arXiv subjects

Amit Chattopadhyay

Publications and source records attributed to Amit Chattopadhyay.

12 recordsLinked to original sources

arXiv2026

Spatio-Temporal Signatures of Intermittency in Helically Rotating Turbulence through Topological Data Analysis

A central challenge in hydrodynamic turbulence is identifying precisely when, and at which length scales, strong turbulent fluctuations (STFs) emerge and develop into intermittent events, which are often obscured by conventional statistical diagnostics. We address this problem by applying a Topological Data Analysis (TDA) framework to reveal the spatiotemporal signatures of intermittency in low-resolution ($128^3$) helically rotating turbulent flows. Vorticity magnitude and length-scale (eddy size) fields are used as scalar observables for TDA: vorticity characterizes rotational dynamics that generate multiscale flow structures, while length-scale fields encode the scales at which intermittent activity arises. Their evolving topology is quantified using persistence diagrams and Wasserstein-distance metrics. Compared with traditional statistical approaches, this framework is more sensitive to localized and short-lived flow variations, enabling clearer detection of intermittent behavior. Pronounced variations in Wasserstein-distance heatmaps provide direct signatures of STFs across space and time. Together, these results demonstrate that TDA offers an effective complementary tool for detecting STFs that lead to intermittency within turbulent regime.

physics.flu-dyn↗

arXiv2026

An Improved FPT Algorithm for Computing the Interleaving Distance between Merge Trees via Path-Preserving Maps

A merge tree is a fundamental topological structure used to capture the sub-level set (and similarly, super-level set) topology in scalar data analysis. The interleaving distance is a theoretically sound, stable metric for comparing merge trees. However, computing this distance exactly is NP-hard. First fixed-parameter tractable (FPT) algorithm for it's exact computation introduces the concept of an $\varepsilon$-good map between two merge trees, where $\varepsilon$ is a candidate value for the interleaving distance. The complexity of their algorithm is $O(2^{2τ}(2τ)^{2τ+2}\cdot n^2\log^3n)$ where $τ$ is the degree-bound parameter and $n$ is the total number of nodes in both the merge trees. Their algorithm exhibits exponential complexity in $τ$, which increases with the increasing value of $\varepsilon$. In the current paper, we propose an improved FPT algorithm for computing the $\varepsilon$-good map between two merge trees. Our algorithm introduces two new parameters, $η_f$ and $η_g$, corresponding to the numbers of leaf nodes in the merge trees $M_f$ and $M_g$, respectively. This parametrization is motivated by the observation that a merge tree can be decomposed into a collection of unique leaf-to-root paths. The proposed algorithm achieves a complexity of $O\!\left(n^2\log n+η_g^{η_f}(η_f+η_g)\, n \log n \right)$. To obtain this reduced complexity, we assume that number of possible $\varepsilon$-good maps from $M_f$ to $M_g$ does not exceed that from $M_g$ to $M_f$. Notably, the parameters $η_f$ and $η_g$ are independent of the choice of $\varepsilon$. Compared to their algorithm, our approach substantially reduces the search space for computing an optimal $\varepsilon$-good map. We also provide a formal proof of correctness for the proposed algorithm.

cs.CG↗

arXiv2026

An Algorithm for Fast and Correct Computation of Reeb Spaces for PL Bivariate Fields

The Reeb space is a fundamental data structure in computational topology that represents the fiber topology of a multi-field (or multiple scalar fields), extending the level set topology of a scalar field. Efficient algorithms have been designed for computing Reeb graphs, however, computing correct Reeb spaces for PL bivariate fields, is a challenging open problem. There are only a few implementable algorithms in the literature for computing Reeb space or its approximation, via range quantization or by computing a Jacobi fiber surface, which are computationally expensive or have correctness issues, i.e., the computed Reeb space may not be topologically equivalent or homeomorphic to the actual Reeb space. In the current paper, we propose a novel algorithm for fast and correct computation of the Reeb space corresponding to a generic PL bivariate field defined on a triangulation $\mathbb{M}$ of a $3$-manifold without boundary, leveraging the fast algorithms for computing Reeb graphs in the literature. Our algorithm is based on the computation of a Multi-Dimensional Reeb Graph (MDRG) which is first proved to be homeomorphic with the Reeb space. For the correct computation of the MDRG, we compute the Jacobi set of the PL bivariate field and its projection into the Reeb space, called the Jacobi structure. Finally, the correct Reeb space is obtained by computing a net-like structure embedded in the Reeb space and then computing its $2$-sheets in the net-like structure. The time complexity of our algorithm is $\mathcal{O}(n^2 + n\, c_{int}\, \log n + nc_L^2)$, where $n$ is the total number of simplices in $\mathbb{M}$, $c_{int}$ is the number of intersection points of the projections of the non-adjacent Jacobi set edges on the range of the bivariate field and $c_L$ is the upper bound on the number of simplices in the link of an edge of $\mathbb{M}$.

cs.CG↗

arXiv2023

Image complexity based fMRI-BOLD visual network categorization across visual datasets using topological descriptors and deep-hybrid learning

This study proposes a new approach that investigates differences in topological characteristics of visual networks, which are constructed using fMRI BOLD time-series corresponding to visual datasets of COCO, ImageNet, and SUN. A publicly available BOLD5000 dataset is utilized that contains fMRI scans while viewing 5254 images of diverse complexities. The objective of this study is to examine how network topology differs in response to distinct visual stimuli from these visual datasets. To achieve this, 0- and 1-dimensional persistence diagrams are computed for each visual network representing COCO, ImageNet, and SUN. For extracting suitable features from topological persistence diagrams, K-means clustering is executed. The extracted K-means cluster features are fed to a novel deep-hybrid model that yields accuracy in the range of 90%-95% in classifying these visual networks. To understand vision, this type of visual network categorization across visual datasets is important as it captures differences in BOLD signals while perceiving images with different contexts and complexities. Furthermore, distinctive topological patterns of visual network associated with each dataset, as revealed from this study, could potentially lead to the development of future neuroimaging biomarkers for diagnosing visual processing disorders like visual agnosia or prosopagnosia, and tracking changes in visual cognition over time.

eess.IV↗

arXiv2023

A Topological Distance between Multi-fields based on Multi-Dimensional Persistence Diagrams

The problem of computing topological distance between two scalar fields based on Reeb graphs or contour trees has been studied and applied successfully to various problems in topological shape matching, data analysis, and visualization. However, generalizing such results for computing distance measures between two multi-fields based on their Reeb spaces is still in its infancy. Towards this, in the current paper we propose a technique to compute an effective distance measure between two multi-fields by computing a novel \emph{multi-dimensional persistence diagram} (MDPD) corresponding to each of the (quantized) Reeb spaces. First, we construct a multi-dimensional Reeb graph (MDRG), which is a hierarchical decomposition of the Reeb space into a collection of Reeb graphs. The MDPD corresponding to each MDRG is then computed based on the persistence diagrams of the component Reeb graphs of the MDRG. Our distance measure extends the Wasserstein distance between two persistence diagrams of Reeb graphs to MDPDs of MDRGs. We prove that the proposed measure is a pseudo-metric and satisfies a stability property. Effectiveness of the proposed distance measure has been demonstrated in (i) shape retrieval contest data - SHREC $2010$ and (ii) Pt-CO bond detection data from computational chemistry. Experimental results show that the proposed distance measure based on the Reeb spaces has more discriminating power in clustering the shapes and detecting the formation of a stable Pt-CO bond as compared to the similar measures between Reeb graphs.

cs.GR↗

arXiv2023

Euler Characteristic Transform Based Topological Loss for Reconstructing 3D Images from Single 2D Slices

The computer vision task of reconstructing 3D images, i.e., shapes, from their single 2D image slices is extremely challenging, more so in the regime of limited data. Deep learning models typically optimize geometric loss functions, which may lead to poor reconstructions as they ignore the structural properties of the shape. To tackle this, we propose a novel topological loss function based on the Euler Characteristic Transform. This loss can be used as an inductive bias to aid the optimization of any neural network toward better reconstructions in the regime of limited data. We show the effectiveness of the proposed loss function by incorporating it into SHAPR, a state-of-the-art shape reconstruction model, and test it on two benchmark datasets, viz., Red Blood Cells and Nuclei datasets. We also show a favourable property, namely injectivity and discuss the stability of the topological loss function based on the Euler Characteristic Transform.

cs.LG↗

arXiv2023

A Topological Distance Measure between Multi-Fields for Classification and Analysis of Shapes and Data

Distance measures play an important role in shape classification and data analysis problems. Topological distances based on Reeb graphs and persistence diagrams have been employed to obtain effective algorithms in shape matching and scalar data analysis. In the current paper, we propose an improved distance measure between two multi-fields by computing a multi-dimensional Reeb graph (MDRG) each of which captures the topology of a multi-field through a hierarchy of Reeb graphs in different dimensions. A hierarchy of persistence diagrams is then constructed by computing a persistence diagram corresponding to each Reeb graph of the MDRG. Based on this representation, we propose a novel distance measure between two MDRGs by extending the bottleneck distance between two Reeb graphs. We show that the proposed measure satisfies the pseudo-metric and stability properties. We examine the effectiveness of the proposed multi-field topology-based measure on two different applications: (1) shape classification and (2) detection of topological features in a time-varying multi-field data. In the shape classification problem, the performance of the proposed measure is compared with the well-known topology-based measures in shape matching. In the second application, we consider a time-varying volumetric multi-field data from the field of computational chemistry where the goal is to detect the site of stable bond formation between Pt and CO molecules. We demonstrate the ability of the proposed distance in classifying each of the sites as occurring before and after the bond stabilization.

cs.LG↗

arXiv2020

A Topological Similarity Measure between Multi-Field Data using Multi-Resolution Reeb Spaces

Searching topological similarity between a pair of shapes or data is an important problem in data analysis and visualization. The problem of computing similarity measures using scalar topology has been studied extensively and proven useful in shape and data matching. Even though multi-field (or multivariate) topology-based techniques reveal richer topological features, research on computing similarity measures using multi-field topology is still in its infancy. In the current paper, we propose a novel similarity measure between two piecewise-linear multi-fields based on their multi-resolution Reeb spaces - a newly developed data-structure that captures the topology of a multi-field. Overall, our method consists of two steps: (i) building a multi-resolution Reeb space corresponding to each of the multi-fields and (ii) proposing a similarity measure for a list of matching pairs (of nodes), obtained by comparing the multi-resolution Reeb spaces. We demonstrate an application of the proposed similarity measure by detecting the nuclear scission point in a time-varying multi-field data from computational physics.

cs.CG↗

arXiv2019

Topological Feature Search in Time-Varying Multifield Data

A wide range of data that appear in scientific experiments and simulations are multivariate or multifield in nature, consisting of multiple scalar fields. Topological feature search of such data aims to reveal important properties useful to the domain scientists. It has been shown in recent works that a single scalar field is insufficient to capture many important topological features in the data, instead one needs to consider topological relationships between multiple scalar fields. In the current paper, we propose a novel method of finding similarity between two multifield data by comparing their respective fiber component distributions. Given a time-varying multifield data, the method computes a metric plot for each pair of histograms at consecutive time stamps to understand the topological changes in the data over time. We validate the method using real and synthetic data. The effectiveness of the proposed method is shown by its ability to capture important topological features that are not always possible to detect using the individual component scalar fields.

cs.CG↗

arXiv2015

Multivariate Topology Simplification

Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multi-field data, topological analysis requires simultaneous advances both mathematically and computationally. We propose a robust multivariate topology simplification method based on ``lip''-pruning from the Reeb Space. Mathematically, we show that the projection of the Jacobi Set of multivariate data into the Reeb Space produces a Jacobi Structure that separates the Reeb Space into simple components. We also show that the dual graph of these components gives rise to a Reeb Skeleton that has properties similar to the scalar contour tree and Reeb Graph, for topologically simple domains. We then introduce a range measure to give a scaling-invariant total ordering of the components or features that can be used for simplification. Computationally, we show how to compute Jacobi Structure, Reeb Skeleton, Range and Geometric Measures in the Joint Contour Net (an approximation of the Reeb Space) and that these can be used for visualisation similar to the contour tree or Reeb Graph.

cs.CG↗

arXiv2015

Certified Computation of planar Morse-Smale Complexes

The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds. However, previous research in this field is incomplete in the case of smooth functions. In the current paper we address the following question: Given an arbitrarily complex Morse-Smale system on a planar domain, is it possible to compute its certified (topologically correct) Morse-Smale complex? Towards this, we develop an algorithm using interval arithmetic to compute certified critical points and separatrices forming the Morse-Smale complexes of smooth functions on bounded planar domain. Our algorithm can also compute geometrically close Morse-Smale complexes.

cs.CG↗

arXiv2013

Interacting Non-equilibrium Systems with Two Temperatures

We investigate a simplified model of two fully connected magnetic systems maintained at different temperatures by virtue of being connected to two independent thermal baths while simultaneously being inter-connected with each other. Using generating functional analysis, commonly used in statistical mechanics, we find exactly soluble expressions for their individual magnetisations that define a two-dimensional non-linear map, the equations of which have the same form as those obtained for densely connected equilibrium systems. Steady states correspond to the fixed points of this map, separating the parameter space into a rich set of non-equilibrium phases that we analyse in asymptotically high and low (non-equilibrium) temperature limits. The theoretical formalism is shown to subvert to the classical non-equilibrium steady state problem for two interacting systems with a non-zero heat transfer between them that catalyses a phase transition between ambient non-equilibrium states.

cond-mat.stat-mech↗