classification of experimental errors, and extensions to LC-MS/MS, LC/LC-MS, and LC/LC-MS/MS data analysis. Due to the presence of randomly occurring rare isotopes within the same protein molecule, several intensity peaks may be detected that all refer to the same peptide. We have developed methods to unite such intensity peaks. This deisotoping step is visually documented by our system, such that misclassification can be detected intuitively. For differential protein expression analysis, we compute and visualize the differences in protein amounts between experiments. In order to compute the differential expression, the experimental data need to be registered. For registration, we perform a nonrigid warping step based on landmarks. The landmarks can be assigned automatically using protein identification methods. We evaluate our methods by comparing protein analysis with and without our interactive visualization-based exploration tool.

Natural-neighbor interpolation methods, such as Sibson's method, are well-known schemes for multivariate data fitting and reconstruction. Despite its many desirable properties, Sibson's method is computationally expensive and difficult to implement, especially when applied to higher-dimensional data. The main reason for both problems is the method's implementation based on a Voronoi diagram of all data points. We describe a discrete approach to evaluating Sibson's interpolant on a regular grid, based solely on finding nearest neighbors and rendering and blending d-dimensional spheres. Our approach does not require us to construct an explicit Voronoi diagram, is easily implemented using commodity three-dimensional graphics hardware, leads to a significant speed increase compared to traditional approaches, and generalizes easily to higher dimensions. For large scattered data sets, we achieve two-dimensional (2D) interpolation at interactive rates and 3D interpolation (3D) with computation times of a few seconds.

We present a flow visualization technique based on rendering geometry in a dense, uniform distribution. Flow is integrated using particle advection. By adopting ideas from texture-based techniques and taking advantage of parallelism and programmability of contemporary graphics hardware, we generate streamlines and pathlines addressing both steady and unsteady flow. Pipelining is used to manage seeding, advection, and expiration of streamlines/ pathlines with constant lifetime. We achieve high numerical accuracy by enforcing short particle lifetimes and employing a fourth-order integration method. The occlusion problem inherent to dense volumetric representations is addressed by applying multi-dimensional transfer functions (MDTFs), restricting particle attenuation to regions of certain physical behavior, or features. Geometry is rendered in graphics hardware using techniques such as depth sorting, illumination, haloing, flow orientation, and depth-based color attenuation to enhance visual perception. We achieve dense geometric three-dimensional flow visualization with interactive frame rates.

Differential protein expression analysis is one of the main challenges in proteomics. It denotes the search for proteins, whose encoding genes are differentially expressed under a given experimental setup. An important task in this context is to identify the differentially expressed proteins or, more generally, all proteins present in the sample. One of the most promising and recently widely used approaches for protein identification is to cleave proteins into peptides, separate the peptides using liquid chromatography, and determine the masses of the separated peptides using mass spectrometry. The resulting data needs to be analyzed and matched against protein sequence databases. The analysis step is typically done by searching for intensity peaks in a large number of 2D graphs. We present an interactive visualization tool for the exploration of liquid-chromatography/mass-spectrometry data in a 3D space, which allows for the understanding of the data in its entirety and a detailed analysis of regions of interest. We compute differential expression over the liquid-chromatography/mass-spectrometry domain and embed it visually in our system. Our exploration tool can treat single liquid-chromatography/mass-spectrometry data sets as well as data acquired using multi-dimensional protein identification technology. For efficiency purposes we perform a peak-preserving data resampling and multiresolution hierarchy generation prior to visualization.

Transfer functions are a standard technique used in volume rendering to assign color and opacity to a volume of a scalar field. Multidimensional transfer functions (MDTFs) have proven to be an effective way to extract specific features with subtle properties. As 3D texture-based methods gain widespread popularity for the visualization of steady and unsteady flow field data, there is a need to define and apply similar MDTFs to interactive 3D flow visualization. We exploit flow field properties such as velocity, gradient, curl, helicity, and divergence using vector calculus methods to define an MDTF that can be used to extract and track features in a flow field. We show how the defined MDTF can be applied to interactive 3D flow visualization by combining them with state-of-the-art texture-based flow visualization of steady and unsteady fields. We demonstrate that MDTFs can be used to help alleviate the problem of occlusion, which is one of the main inherent drawbacks of 3D texture-based flow visualization techniques. In our implementation, we make use of current graphics hardware to obtain interactive frame rates.

Multiresolution methods for representing data at multiple levels of detail are widely used for large-scale two- and three-dimensional data sets. We present a four-dimensional multiresolution approach for time-varying volume data. This approach supports a hierarchy with spatial and temporal scalability. The hierarchical data organization is based on /sup 4//spl radic/2 subdivision. The /sup n//spl radic/2-subdivision scheme only doubles the overall number of grid points in each subdivision step. This fact leads to fine granularity and high adaptivity, which is especially desirable in the spatial dimensions. For high-quality data approximation on each level of detail, we use quadrilinear B-spline wavelets. We present a linear B-spline wavelet lifting scheme based on /sup n//spl radic/2 subdivision to obtain narrow masks for the update rules. Narrow masks provide a basis for out-of-core data exploration techniques and view-dependent visualization of sequences of time steps.