arXiv subjects

Talha Bin Masood

Publications and source records attributed to Talha Bin Masood.

At least 19 recordsLinked to original sources

arXiv2026

Exact Computation of Trait-induced Merge Trees for Bivariate Fields

Trait-induced merge trees (TIMTs) provide a robust topology-based method for selecting and browsing feature level sets in multivariate data by analyzing the distance field induced by a user-specified trait in attribute space. Existing TIMT computations typically sample this distance field at mesh vertices and assume piecewise-linear interpolation, although the Euclidean distance-to-trait function is generally not piecewise linear on the original mesh. As a result, the resulting merge tree may miss zero-valued features and may perturb the locations and values of minima and merge events. We study the exact computation of TIMTs for piecewise-linear bivariate fields, focusing first on point traits. We show that the restricted sublevel sets inside each tetrahedron are convex and therefore have trivial local merge-tree structure, implying that global topological changes arise only through gluing across simplex boundaries. Based on this observation, we construct a weighted graph whose merge tree is isomorphic to the exact merge tree of the induced distance field. We further relate TIMTs to Jacobi sets, showing how nonzero edge events of the TIMT are localized by the singular structure of the underlying bivariate map. We establish a theoretical upper bound on the error of the vertex-sampled linear interpolation, expressed in terms of the maximum length of projected mesh edges in the range. We discuss extensions to line, line-segment, and finite point-set traits, and implement the method robustly using CGAL and VTK, demonstrating results on both synthetic and real-world datasets.

cs.CG↗

arXiv2026

Temporal Tracking of Reeb-Space Sheets

Time-varying bivariate fields arise in many scientific applications, where the relationship between two scalar quantities evolves over time. While topological methods such as merge trees provide an effective framework for identifying and tracking features in univariate data, analogous approaches for bivariate fields remain comparatively underexplored. Reeb spaces extend topological analysis to multivariate data by representing fiber connectivity through a collection of interconnected sheets, making these sheets natural candidates for describing bivariate structures. However, establishing temporal correspondences between sheets is challenging due to the structural complexity of Reeb spaces, sensitivity to noise, and the difficulty of defining meaningful similarity measures across timesteps. We present a framework for tracking Reeb space sheets in time-varying bivariate fields. The method establishes correspondences between sheets in consecutive timesteps using complementary similarity measures defined in the spatial domain and the range space. We evaluate the method on a synthetic torus dataset and two time-varying molecular electronic structure datasets. The results show that Reeb space sheet tracking reveals persistent structures and highlights interesting intervals of temporal change. Overall, the results demonstrate that Reeb space sheets can serve as trackable topological structures and provide a foundation for the visual analysis of time-varying bivariate data.

cs.HC↗

arXiv2026

Electronic structure and oxidation states in high-pressure synthesized isostructural CeCN$_5$ and TbCN$_5$

Understanding the behavior of 4$f$ electrons in materials containing rare earth elements is one of the fundamental questions within condensed matter physics. In this work the electronic properties of isostructural CeCN$_5$ and TbCN$_5$, both recently synthesized at extreme pressure, are investigated using Density Functional Theory (DFT) calculations. We include the on-site Coulomb repulsion between localized 4$f$ states within the static DFT+U framework; the DFT+U results are cross-checked with DFT+dynamical mean-field theory (DMFT) calculations within the quasi-atomic (Hubbard-I) approximation. Despite CeCN$_5$ and TbCN$_5$ being isostructural compounds Ce and Tb show different oxidation states, 4+ and 3+ respectively. This leads to distinctly different electronic properties: the former compound is an insulator, while the latter is a metal. An extra electron which is donated by Ce to the polymeric C-N network is distributed across the network. This leads to a modification of the bond length in CeCN$_5$ compared to TbCN$_5$. Still, the polymeric C-N networks can accommodate the different oxidation states in isostructural lanthanide-carbon-nitrogen (LnCN) compounds. Our results underline that LnCN compounds under high pressure offer a unique platform for probing the interplay between 4$f$-electron behavior and structural complexity.

cond-mat.mtrl-sci↗

arXiv2026

Towards Computing Average Merge Tree Based on the Interleaving Distance

The interleaving distance is a key tool for comparing merge trees, which provide topological summaries of scalar functions. In this work, we define an average merge tree for a pair of merge trees using the interleaving distance. Since such an average is not unique, we propose a method to construct a representative average merge tree. We further prove that the resulting merge tree indeed satisfies a natural notion of averaging for the two given merge trees. To demonstrate the structure of the average merge tree, we include illustrative examples.

cs.CG↗

arXiv2026

Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps

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. 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. 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.

cs.CG↗

arXiv2025

Robust Geometric Predicates for Bivariate Computational Topology

We present theory and practice for robust implementations of bivariate Jacobi set and Reeb space algorithms. Robustness is a fundamental topic in computational geometry that deals with the issues of numerical errors and degenerate cases in algorithm implementations. Computational topology already uses some robustness techniques for the development of scalar field algorithms, such as those for computing critical points, merge trees, contour trees, Reeb graphs, Morse-Smale complexes, and persistent homology. In most cases, robustness can be ensured with floating-point arithmetic, and degenerate cases can be resolved with a standard symbolic perturbation technique called Simulation of Simplicity. However, this becomes much more complex for topological data structures of multifields, such as Jacobi sets and Reeb spaces. The geometric predicates used in their computation require exact arithmetic and a more involved treatment of degenerate cases to ensure correctness. Neither of these challenges has been fully addressed in the literature so far. In this paper, we describe how exact arithmetic and symbolic perturbation schemes can be used to enable robust implementations of bivariate Jacobi set and Reeb space algorithms. In the process, we develop a method for automatically evaluating predicates that can be expressed as large symbolic polynomials, which are difficult to factor appropriately by hand, as is typically done in the computational geometry literature. We provide implementations of all proposed approaches and evaluate their efficiency.

cs.CG↗

arXiv2025

Efficient Heuristic Algorithms for Interleaving Distance between Merge Trees

Merge trees are fundamental structures in topological data analysis. Interleaving distance is a widely accepted metric for comparing merge trees, with applications in visualization and scientific computing. While a greedy algorithm exists for finding the interleaving distance between labeled merge trees with overlapping labels, computing the interleaving distance between unlabeled trees or labeled trees with disjoint labels remains a significant challenge. In this work, we introduce a novel heuristic algorithm for approximating the interleaving distance between labeled merge trees with partial agreement and disagreement. Our method strategically assigns labels primarily to the leaves of the trees to infer structural correspondence. We also introduce an enhanced version of a previous algorithm that offers improved performance. Both algorithms run in polynomial time and provide practical, efficient alternatives for comparing merge trees, particularly in cases involving unlabeled or structurally diverse data. This work contributes a new direction for merge tree analysis and offers promising tools for real-world applications. We demonstrate this application on the simulation of time-varying electron density.

cs.CG↗

arXiv2025

Topology-Aware Volume Fusion for Spectral Computed Tomography via Histograms and Extremum Graph

Photon-Counting Computed Tomography (PCCT) is a novel imaging modality that simultaneously acquires volumetric data at multiple X-ray energy levels, generating separate volumes that capture energy-dependent attenuation properties. Attenuation refers to the reduction in X-ray intensity as it passes through different tissues or materials. This spectral information enhances tissue and material differentiation, enabling more accurate diagnosis and analysis. However, the resulting multivolume datasets are often complex and redundant, making visualization and interpretation challenging. To address these challenges, we propose a method for fusing spectral PCCT data into a single representative volume that enables direct volume rendering and segmentation by leveraging both shared and complementary information across different channels. Our approach starts by computing 2D histograms between pairs of volumes to identify those that exhibit prominent structural features. These histograms reveal relationships and variations that may be difficult to discern from individual volumes alone. Next, we construct an extremum graph from the 2D histogram of two minimally correlated yet complementary volumes-selected to capture both shared and distinct features-thereby maximizing the information content. The graph captures the topological distribution of histogram extrema. By extracting prominent structure within this graph and projecting each grid point in histogram space onto it, we reduce the dimensionality to one, producing a unified volume. This representative volume retains key structural and material characteristics from the original spectral data while significantly reducing the analysis scope from multiple volumes to one. The result is a topology-aware, information-rich fusion of multi-energy CT datasets that facilitates more effective visualization and segmentation.

cs.HC↗

arXiv2025

Continuous Scatterplot and Image Moments for Time-Varying Bivariate Field Analysis of Electronic Structure Evolution

Photoinduced electronic transitions are complex quantum-mechanical processes where electrons move between energy levels due to light absorption. This induces dynamics in electronic structure and nuclear geometry, driving important physical and chemical processes in fields like photobiology, materials design, and medicine. The evolving electronic structure can be characterized by two electron density fields: hole and particle natural transition orbitals (NTOs). Studying these density fields helps understand electronic charge movement between donor and acceptor regions within a molecule. Previous works rely on side-by-side visual comparisons of isosurfaces, statistical approaches, or bivariate field analysis with few instances. We propose a new method to analyze time-varying bivariate fields with many instances, which is relevant for understanding electronic structure changes during light-induced dynamics. Since NTO fields depend on nuclear geometry, the nuclear motion results in numerous time steps to analyze. This paper presents a structured approach to feature-directed visual exploration of time-varying bivariate fields using continuous scatterplots (CSPs) and image moment-based descriptors, tailored for studying evolving electronic structures post-photoexcitation. The CSP of the bivariate field at each time step is represented by a four-length image moment vector. The collection of all vector descriptors forms a point cloud in R^4, visualized using principal component analysis. Selecting appropriate principal components results in a representation of the point cloud as a curve on the plane, aiding tasks such as identifying key time steps, recognizing patterns within the bivariate field, and tracking the temporal evolution. We demonstrate this with two case studies on excited-state molecular dynamics, showing how bivariate field analysis provides application-specific insights.

cs.HC↗

arXiv2025

Multi-field Visualization: Trait design and trait-induced merge trees

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.

cs.LG↗

arXiv2024

Multi-scale Cycle Tracking in Dynamic Planar Graphs

This paper presents a nested tracking framework for analyzing cycles in 2D force networks within granular materials. These materials are composed of interacting particles, whose interactions are described by a force network. Understanding the cycles within these networks at various scales and their evolution under external loads is crucial, as they significantly contribute to the mechanical and kinematic properties of the system. Our approach involves computing a cycle hierarchy by partitioning the 2D domain into segments bounded by cycles in the force network. We can adapt concepts from nested tracking graphs originally developed for merge trees by leveraging the duality between this partitioning and the cycles. We demonstrate the effectiveness of our method on two force networks derived from experiments with photoelastic disks.

cs.GR↗

arXiv2024

Keeping it sparse: Computing Persistent Homology revisited

In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major subroutine to compute barcodes, the main invariant therein. We propose two novel variants of the standard algorithm, called swap and retrospective reductions. We test them on a large collection of data against other known variants to compare their efficiency, and we find that sometimes they provide a considerable speed-up. We also present novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worst-case complexity bound and the almost linear practical behavior of matrix reduction. Finally, we provide several constructions on which one of the variants performs strictly better than the others.

cs.CG↗

arXiv2023

Probabilistic Gradient-Based Extrema Tracking

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.

cs.GR↗

arXiv2023

Multi-field Visualisation via Trait-induced Merge Trees

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.

cs.LG↗

arXiv2023

Continuous Scatterplot Operators for Bivariate Analysis and Study of Electronic Transitions

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.

cs.HC↗

arXiv2022

Designing Feature Vector Representations: A case study from Chemistry

We present a case study investigating feature descriptors in the context of the analysis of chemical multivariate ensemble data. The data of each ensemble member consists of three parts: the design parameters for each ensemble member, field data resulting from the numerical simulations, and physical properties of the molecules. Since feature-based methods have the potential to reduce the data complexity and facilitate comparison and clustering, we are focusing on such methods. However, there are many options to design the feature vector representation and there is no obvious preference. To get a better understanding of the different representations, we analyze their similarities and differences. Thereby, we focus on three characteristics derived from the representations: the distribution of pairwise distances, the clustering tendency, and the rank-order of the pairwise distances. The results of our investigations partially confirmed expected behavior, but also provided some surprising observations that can be used for the future development of feature representations in the chemical domain.

cs.HC↗

arXiv2022

Edit Distance between Merge Trees

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.

cs.CG↗

arXiv2021

Aromatic Hexazine [N6]4- Anion Revealed in the Complex Structure of the High-Pressure Potassium Nitride K9N56

Recent high-pressure synthesis of pentazolates and subsequent stabilization of the aromatic [N5]- anion at atmospheric pressure had an immense impact on nitrogen chemistry. Here, we present the first synthesis of an aromatic hexazine [N6]4- anion realized in high-pressure potassium nitride K9N56 at 46 and 61 GPa. The extremely complex structure of K9N56 was solved based on synchrotron single-crystal X-ray diffraction and corroborated by density functional theory calculations. This result resolves a long-standing question of the aromatic hexazine stability and the possibility of its synthesis.

cond-mat.mtrl-sci↗