Fiber surfaces and fiber lines, being the preimages of a bivariate function to a so-called control polygon in the function’s range, are the adaptation of isosurfaces and isolines to bivariate scalar fields. Previous works have proposed use cases for fiber surface and fiber line extraction using algorithmically generated control polygons with many line segments, but their authors have either not presented results or noted that the computation becomes prohibitive slow. We present an algorithm that speeds up fiber surface extraction by using dual bounding volume hierarchy (BVH) traversal, as well as a variation that combines the benefits of single and dual BVH traversal. We study the influence of the number of line segments in the control polygon on the performance of single and dual BVH traversal algorithms using data sets from various application domains and types of control polygons. We find that dual BVH traversal is several times faster in test cases where single BVH traversal is slow, facilitating the interactive exploration of fiber surfaces in many cases where existing methods are too slow.

Analyzing uncertainty in spatial data is a vital task in many domains, as for example with climate and weather simulation ensembles. Although many methods support the analysis of uncertain 2D data, such as uncertain isocontours or overlaying of statistical information on plots of the actual data, it is still a challenge to get a more detailed overview of 2D data together with its statistical properties. We present cumulative height fields, a visualization method for 2D scalar field ensembles using the marginal empirical distribution function and show preliminary results using volume rendering and slicing for the Max Planck Institute Grand Ensemble.

In an era of quickly growing data set sizes, information reduction methods such as extracting or highlighting characteristic features become more and more important for data analysis. For single scalar fields, topological methods can fill this role by extracting and relating critical points. While such methods are regularly employed to study single scalar fields, it is less well studied how they can be extended to uncertain data, as produced, e.g., by ensemble simulations. Motivated by our previous work on visualization in climate research, we study new methods to characterize critical points in ensembles of 2D scalar fields. Previous work on this topic either assumed or required specific distributions, did not account for uncertainty introduced by approximating the underlying latent distributions by a finite number of fields, or did not allow to answer all our domain experts' questions. In this work, we use Bayesian inference to estimate the probability of critical points, either of the original ensemble or its bootstrapped mean. This does not make any assumptions on the underlying distribution and allows to estimate the sensitivity of the results to finite-sample approximations of the underlying distribution. We use color mapping to depict these probabilities and the stability of their estimation. The resulting images can, e.g., be used to estimate how precise the critical points of the mean-field are. We apply our method to synthetic data to validate its theoretical properties and compare it with other methods in this regard. We also apply our method to the data from our previous work, where it provides a more accurate answer to the domain experts' research questions.

A driving factor of the winter weather in Western Europe is the North Atlantic Oscillation (NAO), manifested by fluctuations in the difference of sea level pressure between the Icelandic Low and the Azores High. Different methods have been developed that describe the strength of this oscillation, but they rely on certain assumptions, e.g., fixed positions of these two pressure systems. It is possible that climate change affects the mean location of both the Low and the High and thus the validity of these descriptive methods. This study is the first to visually analyze large ensemble climate change simulations (the MPI Grand Ensemble) to robustly assess shifts of the drivers of the NAO phenomenon using the uncertain northern hemispheric surface pressure fields. For this, we use a sliding window approach and compute empirical orthogonal functions (EOFs) for each window and ensemble member, then compare the uncertainty of local extrema in the results as well as their temporal evolution across different CO2scenarios. We find systematic northeastward shifts in the location of the pressure systems that correlate with the simulated warming. Applying visualization techniques for this analysis was not straightforward; we reflect and give some lessons learned for the field of visualization.

To assess the reliability of weather forecasts and climate simulations, common practice is to generate large ensembles of numerical simulations. Analyzing such data is challenging and requires pattern and feature detection. For single time-dependent scalar fields, empirical orthogonal functions (EOFs) are a proven means to identify the main variation. In this paper, we present an extension of that concept to time-dependent ensemble data. We applied our methods to two ensemble data sets from climate research in order to investigate the North Atlantic Oscillation (NAO) and East Atlantic (EA) pattern.

Visualization designs typically need to be evaluated with user studies, because their suitability for a particular task is hard to predict. What the field of visualization is currently lacking are theories and models that can be used to explain why certain designs work and others do not. This paper outlines a general framework for modeling visualization processes that can serve as the first step towards such a theory. It surveys related research in mathematical and computational psychology and argues for the use of dynamic Bayesian networks to describe these time-dependent, probabilistic processes. It is discussed how these models could be used to aid in design evaluation. The development of concrete models will be a long process. Thus, the paper outlines a research program sketching how to develop prototypes and their extensions from existing models, controlled experiments, and observational studies.

When point clouds are labeled in information visualization applications, sophisticated guidelines as in cartography do not yet exist. Existing naive strategies may mislead as to which points belong to which label. To inform improved strategies, we studied factors influencing this phenomenon. We derived a class of labeled point cloud representations from existing applications and we defined different models predicting how humans interpret such complex representations, focusing on their geometric properties. We conducted an empirical study, in which participants had to relate dots to labels in order to evaluate how well our models predict. Our results indicate that presence of point clusters, label size, and angle to the label have an effect on participants' judgment as well as that the distance measure types considered perform differently discouraging the use of label centers as reference points.

Abstract This paper presents the state of the art in the area of topology‐based visualization. It describes the process and results of an extensive annotation for generating a definition and terminology for the field. The terminology enabled a typology for topological models which is used to organize research results and the state of the art. Our report discusses relations among topological models and for each model describes research results for the computation, simplification, visualization, and application. The paper identifies themes common to subfields, current frontiers, and unexplored territory in this research area.

Topological and structural analysis of multivariate data is aimed at improving the understanding and usage of such data through identification of intrinsic features and structural relationships among multiple variables. We present two novel methods for simplifying so-called Pareto sets that describe such structural relationships. Such simplification is a precondition for meaningful visualization of structurally rich or noisy data. As a framework for simplification operations, we introduce a decomposition of the data domain into regions of equivalent structural behavior and the reachability graph that describes global connectivity of Pareto extrema. Simplification is then performed as a sequence of edge collapses in this graph; to determine a suitable sequence of such operations, we describe and utilize a comparison measure that reflects the changes to the data that each operation represents. We demonstrate and evaluate our methods on synthetic and real-world examples.

Abstract How can the notion of topological structures for single scalar fields be extended to multifields? In this paper we propose a definition for such structures using the concepts of Pareto optimality and Pareto dominance. Given a set of piecewise‐linear, scalar functions over a common simplical complex of any dimension, our method finds regions of “consensus” among single fields’ critical points and their connectivity relations. We show that our concepts are useful to data analysis on real‐world examples originating from fluid‐flow simulations; in two cases where the consensus of multiple scalar vortex predictors is of interest and in another case where one predictor is studied under different simulation parameters. We also compare the properties of our approach with current alternatives.

Analyzing high-dimensional point clouds is a classical challenge in visual analytics. Traditional techniques, such as projections or axis-based techniques, suffer from projection artifacts, occlusion, and visual complexity. We propose to split data analysis into two parts to address these shortcomings. First, a structural overview phase abstracts data by its density distribution. This phase performs topological analysis to support accurate and nonoverlapping presentation of the high-dimensional cluster structure as a topological landscape profile. Utilizing a landscape metaphor, it presents clusters and their nesting as hills whose height, width, and shape reflect cluster coherence, size, and stability, respectively. A second local analysis phase utilizes this global structural knowledge to select individual clusters or point sets for further, localized data analysis. Focusing on structural entities significantly reduces visual clutter in established geometric visualizations and permits a clearer, more thorough data analysis. This analysis complements the global topological perspective and enables the user to study subspaces or geometric properties, such as shape.

The contour tree compactly describes scalar field topology. From the viewpoint of graph drawing, it is a tree with attributes at vertices and optionally on edges. Standard tree drawing algorithms emphasize structural properties of the tree and neglect the attributes. Applying known techniques to convey this information proves hard and sometimes even impossible. We present several adaptions of popular graph drawing approaches to the problem of contour tree drawing and evaluate them. We identify five esthetic criteria for drawing contour trees and present a novel algorithm for drawing contour trees in the plane that satisfies four of these criteria. Our implementation is fast and effective for contour tree sizes usually used in interactive systems (around 100 branches) and also produces readable pictures for larger trees, as is shown for an 800 branch example.

We present a novel method to visualize multidimensional point clouds. While conventional visualization techniques, like scatterplot matrices or parallel coordinates, have issues with either overplotting of entities or handling many dimensions, we abstract the data using topological methods before presenting it. We assume the input points to be samples of a random variable with a high-dimensional probability distribution which we approximate using kernel density estimates on a suitably reconstructed mesh. From the resulting scalar field we extract the join tree and present it as a topological landscape, a visualization metaphor that utilizes the human capability of understanding natural terrains. In this landscape, dense clusters of points show up as hills. The nesting of hills indicates the nesting of clusters. We augment the landscape with the data points to allow selection and inspection of single points and point sets. We also present optimizations to make our algorithm applicable to large data sets and to allow interactive adaption of our visualization to the kernel window width used in the density estimation.

In this paper, we present a novel three-stage process to visualize the structure of point clouds in arbitrary dimensions. To get insight into the structure and complexity of a data set, we would most preferably just look into it, e.g. by plotting its corresponding point cloud. Unfortunately, for orthogonal scatter plots, this only works up to three dimensions, and other visualizations, like parallel coordinates or scatterplot matrices, also have problems handling many dimensions and visual overlap of data entities. The presented solution tackles the problem of visualizing point clouds indirectly by visualizing the topology of their density distribution. The benefit of this approach is that this topology can be computed in arbitrary dimensions. Similar to examining scatter plots, this gives the important information like the number, size and nesting structure of accumulated regions. We view our approach as an alternative to cluster visualization. To create the visualization, we first estimate the density function using a novel high-dimensional interpolation scheme. Second, we compute that function's topology by means of the join tree, generate a corresponding 3-D terrain using the topological landscape metaphor introduced by Weber et al. (2007), and finally augment that landscape by placing the original data points at suitable locations.

Graphs are a versatile structure and abstraction for binary relationships between objects. To gain insight into such relationships, their corresponding graph can be visualized. In the past, many classes of graphs have been defined, e.g. trees, planar graphs, directed acyclic graphs, and visualization algorithms were proposed for these classes. Although many graphs may only be classified as "general" graphs, they can contain substructures that belong to a certain class. Archambault proposed the TopoLayout framework: rather than draw any arbitrary graph using one method, split the graph into components that are homogeneous with respect to one graph class and then draw each component with an algorithm best suited for this class. Graph products constitute a class that arises frequently in graph theory, but for which no visualization algorithm has been proposed until now. In this paper, we present an algorithm for drawing graph products and the aesthetic criterion graph product's drawings are subject to. We show that the popular High-Dimensional Embedder approach applied to cartesian products already respects this aestetic criterion, but has disadvantages. We also present how our method is integrated as a new component into the TopoLayout framework. Our implementation is used for further research of graph products in a biological context.

The huge amount of different automatic clustering methods emphasizes one thing: there is no optimal clustering method for all possible cases. In certain application domains, like genomics and natural language processing, it is not even clear if any of the already known clustering methods suffice. In such cases, an automatic clustering method is often followed by manual refinement. The refined version may then be used as either an illustration, a reference, or even an input for a rule based or other machine learning algorithm as a new clustering method. In this paper, we describe a novel interaction technique to manual cluster refinement using the metaphor of soap bubbles, represented by special implicit surfaces (blobs). For instance, entities can simply be moved inside and outside of these blobs. A modified force-directed layout process automatically arranges entities equidistant on the screen. The modifications include a reduction to the expected amount of computation per iteration down to O(|V| log |V|+|E|) in order to achieve a high response time for use in an interactive system. We also spend a considerable amount of effort making the display of blobs fast enough for an interactive system.

Dynamical models that explain the formation of spatial structures of RNA molecules have reached a complexity that requires novel visualization methods that help to analyze the validity of these models. Here, we focus on the visualization of so-called folding landscapes of a growing RNA molecule. Folding landscapes describe the energy of a molecule as a function of its spatial configuration; thus they are huge and high dimensional. Their most salient features, however, are encapsulated by their so-called barrier tree that reflects the local minima and their connecting saddle points. For each length of the growing RNA chain there exists a folding landscape. We visualize the sequence of folding landscapes by an animation of the corresponding barrier trees. To generate the animation, we adapt the foresight layout with tolerance algorithm for general dynamic graph layout problems. Since it is very general, we give a detailed description of each phase: constructing a supergraph for the trees, layout of that supergraph using a modified DoT algorithm, and presentation techniques for the final animation.