As data becomes increasingly central across engineering and scientific disciplines, effective visualization is essential for interpreting complex, high-dimensional structures. Dimensionality reduction techniques project high-dimensional data into lower dimensions while aiming to preserve structural properties such as pairwise distances and local neighborhoods. In this paper, we focus on improving homological preservation, that is, the retention of topological features such as connected components and loops, which is critical for maintaining global shape and continuity. We first introduce AdaMapper, a Mapper-based algorithm that leverages persistence diagrams to guide both skeleton construction and landmark selection. AdaMapper incorporates an adaptive refinement strategy that automatically increases cover resolution in regions exhibiting topological loops. We then propose AdaHIsomap, which extends landmark Isomap by incorporating homology-informed landmark selection and augmenting it with random anchor points to better balance distance and homology preservation. We evaluate both methods on a diverse set of datasets, including high-dimensional point clouds, scientific simulations, networks, and image data, and benchmark their performance against state-of-the-art approaches.

Topological descriptors such as contour trees are widely utilized in scientific data analysis and visualization, with applications from materials science to climate simulations. It is desirable to preserve topological descriptors when data compression is part of the scientific workflow for these applications. However, classic error-bounded lossy compressors for volumetric data do not guarantee the preservation of topological descriptors, despite imposing strict pointwise error bounds. In this work, we introduce a general framework for augmenting any lossy compressor to preserve the topology of the data during compression. Specifically, our framework quantifies the adjustments (to the decompressed data) needed to preserve the contour tree and then employs a custom variable-precision encoding scheme to store these adjustments. We demonstrate the utility of our framework in augmenting classic compressors (such as SZ3, TTHRESH, and ZFP) and deep learning-based compressors (such as Neurcomp) with topological guarantees.

This research explores a novel paradigm for preserving topological segmentations in existing error-bounded lossy compressors. Today's lossy compressors rarely consider preserving topologies such as Morse-Smale complexes, and the discrepancies in topology between original and decompressed datasets could potentially result in erroneous interpretations or even incorrect scientific conclusions. In this paper, we focus on preserving Morse-Smale segmentations in 2D/3D piecewise linear scalar fields, targeting the precise reconstruction of minimum/maximum labels induced by the integral line of each vertex. The key is to derive a series of edits during compression time. These edits are applied to the decompressed data, leading to an accurate reconstruction of segmentations while keeping the error within the prescribed error bound. To this end, we develop a workflow to fi x ex trema an d in tegral lines alternatively until convergence within finite iterations. We accelerate each workflow component with shared-memory/GPU parallelism to make the performance practical for coupling with compressors. We demonstrate use cases with fluid dynamics, ocean, and cosmology application datasets with a significant acceleration with an NVIDIA A100 GPU.

In this paper, we present a flexible and probabilistic framework for tracking topological features in time-varying scalar fields using merge trees and partial optimal transport. Merge trees are topological descriptors that record the evolution of connected components in the sublevel sets of scalar fields. We present a new technique for modeling and comparing merge trees using tools from partial optimal transport. In particular, we model a merge tree as a measure network, that is, a network equipped with a probability distribution, and define a notion of distance on the space of merge trees inspired by partial optimal transport. Such a distance offers a new and flexible perspective for encoding intrinsic and extrinsic information in the comparative measures of merge trees. More importantly, it gives rise to a partial matching between topological features in time-varying data, thus enabling flexible topology tracking for scientific simulations. Furthermore, such partial matching may be interpreted as probabilistic coupling between features at adjacent time steps, which gives rise to probabilistic tracking graphs. We derive a stability result for our distance and provide numerous experiments indicating the efficacy of our framework in extracting meaningful feature tracks.

Abstract Atmospheric rivers (ARs) are long, narrow regions of water vapor in the Earth's atmosphere that transport heat and moisture from the tropics to the mid‐latitudes. ARs are often associated with extreme weather events in North America and contribute significantly to water supply and flood risk. However, characterizing ARs has been a major challenge due to the lack of a universal definition and their structural variations. Existing AR detection tools (ARDTs) produce distinct AR boundaries for the same event, making the risk assessment of ARs a difficult task. Understanding these uncertainties is crucial to improving the predictability of AR impacts, including their landfall areas and associated precipitation, which could cause catastrophic flooding and landslides over the coastal regions. In this work, we develop an uncertainty visualization framework that captures boundary and interior uncertainties, i.e., structural variations, of an ensemble of ARs that arise from a set of ARDTs. We first provide a statistical overview of the AR boundaries using the contour boxplots of Whitaker et al. that highlight the structural variations of AR boundaries based on their nesting relationships. We then introduce the topological skeletons of ARs based on Morse complexes that characterize the interior variation of an ensemble of ARs. We propose an uncertainty visualization of these topological skeletons, inspired by MetroSets of Jacobson et al. that emphasizes the agreements and disagreements across the ensemble members. Through case studies and expert feedback, we demonstrate that the two approaches complement each other, and together they could facilitate an effective comparative analysis process and provide a more confident outlook on an AR's shape, area, and onshore impact.

Tropical cyclones (TCs) are among the most destructive weather systems. Realistically and efficiently detecting and tracking TCs are critical for assessing their impacts and risks. In particular, the eye is a signature feature of a mature TC. Therefore, knowing the eyes' locations and movements is crucial for both operational weather forecasts and climate risk assessments. Recently, a multilevel robustness framework has been introduced to study the critical points of time-varying vector fields. The framework quantifies the robustness (i.e., structural stability) of critical points across varying neighborhoods. By relating the multilevel robustness with critical point tracking, the framework has demonstrated its potential in cyclone tracking. An advantage is that it identifies cyclonic features using only 2D wind vector fields, which is encouraging as most tracking algorithms require multiple dynamic and thermodynamic variables at different altitudes. A disadvantage is that the framework does not scale well computationally for datasets containing a large number of cyclones. This paper introduces a topologically robust physics-informed tracking framework (TROPHY) for TC tracking. The main idea is to integrate physical knowledge of TC to drastically improve the computational efficiency of multilevel robustness framework for large-scale climate datasets. First, during preprocessing, we propose a physics-informed feature selection strategy to filter 90% of critical points that are short-lived and have low stability, thus preserving good candidates for TC tracking. Second, during in-processing, we impose constraints during the multilevel robustness computation to focus only on physics-informed neighborhoods of TCs. We apply TROPHY to 30 years of 2D wind fields from reanalysis data in ERA5 and generate a number of TC tracks. In comparison with the observed tracks, we demonstrate that TROPHY can capture TC characteristics (e.g., frequency, intensity, duration, latitudes with maximum intensity, and genesis) that are comparable to and sometimes even better than a well-validated TC tracking algorithm that requires multiple dynamic and thermodynamic scalar fields.

Existing error-bounded lossy compression techniques control the pointwise error during compression to guarantee the integrity of the decompressed data. However, they typically do not explicitly preserve the topological features in data. When performing post hoc analysis with decompressed data using topological methods, preserving topology in the compression process to obtain topologically consistent and correct scientific insights is desirable. In this paper, we introduce TopoSZ, an error-bounded lossy compression method that preserves the topological features in 2D and 3D scalar fields. Specifically, we aim to preserve the types and locations of local extrema as well as the level set relations among critical points captured by contour trees in the decompressed data. The main idea is to derive topological constraints from contour-tree-induced segmentation from the data domain, and incorporate such constraints with a customized error-controlled quantization strategy from the SZ compressor (version 1.4). Our method allows users to control the pointwise error and the loss of topological features during the compression process with a global error bound and a persistence threshold.

Abstract Critical point tracking is a core topic in scientific visualization for understanding the dynamic behaviour of time‐varying vector field data. The topological notion of robustness has been introduced recently to quantify the structural stability of critical points, that is, the robustness of a critical point is the minimum amount of perturbation to the vector field necessary to cancel it. A theoretical basis has been established previously that relates critical point tracking with the notion of robustness, in particular, critical points could be tracked based on their closeness in stability, measured by robustness, instead of just distance proximity within the domain. However, in practice, the computation of classic robustness may produce artifacts when a critical point is close to the boundary of the domain; thus, we do not have a complete picture of the vector field behaviour within its local neighbourhood. To alleviate these issues, we introduce a multilevel robustness framework for the study of 2D time‐varying vector fields. We compute the robustness of critical points across varying neighbourhoods to capture the multiscale nature of the data and to mitigate the boundary effect suffered by the classic robustness computation. We demonstrate via experiments that such a new notion of robustness can be combined seamlessly with existing feature tracking algorithms to improve the visual interpretability of vector fields in terms of feature tracking, selection and comparison for large‐scale scientific simulations. We observe, for the first time, that the minimum multilevel robustness is highly correlated with physical quantities used by domain scientists in studying a real‐world tropical cyclone dataset. Such an observation helps to increase the physical interpretability of robustness.

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.

Morse complexes are gradient-based topological descriptors with close connections to Morse theory. They are widely applicable in scientific visualization as they serve as important abstractions for gaining insights into the topology of scalar fields. Data uncertainty inherent to scalar fields due to randomness in their acquisition and processing, however, limits our understanding of Morse complexes as structural abstractions. We, therefore, explore uncertainty visualization of an ensemble of 2D Morse complexes that arises from scalar fields coupled with data uncertainty. We propose several statistical summary maps as new entities for quantifying structural variations and visualizing positional uncertainties of Morse complexes in ensembles. Specifically, we introduce three types of statistical summary maps - the probabilistic map, the significance map, and the survival map - to characterize the uncertain behaviors of gradient flows. We demonstrate the utility of our proposed approach using wind, flow, and ocean eddy simulation datasets.

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.

Abstract We present a state‐of‐the‐art report on time‐dependent flow topology. We survey representative papers in visualization and provide a taxonomy of existing approaches that generalize flow topology from time‐independent to time‐dependent settings. The approaches are classified based upon four categories: tracking of steady topology, reference frame adaption, pathline classification or clustering, and generalization of critical points. Our unique contributions include introducing a set of desirable mathematical properties to interpret physical meaningfulness for time‐dependent flow visualization, inferring mathematical properties associated with selective research papers, and utilizing such properties for classification. The five most important properties identified in the existing literature include coincidence with the steady case, induction of a partition within the domain, Lagrangian invariance, objectivity, and Galilean invariance.

Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.

Sequence describes the primary structure of a protein, which contains important structural, characteristic, and genetic information and thereby motivates many sequence-based computational approaches to infer protein function. Among them, feature-base approaches attract increased attention because they make prediction from a set of transformed and more biologically meaningful sequence features. However, original features extracted from sequence are usually of high dimensionality and often compromised by irrelevant patterns, therefore dimension reduction is necessary prior to classification for efficient and effective protein function prediction. A protein usually performs several different functions within an organism, which makes protein function prediction amulti-label classificationproblem. In machine learning, multi-label classification deals with problems where each object may belong to more than one class. As a well-known feature reduction method, linear discriminant analysis (LDA) has been successfully applied in many practical applications. It, however, by nature is designed forsingle-label classification, in which each object can belong to exactly one class. Because directly applying LDA in multi-label classification causes ambiguity when computing scatters matrices, we apply a new Multi-label Linear Discriminant Analysis (MLDA) approach to address this problem and meanwhile preserve powerful classification capability inherited from classical LDA. We further extend MLDA by$\ell _1$-normalization to overcome the problem of over-counting data points with multiple labels. In addition, we incorporate biological network data using Laplacian embedding into our method, and assess the reliability of predicted putative functions. Extensive empirical evaluations demonstrate promising results of our methods.

In Drosophila gene expression pattern research, the in situ hybridization (ISH) image has become the standard technique to visualize and study the spatial distribution of RNA. To facilitate the search and comparison of Drosophila gene expression patterns during Drosophila embryogenesis, it is highly desirable to annotate the tissue-level anatomical ontology terms for ISH images. In ISH image annotations, the image content representation is crucial to achieve satisfactory results. However, existing methods mainly focus on improving the classification algorithms and only using simple visual descriptor. If we integrate the effective local and holistic visual descriptors via proper learning method, we can achieve more accurate image annotation results than using individual visual descriptor.