Electronic transitions in molecules due to the absorption or emission of light is a complex quantum mechanical process. Their study plays an important role in the design of novel materials. A common yet challenging task in the study is to determine the nature of electronic transitions, namely which subgroups of the molecule are involved in the transition by donating or accepting electrons, followed by an investigation of the variation in the donor-acceptor behavior for different transitions or conformations of the molecules. In this paper, we present a novel approach for the analysis of a bivariate field and show its applicability to the study of electronic transitions. This approach is based on two novel operators, the continuous scatterplot (CSP) lens operator and the CSP peel operator, that enable effective visual analysis of bivariate fields. Both operators can be applied independently or together to facilitate analysis. The operators motivate the design of control polygon inputs to extract fiber surfaces of interest in the spatial domain. The CSPs are annotated with a quantitative measure to further support the visual analysis. We study different molecular systems and demonstrate how the CSP peel and CSP lens operators help identify and study donor and acceptor characteristics in molecular systems.

This paper presents a prototypical visualization for the analysis of light-induced dynamics in molecules. It utilizes topological distances to find temporal patterns in scalar fields representing the electronic structure of such molecules and to illustrate the evolution of their features. It also provides a means to correlate these findings to the geometric evolution of the molecules.

Diffusion-weighted magnetic resonance imaging (D-MRI) is a technique to measure the diffusion of water, in biological tissues. It is used to detect microscopic patterns, such as neural fibers in the living human brain, with many medical and neuroscience applications e.g. for fiber tracking. In this paper, we consider High-Angular Resolution Diffusion Imaging (HARDI) which provides one of the richest representations of water diffusion. It records the movement of water molecules by measuring diffusion under 64 or more directions. A key challenge is that it generates high-dimensional, large, and complex datasets. In our work, we develop a novel representation that exploits the inherent sparsity of the HARDI signal by approximating it as a linear sum of basic atoms in an overcomplete data-driven dictionary using only a sparse set of coefficients. We show that this approach can be efficiently integrated into the standard q-ball imaging pipeline to compute the diffusion orientation distribution function (ODF). Sparse representations have the potential to reduce the size of the data while also giving some insight into the data. To explore the results, we provide a visualization of the atoms of the dictionary and their frequency in the data to highlight the basic characteristics of the data. We present our proposed pipeline and demonstrate its performance on 5 HARDI datasets.

The recent high-pressure synthesis of pentazolates and the subsequent stabilization of the aromatic [N-5](-) anion at atmospheric pressure have had an immense impact on nitrogen chemistry. Other aromatic nitrogen species have also been actively sought, including the hexaazabenzene N-6 ring. Although a variety of configurations and geometries have been proposed based on ab initio calculations, one that stands out as a likely candidate is the aromatic hexazine anion [N-6](4-). Here we present the synthesis of this species, realized in the high-pressure potassium nitrogen compound K9N56 formed at high pressures (46 and 61 GPa) and high temperature (estimated to be above 2,000 K) by direct reaction between nitrogen and KN3 in a laser-heated diamond anvil cell. The complex structure of K9N56-composed of 520 atoms per unit cell-was solved based on synchrotron single-crystal X-ray diffraction and corroborated by density functional theory calculations. The observed hexazine anion [N-6](4-) is planar and proposed to be aromatic.

Merge trees, a type of topological descriptor, serve to identify and summarize the topological characteristics associated with scalar fields. They present a great potential for the analysis and visualization of time-varying data. First, they give compressed and topology-preserving representations of data instances. Second, their comparisons provide a basis for studying the relations among data instances, such as their distributions, clusters, outliers, and periodicities. A number of comparative measures have been developed for merge trees. However, these measures are often computationally expensive since they implicitly consider all possible correspondences between critical points of the merge trees. In this paper, we perform geometry-aware comparisons of merge trees. The main idea is to decouple the computation of a comparative measure into two steps: a labeling step that generates a correspondence between the critical points of two merge trees, and a comparison step that computes distances between a pair of labeled merge trees by encoding them as matrices. We show that our approach is general, computationally efficient, and practically useful. Our general framework makes it possible to integrate geometric information of the data domain in the labeling process. At the same time, it reduces the computational complexity since not all possible correspondences have to be considered. We demonstrate via experiments that such geometry-aware merge tree comparisons help to detect transitions, clusters, and periodicities of a time-varying dataset, as well as to diagnose and highlight the topological changes between adjacent data instances.

We propose a visual analysis method for the comparison and evaluation of structures in solid-state materials based on the electron density field using topological analysis. The work has been motivated by a material science application, specifically looking for new so-called layered materials whose physical properties are required in many modern technological developments. Due to the incredibly large search space, this is a slow and tedious process, requiring efficient data analysis to characterize and understand the material properties. The core of our proposed analysis pipeline is an abstract bar representation that serves as a concise signature of the material, supporting direct comparison and also an exploration of different material candidates.

In this work, we propose trait-based merge trees a generalization of merge trees to feature level sets, targeting the analysis of tensor field or general multi-variate data. For this, we employ the notion of traits defined in attribute space as introduced in the feature level sets framework. The resulting distance field in attribute space induces a scalar field in the spatial domain that serves as input for topological data analysis. The leaves in the merge tree represent those areas in the input data that are closest to the defined trait and thus most closely resemble the defined feature. Hence, the merge tree yields a hierarchy of features that allows for querying the most relevant and persistent features. The presented method includes different query methods for the tree which enable the highlighting of different aspects. We demonstrate the cross-application capabilities of this approach with three case studies from different domains.

Feature tracking is a common task in visualization applications, where methods based on topological data analysis (TDA) have successfully been applied in the past for feature definition as well as tracking. In this work, we focus on tracking extrema of temporal scalar fields. A family of TDA approaches address this task by establishing one-to-one correspondences between extrema based on discrete gradient vector fields. More specifically, two extrema of subsequent time steps are matched if they fall into their respective ascending and descending manifolds. However, due to this one-to-one assignment, these approaches are prone to fail where, e.g., extrema are located in regions with low gradient magnitude, or are located close to boundaries of the manifolds. Therefore, we propose a probabilistic matching that captures a larger set of possible correspondences via neighborhood sampling, or by computing the overlap of the manifolds. We illustrate the usefulness of the approach with two application cases.

The problem of tracking and visualizing cyclones is still an active area of climate research, since the nature of cyclones varies depending on geospatial location and temporal season, resulting in no clear mathematical definition. Thus, many cyclone tracking methods are tailored to specific datasets and therefore do not support general cyclone extraction across the globe. To address this challenge, we present a conceptual application for exploring cyclone evolution by organizing the extracted cyclone tracks into hierarchical groups. Our approach is based on extrema tracking, and the resulting tracks can be defined in a multi-scale structure by grouping the points based on a novel feature descriptor defined on the merge tree, so-called crown features. Consequently, multiple parameter settings can be visualized and explored in a level-of-detail approach, supporting experts to quickly gain insights on cyclonic formation and evolution. We describe a general cyclone exploration pipeline that consists of four modular building blocks: (1) an extrema tracking method, (2) multiple definitions of cyclones as groups of extrema, including crown features, (3) the correlation of cyclones based on the underlying tracking information, and (4) a hierarchical visualization of the resulting feature tracks and their spatial embedding, allowing exploration on a global and local scale. In order to be as flexible as possible, our pipeline allows for exchanging every module with different techniques, such as other tracking methods and cyclone definitions.

We propose a novel method for the computation of Jacobi sets in 2D domains. The Jacobi set is a topological descriptor based on Morse theory that captures gradient alignments among multiple scalar fields, which is useful for multi-field visualization. Previous Jacobi set computations use piecewise linear approximations on triangulations that result in discretization artifacts like zig-zag patterns. In this paper, we utilize a local bilinear method to obtain a more precise approximation of Jacobi sets by preserving the topology and improving the geometry. Consequently, zig-zag patterns on edges are avoided, resulting in a smoother Jacobi set representation. Our experiments show a better convergence with increasing resolution compared to the piecewise linear method. We utilize this advantage with an efficient local subdivision scheme. Finally, our approach is evaluated qualitatively and quantitatively in comparison with previous methods for different mesh resolutions and across a number of synthetic and real-world examples.