Isosurface Visualization of Data with Nonparametric Models for Uncertainty

EnConVis: A Unified Framework for Ensemble Contour Visualization

Ensemble simulation is a crucial method to handle potential uncertainty in modern simulation and has been widely applied in many disciplines. Many ensemble contour visualization methods have been introduced to facilitate ensemble data analysis. On the basis of deep exploration and summarization of existing techniques and domain requirements, we propose a unified framework of ensemble contour visualization, EnConVis (Ensemble Contour Visualization), which systematically combines state-of-the-art methods. We model ensemble contour visualization as a four-step pipeline consisting of four essential procedures: member filtering, point-wise modeling, uncertainty band extraction, and visual mapping. For each of the four essential procedures, we compare different methods they use, analyze their pros and cons, highlight research gaps, and attempt to fill them. Specifically, we add Kernel Density Estimation in the point-wise modeling procedure and multi-layer extraction in the uncertainty band extraction procedure. This step shows the ensemble data's details accurately and provides abstract levels. We also analyze existing methods from a global perspective. We investigate their mechanisms and compare their effects, on the basis of which, we offer selection guidelines for them. From the overall perspective of this framework, we find choices and combinations that have not been tried before, which can be well compensated by our method. Synthetic data and real-world data are leveraged to verify the efficacy of our method. Domain experts' feedback suggests that our approach helps them better understand ensemble data analysis.

Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)

This article presents a computational framework for the Principal Geodesic Analysis of merge trees (MT-PGA), a novel adaptation of the celebrated Principal Component Analysis (PCA) framework (K. Pearson 1901) to the Wasserstein metric space of merge trees (Pont et al. 2022). We formulate MT-PGA computation as a constrained optimization problem, aiming at adjusting a basis of orthogonal geodesic axes, while minimizing a fitting energy. We introduce an efficient, iterative algorithm which exploits shared-memory parallelism, as well as an analytic expression of the fitting energy gradient, to ensure fast iterations. Our approach also trivially extends to extremum persistence diagrams. Extensive experiments on public ensembles demonstrate the efficiency of our approach - with MT-PGA computations in the orders of minutes for the largest examples. We show the utility of our contributions by extending to merge trees two typical PCA applications. First, we apply MT-PGA to data reduction and reliably compress merge trees by concisely representing them by their first coordinates in the MT-PGA basis. Second, we present a dimensionality reduction framework exploiting the first two directions of the MT-PGA basis to generate two-dimensional layouts of the ensemble. We augment these layouts with persistence correlation views, enabling global and local visual inspections of the feature variability in the ensemble. In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.

Fiber Uncertainty Visualization for Bivariate Data With Parametric and Nonparametric Noise Models

Visualization and analysis of multivariate data and their uncertainty are top research challenges in data visualization. Constructing fiber surfaces is a popular technique for multivariate data visualization that generalizes the idea of level-set visualization for univariate data to multivariate data. In this paper, we present a statistical framework to quantify positional probabilities of fibers extracted from uncertain bivariate fields. Specifically, we extend the state-of-the-art Gaussian models of uncertainty for bivariate data to other parametric distributions (e.g., uniform and Epanechnikov) and more general nonparametric probability distributions (e.g., histograms and kernel density estimation) and derive corresponding spatial probabilities of fibers. In our proposed framework, we leverage Green's theorem for closed-form computation of fiber probabilities when bivariate data are assumed to have independent parametric and nonparametric noise. Additionally, we present a nonparametric approach combined with numerical integration to study the positional probability of fibers when bivariate data are assumed to have correlated noise. For uncertainty analysis, we visualize the derived probability volumes for fibers via volume rendering and extracting level sets based on probability thresholds. We present the utility of our proposed techniques via experiments on synthetic and simulation datasets.

Uncertainty Visualization of 2D Morse Complex Ensembles Using Statistical Summary Maps

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.

Wasserstein Distances, Geodesics and Barycenters of Merge Trees

This paper presents a unified computational framework for the estimation of distances, geodesics and barycenters of merge trees. We extend recent work on the edit distance [104] and introduce a new metric, called the Wasserstein distance between merge trees, which is purposely designed to enable efficient computations of geodesics and barycenters. Specifically, our new distance is strictly equivalent to the $L$2-Wasserstein distance between extremum persistence diagrams, but it is restricted to a smaller solution space, namely, the space of rooted partial isomorphisms between branch decomposition trees. This enables a simple extension of existing optimization frameworks [110] for geodesics and barycenters from persistence diagrams to merge trees. We introduce a task-based algorithm which can be generically applied to distance, geodesic, barycenter or cluster computation. The task-based nature of our approach enables further accelerations with shared-memory parallelism. Extensive experiments on public ensembles and SciVis contest benchmarks demonstrate the efficiency of our approach - with barycenter computations in the orders of minutes for the largest examples - as well as its qualitative ability to generate representative barycenter merge trees, visually summarizing the features of interest found in the ensemble. We show the utility of our contributions with dedicated visualization applications: feature tracking, temporal reduction and ensemble clustering. We provide a lightweight C++ implementation that can be used to reproduce our results.

Direct Volume Rendering with Nonparametric Models of Uncertainty

We present a nonparametric statistical framework for the quantification, analysis, and propagation of data uncertainty in direct volume rendering (DVR). The state-of-the-art statistical DVR framework allows for preserving the transfer function (TF) of the ground truth function when visualizing uncertain data; however, the existing framework is restricted to parametric models of uncertainty. In this paper, we address the limitations of the existing DVR framework by extending the DVR framework for nonparametric distributions. We exploit the quantile interpolation technique to derive probability distributions representing uncertainty in viewing-ray sample intensities in closed form, which allows for accurate and efficient computation. We evaluate our proposed nonparametric statistical models through qualitative and quantitative comparisons with the mean-field and parametric statistical models, such as uniform and Gaussian, as well as Gaussian mixtures. In addition, we present an extension of the state-of-the-art rendering parametric framework to 2D TFs for improved DVR classifications. We show the applicability of our uncertainty quantification framework to ensemble, downsampled, and bivariate versions of scalar field datasets.

Confidence-Controlled Local Isosurfacing

This article presents a novel framework that can generate a high-fidelity isosurface model of X-ray computed tomography (CT) data. CT surfaces with subvoxel precision and smoothness can be simply modeled via isosurfacing, where a single CT value represents an isosurface. However, this inevitably results in geometric distortion of the CT data containing CT artifacts. An alternative is to treat this challenge as a segmentation problem. However, in general, segmentation techniques are not robust against noisy data and require heavy computation to handle the artifacts that occur in three-dimensional CT data. Furthermore, the surfaces generated from segmentation results may contain jagged, overly smooth, or distorted geometries. We present a novel local isosurfacing framework that can address these issues simultaneously. The proposed framework exploits two primary techniques: 1) Canny edge approach for obtaining surface candidate boundary points and evaluating their confidence and 2) screened Poisson optimization for fitting a surface to the boundary points in which the confidence term is incorporated. This combination facilitates local isosurfacing that can produce high-fidelity surface models. We also implement an intuitive user interface to alleviate the burden of selecting the appropriate confidence computing parameters. Our experimental results demonstrate the effectiveness of the proposed framework.

eFESTA: Ensemble Feature Exploration with Surface Density Estimates

We propose surface density estimate (SDE) to model the spatial distribution of surface features-isosurfaces, ridge surfaces, and streamsurfaces-in 3D ensemble simulation data. The inputs of SDE computation are surface features represented as polygon meshes, and no field datasets are required (e.g., scalar fields or vector fields). The SDE is defined as the kernel density estimate of the infinite set of points on the input surfaces and is approximated by accumulating the surface densities of triangular patches. We also propose an algorithm to guide the selection of a proper kernel bandwidth for SDE computation. An ensemble Feature Exploration method based on Surface densiTy EstimAtes (eFESTA) is then proposed to extract and visualize the major trends of ensemble surface features. For an ensemble of surface features, each surface is first transformed into a density field based on its contribution to the SDE, and the resulting density fields are organized into a hierarchical representation based on the pairwise distances between them. The hierarchical representation is then used to guide visual exploration of the density fields as well as the underlying surface features. We demonstrate the application of our method using isosurface in ensemble scalar fields, Lagrangian coherent structures in uncertain unsteady flows, and streamsurfaces in ensemble fluid flows.

Visualization and Visual Analysis of Ensemble Data: A Survey

Over the last decade, ensemble visualization has witnessed a significant development due to the wide availability of ensemble data, and the increasing visualization needs from a variety of disciplines. From the data analysis point of view, it can be observed that many ensemble visualization works focus on the same facet of ensemble data, use similar data aggregation or uncertainty modeling methods. However, the lack of reflections on those essential commonalities and a systematic overview of those works prevents visualization researchers from effectively identifying new or unsolved problems and planning for further developments. In this paper, we take a holistic perspective and provide a survey of ensemble visualization. Specifically, we study ensemble visualization works in the recent decade, and categorize them from two perspectives: (1) their proposed visualization techniques; and (2) their involved analytic tasks. For the first perspective, we focus on elaborating how conventional visualization techniques (e.g., surface, volume visualization techniques) have been adapted to ensemble data; for the second perspective, we emphasize how analytic tasks (e.g., comparison, clustering) have been performed differently for ensemble data. From the study of ensemble visualization literature, we have also identified several research trends, as well as some future research opportunities.

Progressive Wasserstein Barycenters of Persistence Diagrams

This paper presents an efficient algorithm for the progressive approximation of Wasserstein barycenters of persistence diagrams, with applications to the visual analysis of ensemble data. Given a set of scalar fields, our approach enables the computation of a persistence diagram which is representative of the set, and which visually conveys the number, data ranges and saliences of the main features of interest found in the set. Such representative diagrams are obtained by computing explicitly the discrete Wasserstein barycenter of the set of persistence diagrams, a notoriously computationally intensive task. In particular, we revisit efficient algorithms for Wasserstein distance approximation [12,51] to extend previous work on barycenter estimation [94]. We present a new fast algorithm, which progressively approximates the barycenter by iteratively increasing the computation accuracy as well as the number of persistent features in the output diagram. Such a progressivity drastically improves convergence in practice and allows to design an interruptible algorithm, capable of respecting computation time constraints. This enables the approximation of Wasserstein barycenters within interactive times. We present an application to ensemble clustering where we revisit the k-means algorithm to exploit our barycenters and compute, within execution time constraints, meaningful clusters of ensemble data along with their barycenter diagram. Extensive experiments on synthetic and real-life data sets report that our algorithm converges to barycenters that are qualitatively meaningful with regard to the applications, and quantitatively comparable to previous techniques, while offering an order of magnitude speedup when run until convergence (without time constraint). Our algorithm can be trivially parallelized to provide additional speedups in practice on standard workstations. We provide a lightweight C++ implementation of our approach that can be used to reproduce our results.

Volumetric Feature-Based Classification and Visibility Analysis for Transfer Function Design

Transfer function (TF) design is a central topic in direct volume rendering. The TF fundamentally translates data values into optical properties to reveal relevant features present in the volumetric data. We propose a semi-automatic TF design scheme which consists of two steps: First, we present a clustering process within 1D/2D TF domain based on the proximities of the respective volumetric features in the spatial domain. The presented approach provides an interactive tool that aids users in exploring clusters and identifying features of interest (FOI). Second, our method automatically generates a TF by iteratively refining the optical properties for the selected features using a novel feature visibility measurement. The proposed visibility measurement leverages the similarities of features to enhance their visibilities in DVR images. Compared to the conventional visibility measurement, the proposed feature visibility is able to efficiently sense opacity changes and precisely evaluate the impact of selected features on resulting visualizations. Our experiments validate the effectiveness of the proposed approach by demonstrating the advantages of integrating feature similarity into the visibility computations. We examine a number of datasets to establish the utility of our approach for semi-automatic TF design.

A statistical framework for quantification and visualisation of positional uncertainty in deep brain stimulation electrodes

Deep brain stimulation (DBS) is an established therapy for treating patients with movement disorders such as Parkinson's disease. Patient-specific computational modelling and visualisation have been shown to play a key role in surgical and therapeutic decisions for DBS. The computational models use brain imaging, such as magnetic resonance (MR) and computed tomography (CT), to determine the DBS electrode positions within the patient's head. The finite resolution of brain imaging, however, introduces uncertainty in electrode positions. The DBS stimulation settings for optimal patient response are sensitive to the relative positioning of DBS electrodes to a specific neural substrate (white/grey matter). In our contribution, we study positional uncertainty in the DBS electrodes for imaging with finite resolution. In a three-step approach, we first derive a closed-form mathematical model characterising the geometry of the DBS electrodes. Second, we devise a statistical framework for quantifying the uncertainty in the positional attributes of the DBS electrodes, namely the direction of longitudinal axis and the contact-centre positions at subvoxel levels. The statistical framework leverages the analytical model derived in step one and a Bayesian probabilistic model for uncertainty quantification. Finally, the uncertainty in contact-centre positions is interactively visualised through volume rendering and isosurfacing techniques. We demonstrate the efficacy of our contribution through experiments on synthetic and real datasets. We show that the spatial variations in true electrode positions are significant for finite resolution imaging, and interactive visualisation can be instrumental in exploring probabilistic positional variations in the DBS lead.

Persistence Atlas for Critical Point Variability in Ensembles

This paper presents a new approach for the visualization and analysis of the spatial variability of features of interest represented by critical points in ensemble data. Our framework, called Persistence Atlas, enables the visualization of the dominant spatial patterns of critical points, along with statistics regarding their occurrence in the ensemble. The persistence atlas represents in the geometrical domain each dominant pattern in the form of a confidence map for the appearance of critical points. As a by-product, our method also provides 2-dimensional layouts of the entire ensemble, highlighting the main trends at a global level. Our approach is based on the new notion of Persistence Map, a measure of the geometrical density in critical points which leverages the robustness to noise of topological persistence to better emphasize salient features. We show how to leverage spectral embedding to represent the ensemble members as points in a low-dimensional Euclidean space, where distances between points measure the dissimilarities between critical point layouts and where statistical tasks, such as clustering, can be easily carried out. Further, we show how the notion of mandatory critical point can be leveraged to evaluate for each cluster confidence regions for the appearance of critical points. Most of the steps of this framework can be trivially parallelized and we show how to efficiently implement them. Extensive experiments demonstrate the relevance of our approach. The accuracy of the confidence regions provided by the persistence atlas is quantitatively evaluated and compared to a baseline strategy using an off-the-shelf clustering approach. We illustrate the importance of the persistence atlas in a variety of real-life datasets, where clear trends in feature layouts are identified and analyzed. We provide a lightweight VTK-based C++ implementation of our approach that can be used for reproduction purposes.

Probabilistic Asymptotic Decider for Topological Ambiguity Resolution in Level-Set Extraction for Uncertain 2D Data

We present a framework for the analysis of uncertainty in isocontour extraction. The marching squares (MS) algorithm for isocontour reconstruction generates a linear topology that is consistent with hyperbolic curves of a piecewise bilinear interpolation. The saddle points of the bilinear interpolant cause topological ambiguity in isocontour extraction. The midpoint decider and the asymptotic decider are well-known mathematical techniques for resolving topological ambiguities. The latter technique investigates the data values at the cell saddle points for ambiguity resolution. The uncertainty in data, however, leads to uncertainty in underlying bilinear interpolation functions for the MS algorithm, and hence, their saddle points. In our work, we study the behavior of the asymptotic decider when data at grid vertices is uncertain. First, we derive closed-form distributions characterizing variations in the saddle point values for uncertain bilinear interpolants. The derivation assumes uniform and nonparametric noise models, and it exploits the concept of ratio distribution for analytic formulations. Next, the probabilistic asymptotic decider is devised for ambiguity resolution in uncertain data using distributions of the saddle point values derived in the first step. Finally, the confidence in probabilistic topological decisions is visualized using a colormapping technique. We demonstrate the higher accuracy and stability of the probabilistic asymptotic decider in uncertain data with regard to existing decision frameworks, such as deciders in the mean field and the probabilistic midpoint decider, through the isocontour visualization of synthetic and real datasets.

An Interactive Framework for Visualization of Weather Forecast Ensembles

Numerical Weather Prediction (NWP) ensembles are commonly used to assess the uncertainty and confidence in weather forecasts. Spaghetti plots are conventional tools for meteorologists to directly examine the uncertainty exhibited by ensembles, where they simultaneously visualize isocontours of all ensemble members. To avoid visual clutter in practical usages, one needs to select a small number of informative isovalues for visual analysis. Moreover, due to the complex topology and variation of ensemble isocontours, it is often a challenging task to interpret the spaghetti plot for even a single isovalue in large ensembles. In this paper, we propose an interactive framework for uncertainty visualization of weather forecast ensembles that significantly improves and expands the utility of spaghetti plots in ensemble analysis. Complementary to state-of-the-art methods, our approach provides a complete framework for visual exploration of ensemble isocontours, including isovalue selection, interactive isocontour variability exploration, and interactive sub-region selection and re-analysis. Our framework is built upon the high-density clustering paradigm, where the mode structure of the density function is represented as a hierarchy of nested subsets of the data. We generalize the high-density clustering for isocontours and propose a bandwidth selection method for estimating the density function of ensemble isocontours. We present novel visualizations based on high-density clustering results, called the mode plot and the simplified spaghetti plot. The proposed mode plot visually encodes the structure provided by the high-density clustering result and summarizes the distribution of ensemble isocontours. It also enables the selection of subsets of interesting isocontours, which are interactively highlighted in a linked spaghetti plot for providing spatial context. To provide an interpretable overview of the positional variability of isocontours, our system allows for selection of informative isovalues from the simplified spaghetti plot. Due to the spatial variability of ensemble isocontours, the system allows for interactive selection and focus on sub-regions for local uncertainty and clustering re-analysis. We examine a number of ensemble datasets to establish the utility of our approach and discuss its advantages over state-of-the-art visual analysis tools for ensemble data.

CoDDA: A Flexible Copula-based Distribution Driven Analysis Framework for Large-Scale Multivariate Data

CoDDA (Copula-based Distribution Driven Analysis) is a flexible framework for large-scale multivariate datasets. A common strategy to deal with large-scale scientific simulation data is to partition the simulation domain and create statistical data summaries. Instead of storing the high-resolution raw data from the simulation, storing the compact statistical data summaries results in reduced storage overhead and alleviated I/O bottleneck. Such summaries, often represented in the form of statistical probability distributions, can serve various post-hoc analysis and visualization tasks. However, for multivariate simulation data using standard multivariate distributions for creating data summaries is not feasible. They are either storage inefficient or are computationally expensive to be estimated in simulation time (in situ) for large number of variables. In this work, using copula functions, we propose a flexible multivariate distribution-based data modeling and analysis framework that offers significant data reduction and can be used in an in situ environment. The framework also facilitates in storing the associated spatial information along with the multivariate distributions in an efficient representation. Using the proposed multivariate data summaries, we perform various multivariate post-hoc analyses like query-driven visualization and sampling-based visualization. We evaluate our proposed method on multiple real-world multivariate scientific datasets. To demonstrate the efficacy of our framework in an in situ environment, we apply it on a large-scale flow simulation.

Uncertainty Visualization Using Copula-Based Analysis in Mixed Distribution Models

Distributions are often used to model uncertainty in many scientific datasets. To preserve the correlation among the spatially sampled grid locations in the dataset, various standard multivariate distribution models have been proposed in visualization literature. These models treat each grid location as a univariate random variable which models the uncertainty at that location. Standard multivariate distributions (both parametric and nonparametric) assume that all the univariate marginals are of the same type/family of distribution. But in reality, different grid locations show different statistical behavior which may not be modeled best by the same type of distribution. In this paper, we propose a new multivariate uncertainty modeling strategy to address the needs of uncertainty modeling in scientific datasets. Our proposed method is based on a statistically sound multivariate technique called Copula, which makes it possible to separate the process of estimating the univariate marginals and the process of modeling dependency, unlike the standard multivariate distributions. The modeling flexibility offered by our proposed method makes it possible to design distribution fields which can have different types of distribution (Gaussian, Histogram, KDE etc.) at the grid locations, while maintaining the correlation structure at the same time. Depending on the results of various standard statistical tests, we can choose an optimal distribution representation at each location, resulting in a more cost efficient modeling without significantly sacrificing on the analysis quality. To demonstrate the efficacy of our proposed modeling strategy, we extract and visualize uncertain features like isocontours and vortices in various real world datasets. We also study various modeling criterion to help users in the task of univariate model selection.

A Statistical Direct Volume Rendering Framework for Visualization of Uncertain Data

With uncertainty present in almost all modalities of data acquisition, reduction, transformation, and representation, there is a growing demand for mathematical analysis of uncertainty propagation in data processing pipelines. In this paper, we present a statistical framework for quantification of uncertainty and its propagation in the main stages of the visualization pipeline. We propose a novel generalization of Irwin-Hall distributions from the statistical viewpoint of splines and box-splines, that enables interpolation of random variables. Moreover, we introduce a probabilistic transfer function classification model that allows for incorporating probability density functions into the volume rendering integral. Our statistical framework allows for incorporating distributions from various sources of uncertainty which makes it suitable in a wide range of visualization applications. We demonstrate effectiveness of our approach in visualization of ensemble data, visualizing large datasets at reduced scale, iso-surface extraction, and visualization of noisy data.

Uncertainty Footprint: Visualization of Nonuniform Behavior of Iterative Algorithms Applied to 4D Cell Tracking

Research on microscopy data from developing biological samples usually requires tracking individual cells over time. When cells are three-dimensionally and densely packed in a time-dependent scan of volumes, tracking results can become unreliable and uncertain. Not only are cell segmentation results often inaccurate to start with, but it also lacks a simple method to evaluate the tracking outcome. Previous cell tracking methods have been validated against benchmark data from real scans or artificial data, whose ground truth results are established by manual work or simulation. However, the wide variety of real-world data makes an exhaustive validation impossible. Established cell tracking tools often fail on new data, whose issues are also difficult to diagnose with only manual examinations. Therefore, data-independent tracking evaluation methods are desired for an explosion of microscopy data with increasing scale and resolution. In this paper, we propose the uncertainty footprint, an uncertainty quantification and visualization technique that examines nonuniformity at local convergence for an iterative evaluation process on a spatial domain supported by partially overlapping bases. We demonstrate that the patterns revealed by the uncertainty footprint indicate data processing quality in two algorithms from a typical cell tracking workflow - cell identification and association. A detailed analysis of the patterns further allows us to diagnose issues and design methods for improvements. A 4D cell tracking workflow equipped with the uncertainty footprint is capable of self diagnosis and correction for a higher accuracy than previous methods whose evaluation is limited by manual examinations.

In Situ Distribution Guided Analysis and Visualization of Transonic Jet Engine Simulations

Study of flow instability in turbine engine compressors is crucial to understand the inception and evolution of engine stall. Aerodynamics experts have been working on detecting the early signs of stall in order to devise novel stall suppression technologies. A state-of-the-art Navier-Stokes based, time-accurate computational fluid dynamics simulator, TURBO, has been developed in NASA to enhance the understanding of flow phenomena undergoing rotating stall. Despite the proven high modeling accuracy of TURBO, the excessive simulation data prohibits post-hoc analysis in both storage and I/O time. To address these issues and allow the expert to perform scalable stall analysis, we have designed an in situ distribution guided stall analysis technique. Our method summarizes statistics of important properties of the simulation data in situ using a probabilistic data modeling scheme. This data summarization enables statistical anomaly detection for flow instability in post analysis, which reveals the spatiotemporal trends of rotating stall for the expert to conceive new hypotheses. Furthermore, the verification of the hypotheses and exploratory visualization using the summarized data are realized using probabilistic visualization techniques such as uncertain isocontouring. Positive feedback from the domain scientist has indicated the efficacy of our system in exploratory stall analysis.