Christian Heine’s research while affiliated with Leipzig University and other places

Jacobi sets are an important tool to study the relationship between functions. Defined as the set of all points where the function's gradients are linearly dependent, Jacobi sets extend the notion of critical point to multifields. In practice, Jacobi sets for piecewise-linear approximations of smooth functions can become very complex and large due to noise and numerical errors. Existing methods that simplify Jacobi sets exist, but either do not address how the functions' values have to change in order to have simpler Jacobi sets or remain purely theoretical. In this paper, we present a method that modifies 2D bivariate scalar fields such that Jacobi set components that are due to noise are removed, while preserving the essential structures of the fields. The method uses the Jacobi set to decompose the domain, stores the and weighs the resulting regions in a neighborhood graph, which is then used to determine which regions to join by collapsing the image of the region's cells. We investigate the influence of different tie-breaks when building the neighborhood graphs and the treatment of collapsed cells. We apply our algorithm to a range of datasets, both analytical and real-world and compare its performance to simple data smoothing.

Critical points mark locations in the domain where the level-set topology of a scalar function undergoes fundamental changes and thus indicate potentially interesting features in the data. Established methods exist to locate and relate such points in a deterministic setting, but it is less well understood how the concept of critical points can be extended to the analysis of uncertain data. Most methods for this task aim at finding likely locations of critical points or estimate the probability of their occurrence locally but do not indicate if critical points at potentially different locations in different realizations of a stochastic process are manifestations of the same feature, which is required to characterize the spatial uncertainty of critical points. Previous work on relating critical points across different realizations reported challenges for interpreting the resulting spatial distribution of critical points but did not investigate the causes. In this work, we provide a mathematical formulation of the problem of finding critical points with spatial uncertainty and computing their spatial distribution, which leads us to the notion of uncertain critical points. We analyze the theoretical properties of these structures and highlight connections to existing works for special classes of uncertain fields. We derive conditions under which well-interpretable results can be obtained and discuss the implications of those restrictions for the field of visualization. We demonstrate that the discussed limitations are not purely academic but also arise in real-world data.

Extracting level sets from scalar data is a fundamental operation in visualization with many applications. Recently, the concept of level set extraction has been extended to bivariate scalar fields. Prior work on vector field equivalence, wherein an analyst marks a region in the domain and is shown other regions in the domain with similar vector values, pointed out the need to make this extraction operation fast, so that analysts can work interactively. To date, the fast extraction of level sets from bivariate scalar fields has not been researched as extensively as for the univariate case. In this paper, we present a novel algorithm that extracts fiber lines, i.e., the preimages of so called control polygons (FSCP), for bivariate 2D data by joint traversal of bounding volume hierarchies for both grid and FSCP elements. We performed an extensive evaluation, comparing our method to a two-dimensional adaptation of the method proposed by Klacansky et al., as well as to the naive approach for fiber line extraction. The evaluation incorporates a vast array of configurations in several datasets. We found that our method provides a speedup of several orders of magnitudes compared to the naive algorithm and requires two thirds of the computation time compared to Klacansky et al. adapted for 2D.

An important task in visualization is the extraction and highlighting of dominant features in data to support users in their analysis process. Topological methods are a well-known means of identifying such features in deterministic fields. However, many real-world phenomena studied today are the result of a chaotic system that cannot be fully described by a single simulation. Instead, the variability of such systems is usually captured with ensemble simulations that produce a variety of possible outcomes of the simulated process. The topological analysis of such ensemble data sets and uncertain data, in general, is less well studied. In this work, we present an approach for the computation and visual representation of confidence intervals for the occurrence probabilities of critical points in ensemble data sets. We demonstrate the added value of our approach over existing methods for critical point prediction in uncertain data on a synthetic data set and show its applicability to a data set from climate research.

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.