We present an exact and efficient algorithm for computing the Reeb space of a bivariate PL map. The Reeb space is a topological structure that generalizes the Reeb graph to the setting of multiple scalar-valued functions defined over a shared domain, a situation that frequently arises in practical applications. While the Reeb graph has become a standard tool in computer graphics, shape analysis, and scientific visualization, the Reeb space is still in the early stages of adoption. Although several algorithms for computing the Reeb space have been proposed, none offer an implementation that is both exact and efficient, which has substantially limited its practical use. To address this gap, we introduce singular arrange and traverse, a new algorithm built upon the arrange and traverse framework [Hristov et al., 2025]. Our method exploits the fact that, in the bivariate case, only singular edges contribute to the structure of Reeb space, allowing us to ignore many regular edges [Tierny and Carr, 2017]. This observation results in substantial efficiency gains on datasets where most edges are regular, which is common in many numerical simulations of physical systems. We provide an implementation of our method and benchmark it against the original arrange and traverse algorithm, showing performance gains of up to four orders of magnitude on real-world datasets.

Abstract Uncertainty in experimental measurements makes it challenging to determine which features are intrinsic to the phenomenon and which are most stable and reliable. Granular materials, such as sand, form a complex system in which the forces between individual particles influence the material's macroscopic behavior. However, these forces are also subject to uncertainty, as repeated measurements can yield different results. In this paper, we investigate how to model and visually analyze the uncertain structure of forces in granular materials. We adopt different perspectives on uncertainty by considering it as variance, probability, or additional variable. For a nuanced analysis of granular material data, we propose combining visualizations that represent these perspectives into a single visual analytics approach. We integrate uncertainty‐aware spatial visualizations that convey the probability of features, visualizations of derived measures and their variance over changes in the packing fraction, and overviews of varying probability thresholds. Finally, we evaluate our proposed approach in a case study conducted together with geotechnical engineers for the example of a 2D ensemble of photoelastic disks.

This paper presents an interactive analysis framework for exploring data from photoelastic disk experiments, which serve as a model for two-dimensional granular materials. Granular materials, composed of discrete particles such as sand or gravel, exhibit behaviors resembling fluid or solid states depending on the system configuration. These behaviors arise from interparticle contact forces, which form complex force networks that govern the material's macroscopic behavior. Our framework is specifically designed to analyze such 2D ensembles of dynamic force networks, enabling the identification and characterization of their underlying structures. The framework is built around a topology-based, multiscale data segmentation in terms of force chains and cycles. The analysis methods are structured across three levels: (1) multiscale analysis of individual instances under specific loading conditions, (2) detailed exploration of single experiments encompassing a series of loading and unloading cycles, and (3) comparative analysis across experiments conducted under similar and differing setups. We demonstrate the capabilities of our framework with a case study for each of these levels.

Abstract We present the first combinatorial algorithm for efficiently computing the Reeb space in all dimensions. The Reeb space is a higher‐dimensional generalization of the Reeb graph, which is standard practice in the analysis of scalar fields, along with other computational topology tools such as persistent homology and the Morse‐Smale complex. One significant limitation of topological tools for scalar fields is that data often involves multiple variables, where joint analysis is more insightful. Generalizing topological data structures to multivariate data has proven challenging and the Reeb space is one of the few available options. However, none of the existing algorithms can efficiently compute the Reeb space in arbitrary dimensions and there are no available implementations which are robust with respect to numerical errors. We propose a new algorithm for computing the Reeb space of a generic piecewise linear map over a simplicial mesh of any dimension called arrange and traverse. We implement a robust specialization of our algorithm for tetrahedral meshes and evaluate it on real‐life data.

Feature level sets (FLS) have shown significant potential in the analysis of multi-field data by using traits defined in attribute space to specify features in the domain. In this work, we address key challenges in the practical use of FLS: trait design and feature selection for rendering. To simplify trait design, we propose a Cartesian decomposition of traits into simpler components, making the process more intuitive and computationally efficient. Additionally, we utilize dictionary learning results to automatically suggest point traits. To enhance feature selection, we introduce trait-induced merge trees (TIMTs), a generalization of merge trees for feature level sets, aimed at topologically analyzing tensor fields or general multi-variate data. The leaves in the TIMT represent areas in the input data that are closest to the defined trait, thereby most closely resembling the defined feature. This merge tree provides a hierarchy of features, enabling the querying of the most relevant and persistent features. Our method includes various query techniques for the tree, allowing the highlighting of different aspects. We demonstrate the cross-application capabilities of this approach through five case studies from different domains.

Photoinduced electronic transitions are complex quantum-mechanical processes where electrons move between energy levels due to the absorption of light. This induces dynamics i.e., coupled changes in the electronic structure and nuclear geometry, that drive physical and chemical processes of importance in diverse fields ranging from photobiology and materials design to medicine. The evolving electronic structure can be characterized by two electron density fields: hole and particle natural transition orbitals (NTOs). A study of the two density fields helps understand the movement of electronic charge from one part of the molecule to another, specifically the donor and acceptor regions. Previous works in this area rely on side-by-side visual comparisons of isosurfaces, statistical approaches, or visual analysis of bivariate fields restricted to limited time instances. We propose a new method to analyze time-varying bivariate fields with a large number of instances, as pertinent to understand electronic structure changes during light-induced dynamics. Since the NTO fields depend on the nuclear geometry, the nuclear motion leads to a large number of bivariate field instances. Structures like tracking graphs have been used to analyze time-varying univariate fields. This article presents a structured and practical approach to feature-directed visual exploration of time-varying bivariate fields using continuous scatterplots (CSPs) and image moment-based descriptors, tailored for studying the evolving electronic structure following photoexcitation. The CSP of the bivariate field at every time step is represented using an image moment vector of length 4. The collection of all image moment vector descriptors is considered as a point cloud in $\mathbb {R}^{4}$R4 and visualized using principal component analysis. Choosing an appropriate pair of principal components results in a representation of the point cloud as a curve on the plane. This representation supports tasks such as identifying interesting time steps, identifying patterns within the bivariate field, and tracking their evolution over time. We present two case studies on excited-state dynamics in molecular systems that demonstrate how the time-varying bivariate field analysis helps provide application-specific insights.

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 article, 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.

Merge trees, a type of topological descriptors, serve to identify and summarize the topological characteristics associated with scalar fields. They have great potential for analyzing and visualizing 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 using labeled interleaving distances. 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 framework makes it possible to integrate geometric information of the data domain in the labeling process. At the same time, the framework 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 time-varying datasets, as well as to diagnose and highlight the topological changes between adjacent data instances.

We present a visual analysis environment based on a multi-scale partitioning of a 2d domain into regions bounded by cycles in weighted planar embedded graphs. The work has been inspired by an application in granular materials research, where the question of scale plays a fundamental role in the analysis of material properties. We propose an efficient algorithm to extract the hierarchical cycle structure using persistent homology. The core of the algorithm is a filtration on a dual graph exploiting Alexander’s duality. The resulting partitioning is the basis for the derivation of statistical properties that can be explored in a visual environment. We demonstrate the proposed pipeline on a few synthetic and one real-world data set.

Abstract We present a pipeline for the interactive visual analysis and exploration of molecular electronic transition ensembles. Each ensemble member is specified by a molecular configuration, the charge transfer between two molecular states, and a set of physical properties. The pipeline is targeted towards theoretical chemists, supporting them in comparing and characterizing electronic transitions by combining automatic and interactive visual analysis. A quantitative feature vector characterizing the electron charge transfer serves as the basis for hierarchical clustering as well as for the visual representations. The interface for the visual exploration consists of four components. A dendrogram provides an overview of the ensemble. It is augmented with a level of detail glyph for each cluster. A scatterplot using dimensionality reduction provides a second visualization, highlighting ensemble outliers. Parallel coordinates show the correlation with physical parameters. A spatial representation of selected ensemble members supports an in‐depth inspection of transitions in a form that is familiar to chemists. All views are linked and can be used to filter and select ensemble members. The usefulness of the pipeline is shown in three different case studies.

Topological and geometric segmentation methods provide powerful concepts for detailed field analysis and visualization. However, when it comes to a quantitative analysis that requires highly accurate geometric segmentation, there is a large discrepancy between the promising theory and the available computational approaches. In this paper, we compare and evaluate various segmentation methods with the aim to identify and quantify the extent of these discrepancies. Thereby, we focus on an application from quantum chemistry: the analysis of electron density fields. It is a scalar quantity that can be experimentally measured or theoretically computed. In the evaluation we consider methods originating from the domain of quantum chemistry and computational topology. We apply the methods to the charge density of a set of crystals and molecules. Therefore, we segment the volumes into atomic regions and derive and compare quantitative measures such as total charge and dipole moments from these regions. As a result, we conclude that an accurate geometry determination can be crucial for correctly segmenting and analyzing a scalar field, here demonstrated on the electron density field.

Abstract The study of electronic transitions within a molecule connected to the absorption or emission of light is a common task in the process of the design of new materials. The transitions are complex quantum mechanical processes and a detailed analysis requires a breakdown of these processes into components that can be interpreted via characteristic chemical properties. We approach these tasks by providing a detailed analysis of the electron density field. This entails methods to quantify and visualize electron localization and transfer from molecular subgroups combining spatial and abstract representations. The core of our method uses geometric segmentation of the electronic density field coupled with a graph‐theoretic formulation of charge transfer between molecular subgroups. The design of the methods has been guided by the goal of providing a generic and objective analysis following fundamental concepts. We illustrate the proposed approach using several case studies involving the study of electronic transitions in different molecular systems.

Abstract In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an essential role in capturing the shape of scalar field data. We present a state‐of‐the‐art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time‐varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.

The alpha complex, a subset of the Delaunay triangulation, has been extensively used as the underlying representation for biomolecular structures. We propose a GPU-based parallel algorithm for the computation of the alpha complex, which exploits the knowledge of typical spatial distribution and sizes of atoms in a biomolecule. Unlike existing methods, this algorithm does not require prior construction of the Delaunay triangulation. The algorithm computes the alpha complex in two stages. The first stage proceeds in a bottom-up fashion and computes a superset of the edges, triangles, and tetrahedra belonging to the alpha complex. The false positives from this estimation stage are removed in a subsequent pruning stage to obtain the correct alpha complex. Computational experiments on several biomolecules demonstrate the superior performance of the algorithm, up to a factor of 50 when compared to existing methods that are optimized for biomolecules.

Topological structures such as the merge tree provide an abstract and succinct representation of scalar fields. They facilitate effective visualization and interactive exploration of feature-rich data. A merge tree captures the topology of sub-level and super-level sets in a scalar field. Estimating the similarity between merge trees is an important problem with applications to feature-directed visualization of time-varying data. We present an approach based on tree edit distance to compare merge trees. The comparison measure satisfies metric properties, it can be computed efficiently, and the cost model for the edit operations is both intuitive and captures well-known properties of merge trees. Experimental results on time-varying scalar fields, 3D cryo electron microscopy data, shape data, and various synthetic datasets show the utility of the edit distance towards a feature-driven analysis of scalar fields.

Abstract In the field of organic electronics, understanding complex material morphologies and their role in efficient charge transport in solar cells is extremely important. Related processes are studied using the Ising model and Kinetic Monte Carlo simulations resulting in large ensembles of stochastic trajectories. Naive visualization of these trajectories, individually or as a whole, does not lead to new knowledge discovery through exploration. In this paper, we present novel visualization and exploration methods to analyze this complex dynamic data, which provide succinct and meaningful abstractions leading to scientific insights. We propose a morphology abstraction yielding a network composed of material pockets and the interfaces, which serves as backbone for the visualization of the charge diffusion. The trajectory network is created using a novel way of implicitly attracting the trajectories to the skeleton of the morphology relying on a relaxation process. Each individual trajectory is then represented as a connected sequence of nodes in the skeleton. The final network summarizes all of these sequences in a single aggregated network. We apply our method to three different morphologies and demonstrate its suitability for exploring this kind of data.

Study of cavities and channels in molecular structure is a crucial step in understanding the function of biomolecules. Current tools and techniques for extracting these structural features are sensitive to uncertainties in atomic position and radii. In this paper, we study the problem of cavity extraction in biomolecules while taking into account such uncertainties. We propose an approach that connects user-specified cavities by computing an optimal conduit within the region occupied by the molecule. The conduit is computed using a topological representation of the occupied and empty regions and is guaranteed to satisfy well defined geometric optimality criteria. Visualization of the set of all cavities with multiple linked views serves as a useful interface for interactive extraction of stable cavities. We demonstrate the utility of the proposed method in successfully identifying biologically significant pathways between molecular cavities using several case studies.

BACKGROUND: Understanding channel structures that lead to active sites or traverse the molecule is important in the study of molecular functions such as ion, ligand, and small molecule transport. Efficient methods for extracting, storing, and analyzing protein channels are required to support such studies. Further, there is a need for an integrated framework that supports computation of the channels, interactive exploration of their structure, and detailed visual analysis of their properties. RESULTS: We describe a method for molecular channel extraction based on the alpha complex representation. The method computes geometrically feasible channels, stores both the volume occupied by the channel and its centerline in a unified representation, and reports significant channels. The representation also supports efficient computation of channel profiles that help understand channel properties. We describe methods for effective visualization of the channels and their profiles. These methods and the visual analysis framework are implemented in a software tool, CHEXVIS. We apply the method on a number of known channel containing proteins to extract pore features. Results from these experiments on several proteins show that CHEXVIS performance is comparable to, and in some cases, better than existing channel extraction techniques. Using several case studies, we demonstrate how CHEXVIS can be used to study channels, extract their properties and gain insights into molecular function. CONCLUSION: CHEXVIS supports the visual exploration of multiple channels together with their geometric and physico-chemical properties thereby enabling the understanding of the basic biology of transport through protein channels. The CHEXVIS web-server is freely available at http://vgl.serc.iisc.ernet.in/chexvis/ . The web-server is supported on all modern browsers with latest Java plug-in.