arXiv subjects

Vijay Natarajan

Publications and source records attributed to Vijay Natarajan.

At least 19 recordsLinked to original sources

arXiv2025

Topological signatures of jamming in granular force networks via persistent homology

Granular materials exhibit intricate force networks, often concentrated in chains rather than being uniformly distributed. These meso-scale structures are complex to characterize, owing to their heterogeneous and anisotropic nature. We present a topological analysis in which we characterize force networks by studying the connectivity of contact forces in granular materials. We obtain the force network data during the isotropic compression of a 2D granular ensemble, which comprises of bidisperse photoelastic disks. The force chains are visualized using a bright-field polariscope, which are then analyzed using photoelasticity techniques to get contact force information across all the particles in the granular ensemble. The forces and contact network are studied using topological descriptors. We analyze the evolution of these descriptors using methods developed in the areas of computational topology and persistent homology. Specifically, we evaluate the number of homology class generators to capture critical changes in network connectivity with the evolution of packing fraction. Building on previous applications of persistent homology, our topological analysis of the force network provides valuable insights by distinguishing different phases of particle interactions and revealing unique structural transitions around jamming and thus help enhance our understanding of the stability and jamming dynamics in granular systems.

cond-mat.soft↗

arXiv2025

Analyzing Time-Varying Scalar Fields using Piecewise-Linear Morse-Cerf Theory

Morse-Cerf theory considers a one-parameter family of smooth functions defined on a manifold and studies the evolution of their critical points with the parameter. This paper presents an adaptation of Morse-Cerf theory to a family of piecewise-linear (PL) functions. The vertex diagram and Cerf diagram are introduced as representations of the evolution of critical points of the PL function. The characterization of a crossing in the vertex diagram based on the homology of the lower links of vertices leads to the definition of a topological descriptor for time-varying scalar fields. An algorithm for computing the Cerf diagram and a measure for comparing two Cerf diagrams are also described together with experimental results on time-varying scalar fields.

cs.GR↗

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

A Scalable System for Visual Analysis of Ocean Data

Oceanographers rely on visual analysis to interpret model simulations, identify events and phenomena, and track dynamic ocean processes. The ever increasing resolution and complexity of ocean data due to its dynamic nature and multivariate relationships demands a scalable and adaptable visualization tool for interactive exploration. We introduce pyParaOcean, a scalable and interactive visualization system designed specifically for ocean data analysis. pyParaOcean offers specialized modules for common oceanographic analysis tasks, including eddy identification and salinity movement tracking. These modules seamlessly integrate with ParaView as filters, ensuring a user-friendly and easy-to-use system while leveraging the parallelization capabilities of ParaView and a plethora of inbuilt general-purpose visualization functionalities. The creation of an auxiliary dataset stored as a Cinema database helps address I/O and network bandwidth bottlenecks while supporting the generation of quick overview visualizations. We present a case study on the Bay of Bengal (BoB) to demonstrate the utility of the system and scaling studies to evaluate the efficiency of the system.

cs.GR↗

arXiv2025

Topological analysis reveals multiple pathways in molecular dynamics

Molecular Dynamics simulations are essential tools for understanding the dynamic behavior of biomolecules, yet extracting meaningful molecular pathways from these simulations remains challenging due to the vast amount of generated data. In this work, we present Molecular Kinetics via Topology (MoKiTo), a novel approach that combines the ISOKANN algorithm to determine the reaction coordinate of a molecular system with a topological analysis inspired by the Mapper algorithm. Our strategy efficiently identifies and characterizes distinct molecular pathways, enabling the detection and visualization of critical conformational transitions and rare events. This method offers deeper insights into molecular mechanisms, facilitating the design of targeted interventions in drug discovery and protein engineering.

physics.chem-ph↗

arXiv2024

Time-varying Extremum Graphs

We introduce time-varying extremum graph (TVEG), a topological structure to support visualization and analysis of a time-varying scalar field. The extremum graph is a substructure of the Morse-Smale complex. It captures the adjacency relationship between cells in the Morse decomposition of a scalar field. We define the TVEG as a time-varying extension of the extremum graph and demonstrate how it captures salient feature tracks within a dynamic scalar field. We formulate the construction of the TVEG as an optimization problem and describe an algorithm for computing the graph. We also demonstrate the capabilities of \TVEG towards identification and exploration of topological events such as deletion, generation, split, and merge within a dynamic scalar field via comprehensive case studies including a viscous fingers and a 3D von Kármán vortex street dataset.

cs.GR↗

arXiv2024

Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields

The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. It captures the regions of topological changes in the bivariate field. The Jacobi set is a bivariate analog of critical points, and may correspond to features of interest. In the specific case of time-varying fields and when one of the scalar fields is time, the Jacobi set corresponds to temporal tracks of critical points, and serves as a feature-tracking graph. The Jacobi set of a bivariate field or a time-varying scalar field is complex, resulting in cluttered visualizations that are difficult to analyze. This paper addresses the problem of Jacobi set simplification. Specifically, we use the time-varying scalar field scenario to introduce a method that computes a reduced Jacobi set. The method is based on a stability measure called robustness that was originally developed for vector fields and helps capture the structural stability of critical points. We also present a mathematical analysis for the method, and describe an implementation for 2D time-varying scalar fields. Applications to both synthetic and real-world datasets demonstrate the effectiveness of the method for tracking features.

math.NA↗

arXiv2024

Fast Algorithms for Minimum Homology Basis

We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the $1$-dimensional homology classes with $\mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al., runs in $O(N m^{ω-1} + n m g)$ time, where $N$ denotes the total number of simplices in $K$, $m$ denotes the number of edges in $K$, $n$ denotes the number of vertices in $K$, $g$ denotes the rank of the $1$-homology group of $K$, and $ω$ denotes the exponent of matrix multiplication. In this paper, we present three conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex $K$. The first algorithm runs in $\tilde{O}(m^ω)$ time, the second algorithm runs in $O(N m^{ω-1})$ time and the third algorithm runs in $\tilde{O}(N^2 g + N m g{^2} + m g{^3})$ time which is nearly quadratic time when $g=O(1)$. We also study the problem of finding a minimum cycle basis in an undirected graph $G$ with $n$ vertices and $m$ edges. The best known algorithm for this problem runs in $O(m^ω)$ time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in $\tilde{O}(m^ω)$ time. We also provide a practical implementation of computing the minimum homology basis for general weighted complexes. The implementation is broadly based on the algorithmic ideas described in this paper, differing in its use of practical subroutines. Of these subroutines, the more costly step makes use of a parallel implementation, thus potentially addressing the issue of scale. We compare results against the currently known state of the art implementation (ShortLoop).

cs.CG↗

arXiv2024

Geometric Localization of Homology Cycles

Computing an optimal cycle in a given homology class, also referred to as the homology localization problem, is known to be an NP-hard problem in general. Furthermore, there is currently no known optimality criterion that localizes classes geometrically and admits a stability property under the setting of persistent homology. We present a geometric optimization of the cycles that is computable in polynomial time and is stable in an approximate sense. Tailoring our search criterion to different settings, we obtain various optimization problems like optimal homologous cycle, minimum homology basis, and minimum persistent homology basis. In practice, the (trivial) exact algorithm is computationally expensive despite having a worst case polynomial runtime. Therefore, we design approximation algorithms for the above problems and study their performance experimentally. These algorithms have reasonable runtimes for moderate sized datasets and the cycles computed by these algorithms are consistently of high quality as demonstrated via experiments on multiple datasets.

cs.CG↗

arXiv2023

pyParaOcean: A System for Visual Analysis of Ocean Data

Visual analysis is well adopted within the field of oceanography for the analysis of model simulations, detection of different phenomena and events, and tracking of dynamic processes. With increasing data sizes and the availability of multivariate dynamic data, there is a growing need for scalable and extensible tools for visualization and interactive exploration. We describe pyParaOcean, a visualization system that supports several tasks routinely used in the visual analysis of ocean data. The system is available as a plugin to Paraview and is hence able to leverage its distributed computing capabilities and its rich set of generic analysis and visualization functionalities. pyParaOcean provides modules to support different visual analysis tasks specific to ocean data, such as eddy identification and salinity movement tracking. These modules are available as Paraview filters and this seamless integration results in a system that is easy to install and use. A case study on the Bay of Bengal illustrates the utility of the system for the study of ocean phenomena and processes.

cs.GR↗

arXiv2023

TACHYON: Efficient Shared Memory Parallel Computation of Extremum Graphs

The extremum graph is a succinct representation of the Morse decomposition of a scalar field. It has increasingly become a useful data structure that supports topological feature directed visualization of 2D / 3D scalar fields, and enables dimensionality reduction together with exploratory analysis of high dimensional scalar fields. Current methods that employ the extremum graph compute it either using a simple sequential algorithm for computing the Morse decomposition or by computing the more detailed Morse-Smale complex. Both approaches are typically limited to two and three dimensional scalar fields. We describe a GPU-CPU hybrid parallel algorithm for computing the extremum graph of scalar fields in all dimensions. The proposed shared memory algorithm utilizes both fine grained parallelism and task parallelism to achieve efficiency. An open source software library, TACHYON, that implements the algorithm exhibits superior performance and good scaling behavior.

cs.GR↗

arXiv2023

A GPU Parallel Algorithm for Computing Morse-Smale Complexes

The Morse-Smale complex is a well studied topological structure that represents the gradient flow behavior between critical points of a scalar function. It supports multi-scale topological analysis and visualization of feature-rich scientific data. Several parallel algorithms have been proposed towards the fast computation of the 3D Morse-Smale complex. Its computation continues to pose significant algorithmic challenges. In particular, the non-trivial structure of the connections between the saddle critical points are not amenable to parallel computation. This paper describes a fine grained parallel algorithm for computing the Morse-Smale complex and a GPU implementation gMSC. The algorithm first determines the saddle-saddle reachability via a transformation into a sequence of vector operations, and next computes the paths between saddles by transforming it into a sequence of matrix operations. Computational experiments show that the method achieves up to 8.6x speedup over pyms3d and 6x speedup over TTK, the current shared memory implementations. The paper also presents a comprehensive experimental analysis of different steps of the algorithm and reports on their contribution towards runtime performance. Finally, it introduces a CPU based data parallel algorithm for simplifying the Morse-Smale complex via iterative critical point pair cancellation.

cs.GR↗

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

Jacobi Set Driven Search for Flexible Fiber Surface Extraction

Isosurfaces are an important tool for analysis and visualization of univariate scalar fields. Earlier works have demonstrated the presence of interesting isosurfaces at isovalues close to critical values. This motivated the development of efficient methods for computing individual components of isosurfaces restricted to a region of interest. Generalization of isosurfaces to fiber surfaces and critical points to Jacobi sets has resulted in new approaches for analyzing bivariate scalar fields. Unlike isosurfaces, there exists no output sensitive method for computing fiber surfaces. Existing methods traverse through all the tetrahedra in the domain. In this paper, we propose the use of the Jacobi set to identify fiber surface components of interest and present an output sensitive approach for its computation. The Jacobi edges are used to initiate the search towards seed tetrahedra that contain the fiber surface, thereby reducing the search space. This approach also leads to effective analysis of the bivariate field by supporting the identification of relevant fiber surfaces near Jacobi edges.

cs.GR↗

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

Comparative Analysis of Merge Trees using Local Tree Edit Distance

Comparative analysis of scalar fields is an important problem with various applications including feature-directed visualization and feature tracking in time-varying data. Comparing topological structures that are abstract and succinct representations of the scalar fields lead to faster and meaningful comparison. While there are many distance or similarity measures to compare topological structures in a global context, there are no known measures for comparing topological structures locally. While the global measures have many applications, they do not directly lend themselves to fine-grained analysis across multiple scales. We define a local variant of the tree edit distance and apply it towards local comparative analysis of merge trees with support for finer analysis. We also present experimental results on time-varying scalar fields, 3D cryo-electron microscopy data, and other synthetic data sets to show the utility of this approach in applications like symmetry detection and feature tracking.

cs.GR↗

arXiv2021

Segmentation Driven Peeling for Visual Analysis of Electronic Transitions

Electronic transitions in molecules due to 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 those electronic transitions, i.e. 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 towards the study of electronic transitions based on the visual analysis of a bivariate field, namely the electron density in the hole and particle Natural Transition Orbital (NTO). The visual analysis focuses on the continuous scatter plots (CSPs) of the bivariate field linked to their spatial domain. The method supports selections in the CSP visualized as fiber surfaces in the spatial domain, the grouping of atoms, and segmentation of the density fields to peel the CSP. This peeling operator is central to the visual analysis process and helps identify donors and acceptors. We study different molecular systems, identifying local excitation and charge transfer excitations to demonstrate the utility of the method.

physics.chem-ph↗

arXiv2021

Visual Analysis of Electronic Densities and Transitions in Molecules

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.

cs.HC↗