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.

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.

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.

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.

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.

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.