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A Mathematical Foundation for the Spatial Uncertainty of Critical Points in Probabilistic Scalar Fields
... Embedding of a characteristic feature may be helpful in reducing the amount of information. Fortunately, the literature offers a wide range of methods for visualizing features of scalar fields with uncertainty, e. g., [20,42,48,56]. This approach is applied in Fig. 4 visualizing the difference between both data sets as contours, while the LCP [40] of both data sets is superposed into the explicit encoded visualization. ...
Scientists working with uncertain data, such as climate simulations, medical images, or ensembles of physical simulations, regularly confront the problem of comparing observations, e.g., to identify similarities, differences, or patterns. Current approaches in comparative visualization of uncertain scalar fields mainly rely on juxtaposition of both data and uncertainties, where each is represented using, e.g., color mapping or volume rendering. While interpretation of uncertain scalar data from visual encodings is already cognitively challenging, comparison of uncertain fields without explicit visualization support adds a further layer of complexity. In this paper, we present a theoretical framework to devise and describe a class of techniques that directly visualize differences between two or more uncertain scalar fields in a single image. We model each such technique as a combination of one or more interpolation stages, with the application of distance measures on random variables to the resulting distributions, and an appropriate visual encoding. Our framework captures existing methods and lends itself well to formulating new comparative visualization techniques for uncertain data for different visualization scenarios. Furthermore, by modeling uncertain scalar field differences as random variables themselves, we enable additional opportunities for comparison. We demonstrate the usefulness of our framework and its properties by applying it to effective comparative visualization techniques for several synthetic and real-world data sets.
... Vietinghoff et al. [63] derived the critical point probability using the Bayesian inference and derived confidence intervals [61]. Recently, Vietinghoff et al. developed a novel mathematical framework [62] that quantified uncertainty in critical points by analyzing the variation in manifestation of the same critical points occurring across realizations of the ensemble. ...
This paper presents a novel end-to-end framework for closed-form computation and visualization of critical point uncertainty in 2D uncertain scalar fields. Critical points are fundamental topological descriptors used in the visualization and analysis of scalar fields. The uncertainty inherent in data (e.g., observational and experimental data, approximations in simulations, and compression), however, creates uncertainty regarding critical point positions. Uncertainty in critical point positions, therefore, cannot be ignored, given their impact on downstream data analysis tasks. In this work, we study uncertainty in critical points as a function of uncertainty in data modeled with probability distributions. Although Monte Carlo (MC) sampling techniques have been used in prior studies to quantify critical point uncertainty, they are often expensive and are infrequently used in production-quality visualization software. We, therefore, propose a new end-to-end framework to address these challenges that comprises a threefold contribution. First, we derive the critical point uncertainty in closed form, which is more accurate and efficient than the conventional MC sampling methods. Specifically, we provide the closed-form and semianalytical (a mix of closed-form and MC methods) solutions for parametric (e.g., uniform, Epanechnikov) and nonparametric models (e.g., histograms) with finite support. Second, we accelerate critical point probability computations using a parallel implementation with the VTK-m library, which is platform portable. Finally, we demonstrate the integration of our implementation with the ParaView software system to demonstrate near-real-time results for real datasets.
This paper presents a novel end-to-end framework for closed-form computation and visualization of critical point uncertainty in 2D uncertain scalar fields. Critical points are fundamental topological descriptors used in the visualization and analysis of scalar fields. The uncertainty inherent in data (e.g., observational and experimental data, approximations in simulations, and compression), however, creates uncertainty regarding critical point positions. Uncertainty in critical point positions, therefore, cannot be ignored, given their impact on downstream data analysis tasks. In this work, we study uncertainty in critical points as a function of uncertainty in data modeled with probability distributions. Although Monte Carlo (MC) sampling techniques have been used in prior studies to quantify critical point uncertainty, they are often expensive and are infrequently used in production-quality visualization software. We, therefore, propose a new end-to-end framework to address these challenges that comprises a threefold contribution. First, we derive the critical point uncertainty in closed form, which is more accurate and efficient than the conventional MC sampling methods. Specifically, we provide the closed-form and semianalytical (a mix of closed-form and MC methods) solutions for parametric (e.g., uniform, Epanechnikov) and nonparametric models (e.g., histograms) with finite support. Second, we accelerate critical point probability computations using a parallel implementation with the VTK-m library, which is platform portable. Finally, we demonstrate the integration of our implementation with the ParaView software system to demonstrate near-real-time results for real datasets.
With data comes uncertainty, which is a widespread and frequent phenomenon in data science and analysis. The amount of information available to us is growing exponentially, owing to never-ending technological advancements. Data visualization is one of the ways to convey that information effectively. Since the error is intrinsic to data, users cannot ignore it in visualization. Failing to observe it in visualization can lead to flawed decision-making by data analysts. Data scientists know that missing out on uncertainty in data visualization can lead to misleading conclusions about data accuracy. In most cases, visualization approaches assume that the information represented is free from any error or unreliability; however, this is rarely true. The goal of uncertainty visualization is to minimize the errors in judgment and represent the information as accurately as possible. This survey discusses state-of-the-art approaches to uncertainty visualization, along with the concept of uncertainty and its sources. From the study of uncertainty visualization literature, we identified popular techniques accompanied by their merits and shortcomings. We also briefly discuss several uncertainty visualization evaluation strategies. Finally, we present possible future research directions in uncertainty visualization, along with the conclusion.
We describe a novel technique for the simultaneous visualization of multiple scalar fields, e.g. representing the members of an ensemble, based on their contour trees. Using tree alignments, a graph‐theoretic concept similar to edit distance mappings, we identify commonalities across multiple contour trees and leverage these to obtain a layout that can represent all trees simultaneously in an easy‐to‐interpret, minimally‐cluttered manner. We describe a heuristic algorithm to compute tree alignments for a given similarity metric, and give an algorithm to compute a joint layout of the resulting aligned contour trees. We apply our approach to the visualization of scalar field ensembles, discuss basic visualization and interaction possibilities, and demonstrate results on several analytic and real‐world examples.
Abstract The Max Planck Institute Grand Ensemble (MPI‐GE) is the largest ensemble of a single comprehensive climate model currently available, with 100 members for the historical simulations (1850–2005) and four forcing scenarios. It is currently the only large ensemble available that includes scenario representative concentration pathway (RCP) 2.6 and a 1% CO2 scenario. These advantages make MPI‐GE a powerful tool. We present an overview of MPI‐GE, its components, and detail the experiments completed. We demonstrate how to separate the forced response from internal variability in a large ensemble. This separation allows the quantification of both the forced signal under climate change and the internal variability to unprecedented precision. We then demonstrate multiple ways to evaluate MPI‐GE and put observations in the context of a large ensemble, including a novel approach for comparing model internal variability with estimated observed variability. Finally, we present four novel analyses, which can only be completed using a large ensemble. First, we address whether temperature and precipitation have a pathway dependence using the forcing scenarios. Second, the forced signal of the highly noisy atmospheric circulation is computed, and different drivers are identified to be important for the North Pacific and North Atlantic regions. Third, we use the ensemble dimension to investigate the time dependency of Atlantic Meridional Overturning Circulation variability changes under global warming. Last, sea level pressure is used as an example to demonstrate how MPI‐GE can be utilized to estimate the ensemble size needed for a given scientific problem and provide insights for future ensemble projects.
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.
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 adopted 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.
We present an algorithm for tracking regions in time-dependent scalar fields that uses global knowledge from all time steps for determining the tracks. The regions are defined using merge trees, thereby representing a hierarchical segmentation of the data in each time step. The similarity of regions of two consecutive time steps is measured using their volumetric overlap and a histogram difference. The main ingredient of our method is a directed acyclic graph that records all relevant similarity information as follows: the regions of all time steps are the nodes of the graph, the edges represent possible short feature tracks between consecutive time steps, and the edge weights are given by the similarity of the connected regions. We compute a feature track as the global solution of a shortest path problem in the graph. We use these results to steer the – to the best of our knowledge – first algorithm for spatio-temporal feature similarity estimation. Our algorithm works for 2D and 3D time-dependent scalar fields. We compare our results to previous work, showcase its robustness to noise, and exemplify its utility using several real-world data sets.
Uncertainty is a common and crucial issue in scientific data. The exploration and analysis of three-dimensional (3D) and large two-dimensional (2D) data with uncertainty information demand an effective visualization augmented with both user interaction and relevant context. The contour tree has been exploited as an efficient data structure to guide exploratory visualization. This paper proposes an interactive visualization tool for exploring data with quantitative uncertainty representations. First, we introduce a balanced planar hierarchical contour tree layout integrated with tree view interaction, allowing users to quickly navigate between levels of detail for contours of large data. Further, uncertainty information is attached to a planar contour tree layout to avoid the visual cluttering and occlusion in viewing uncertainty in 3D data or large 2D data. For the first time, the uncertainty information is explored as a combination of the data-level uncertainty which represents the uncertainty concerning the numerical values of the data, the contour variability which quantifies the positional variation of contours, and the topology variability which reveals the topological variation of contour trees. This information provides a new insight into how the uncertainty exists with and relates to the features of the data. The experimental results show that this new visualization facilitates a quick and accurate selection of prominent contours with high or low uncertainty and variability.
Quantifying uncertainty is an increasingly important topic across many domains. The uncertainties present in data come with many diverse representations having originated from a wide variety of disci-plines. Communicating these uncertainties is a task often left to visu-alization without clear connection between the quantification and vi-sualization. In this paper, we first identify frequently occurring types of uncertainty. Second, we connect those uncertainty representations to ones commonly used in visualization. We then look at various approaches to visualizing this uncertainty by partitioning the work based on the di-mensionality of the data and the dimensionality of the uncertainty. We also discuss noteworthy exceptions to our taxonomy along with future research directions for the uncertainty visualization community.
This paper introduces a novel, non-local characterization of critical points and their global relation in 2D uncertain scalar fields. The characterization is based on the analysis of the support of the probability density functions (PDF) of the input data. Given two scalar fields representing reliable estimations of the bounds of this support, our strategy identifies mandatory critical points: spatial regions and function ranges where critical points have to occur in any realization of the input. The algorithm provides a global pairing scheme for mandatory critical points which is used to construct mandatory join and split trees. These trees enable a visual exploration of the common topological structure of all possible realizations of the uncertain data. To allow multi-scale visualization, we introduce a simplification scheme for mandatory critical point pairs revealing the most dominant features. Our technique is purely combinatorial and handles parametric distribution models and ensemble data. It does not depend on any computational parameter and does not suffer from numerical inaccuracy or global inconsistency. The algorithm exploits ideas of the established join/split tree computation. It is therefore simple to implement, and its complexity is output-sensitive. We illustrate, evaluate, and verify our method on synthetic and real-world data.
The relation between two Morse functions defined on a smooth, compact and orientable 2-manifold can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain, where the gradients of the two functions are aligned. Both the Jacobi set itself as well as the segmentation of the domain it induces, have shown to be useful in various applications. Unfortunately in practice, functions often contain noise and discretization artifacts, causing their Jacobi set to become unmanageably large and complex. Although there exist techniques to simplify Jacobi sets, they are unsuitable for most applications as they lack fine-grained control over the process, and heavily restrict the type of simplifications possible.
This paper introduces the theoretical foundations of a new simplification framework for Jacobi sets. We present a new interpretation of Jacobi set simplification based on the perspective of domain segmentation. Generalizing the cancellation of critical points from scalar functions to Jacobi sets, we focus on simplifications that can be realized by smooth approximations of the corresponding functions, and show how this implies simultaneous simplification of contiguous subsets of the Jacobi set. Using these extended cancellations as atomic operations, we introduce an algorithm to successively cancel subsets of the Jacobi set with minimal modifications to some user-defined metric. We show that for simply connected domains, our algorithm reduces a given Jacobi set to its minimal configuration, that is, one with no birth-death points (a birth-death point is a specific type of singularity within the Jacobi set where the level sets of the two functions and the Jacobi set have a common normal direction).
Exploring and analyzing the temporal evolution of features in large-scale time-varying datasets is a common problem in many areas of science and engineering. One natural representation of such data is tracking graphs, i.e., constrained graph layouts that use one spatial dimension to indicate time and show the “tracks” of each feature as it evolves, merges or disappears. However, for practical data sets creating the corresponding optimal graph layouts that minimize the number of intersections can take hours to compute with existing techniques. Furthermore, the resulting graphs are often unmanageably large and complex even with an ideal layout. Finally, due to the cost of the layout, changing the feature definition, e.g. by changing an iso-value, or analyzing properly adjusted sub-graphs is infeasible. To address these challenges, this paper presents a new framework that couples hierarchical feature definitions with progressive graph layout algorithms to provide an interactive exploration of dynamically constructed tracking graphs. Our system enables users to change feature definitions on-the-fly and filter features using arbitrary attributes while providing an interactive view of the resulting tracking graphs. Furthermore, the graph display is integrated into a linked view system that provides a traditional 3D view of the current set of features and allows a cross-linked selection to enable a fully flexible spatio-temporal exploration of data. We demonstrate the utility of our approach with several large-scale scientific simulations from combustion science.
Given a distribution on persistence diagrams and observations
we introduce an algorithm in this paper
that computes a Fr\'echet mean from the set of diagrams . We prove
the algorithm converges to a local minima. If the underlying measure is
a combination of Dirac masses
then we prove a law of large numbers result for a Fr\'echet mean computed by
the algorithm given observations drawn iid from . We illustrate the
convergence of an empirical mean computed by the algorithm to a population mean
by simulations from Gaussian random fields.
Uncertainty is ubiquitous in science, engineering and medicine. Drawing conclusions from uncertain data is the normal case, not an exception. While the field of statistical graphics is well established, only a few 2D and 3D visualization and feature extraction methods have been devised that consider uncertainty. We present mathematical formulations for uncertain equivalents of isocontours based on standard probability theory and statistics and employ them in interactive visualization methods. As input data, we consider discretized uncertain scalar fields and model these as random fields. To create a continuous representation suitable for visualization we introduce interpolated probability density functions. Furthermore, we introduce numerical condition as a general means in feature-based visualization. The condition number-which potentially diverges in the isocontour problem-describes how errors in the input data are amplified in feature computation. We show how the average numerical condition of isocontours aids the selection of thresholds that correspond to robust isocontours. Additionally, we introduce the isocontour density and the level crossing probability field; these two measures for the spatial distribution of uncertain isocontours are directly based on the probabilistic model of the input data. Finally, we adapt interactive visualization methods to evaluate and display these measures and apply them to 2D and 3D data sets.
We consider a uniform distribution on the set of moments of order corresponding to probability measures on the interval [0,1]. To each (random) vector of moments in we consider the corresponding uniquely determined monic (random) orthogonal polynomial of degree n and study the asymptotic properties of its roots if .
This paper 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 [87] to the Wasserstein metric space of merge trees [93]. 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.
In an era of quickly growing data set sizes, information reduction methods such as extracting or highlighting characteristic features become more and more important for data analysis. For single scalar fields, topological methods can fill this role by extracting and relating critical points. While such methods are regularly employed to study single scalar fields, it is less well studied how they can be extended to uncertain data, as produced, e.g., by ensemble simulations. Motivated by our previous work on visualization in climate research, we study new methods to characterize critical points in ensembles of 2D scalar fields. Previous work on this topic either assumed or required specific distributions, did not account for uncertainty introduced by approximating the underlying latent distributions by a finite number of fields, or did not allow to answer all our domain experts' questions. In this work, we use Bayesian inference to estimate the probability of critical points, either of the original ensemble or its bootstrapped mean. This does not make any assumptions on the underlying distribution and allows to estimate the sensitivity of the results to finite-sample approximations of the underlying distribution. We use color mapping to depict these probabilities and the stability of their estimation. The resulting images can, e.g., be used to estimate how precise the critical points of the mean-field are. We apply our method to synthetic data to validate its theoretical properties and compare it with other methods in this regard. We also apply our method to the data from our previous work, where it provides a more accurate answer to the domain experts' research questions.
This paper presents a framework for topological feature analysis in time-dependent climate ensembles. Important climate indices such as the El Nino Southern Oscillation Index (ENSO) or the North Atlantic Oscillation Index (NAO) are usually derived by evaluating scalar fields at fixed locations or regions where these extremal values occur most frequently today. However, under climate change, dynamic circulation changes are likely to cause shifts in the intensity, frequency and location of underlying physical phenomena. In case of the NAO for instance, climatologists are interested in the position and intensity of the Icelandic Low and the Azores High as their interplay strongly influences the European climate, especially during the winter season. To robustly extract and track such highly variable features without a-priori region information, we present a topology-based method for dynamic extraction of such uncertain critical points on a global scale. The system additionally integrates techniques to visualize the variability within the ensemble and correlations between features. To demonstrate the utility of our VTK+TTK-based software framework, we explore a 150-year climate projection consisting of 100 ensemble members and particularly concentrate on sea level pressure fields.
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 L2-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.
The Foundations of Computational Mathematics meetings are a platform for cross-fertilization between numerical analysis, mathematics and computer science. This volume, first published in 2004, contains the plenary presentations, given by some of the leading authorities in the world, and topics surveyed range from optimization to computer algebra, image processing to differential equations, quantum complexity to geometry. The volume will be essential reading for all those wishing to be informed of the state-of-the-art in computational mathematics.
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.
In this paper, we present an approach to analyze 1D parameter spaces of time-dependent flow simulation ensembles. By extending the concept of the finite-time Lyapunov exponent to the ensemble domain, i.e., to the parameter that gives rise to the ensemble, we obtain a tool for quantitative analysis of parameter sensitivity both in space and time. We exemplify our approach using 2D synthetic examples and computational fluid dynamics 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.
Tracking graphs are a well established tool in topological analysis to visualize the evolution of components and their properties over time, i.e., when components appear, disappear, merge, and split. However, tracking graphs are limited to a single level threshold and the graphs may vary substantially even under small changes to the threshold. To examine the evolution of features for varying levels, users have to compare multiple tracking graphs without a direct visual link between them. We propose a novel, interactive, nested graph visualization based on the fact that the tracked superlevel set components for different levels are related to each other through their nesting hierarchy. This approach allows us to set multiple tracking graphs in context to each other and enables users to effectively follow the evolution of components for different levels simultaneously. We demonstrate the effectiveness of our approach on datasets from finite pointset methods, computational fluid dynamics, and cosmology simulations.
We introduce a new method that identifies and tracks features in arbitrary dimensions using the merge tree—a structure for identifying topological features based on thresholding in scalar fields. This method analyzes the evolution of features of the function by tracking changes in the merge tree and relates features by matching subtrees between consecutive time steps. Using the time-varying merge tree, we present a structural visualization of the changing function that illustrates both features and their temporal evolution. We demonstrate the utility of our approach by applying it to temporal cluster analysis of high-dimensional point clouds.
We study contour trees of terrains, which encode the topological changes of the level set of the height value ℓ as we raise ℓ from -∞ to +∞ on the terrains, in the presence of uncertainty in data. We assume that the terrain is represented by a piecewise-linear height function over a planar triangulation M, by specifying the height of each vertex. We study the case when M is fixed and the uncertainty lies in the height of each vertex in the triangulation, which is described by a probability distribution. We present efficient sampling-based Monte Carlo methods for estimating, with high probability, (i) the probability that two points lie on the same edge of the contour tree, within additive error; (ii) the expected distance of two points p, q and the probability that the distance of p, q is at least ℓ on the contour tree, within additive error, where the distance of p, q on a contour tree is defined to be the difference between the maximum height and the minimum height on the unique path from p to q on the contour tree. The main technical contribution of the paper is to prove that a small number of samples are sufficient to estimate these quantities. We present two applications of these algorithms, and also some experimental results to demonstrate the effectiveness of our approach.
This paper presents the state of the art in the area of topology-based visualization. It describes the process and results of an extensive annotation for generating a definition and terminology for the field. The terminology enabled a typology for topological models which is used to organize research results and the state of the art. Our report discusses relations among topological models and for each model describes research results for the computation, simplification, visualization, and application. The paper identifies themes common to subfields, current frontiers, and unexplored territory in this research area.
Simulations and measurements often result in scalar fields with uncertainty due to errors or output sensitivity estimates. Methods for analyzing topological features of such fields usually are not capable of handling all aspects of the data. They either are not deterministic due to using Monte Carlo approaches, approximate the data with confidence intervals, or miss out on incorporating important properties, such as correlation. In this paper, we focus on the analysis of critical points of Gaussian-distributed scalar fields. We introduce methods to deterministically extract critical points, approximate their probability with high precision, and even capture relations between them resulting in an abstract graph representation. Unlike many other methods, we incorporate all information contained in the data including global correlation. Our work therefore is a first step towards a reliable and complete description of topological features of Gaussian-distributed scalar fields.
Most visualization techniques have been designed on the assumption that the data to be represented are free from uncertainty. Yet this is rarely the case. Recently the visualization community has risen to the challenge of incorporating an indication of uncertainty into visual representations, and in this article we review their work. We place the work in the context of a reference model for data visualization, that sees data pass through a pipeline of processes. This allows us to distinguish the visualization of uncertainty—which considers how we depict uncertainty specified with the data—and the uncertainty of visualization—which considers how much inaccuracy occurs as we process data through the pipeline. It has taken some time for uncertain visualization methods to be developed, and we explore why uncertainty visualization is hard—one explanation is that we typically need to find another display dimension and we may have used these up already! To organize the material we return to a typology developed by one of us in the early days of visualization, and make use of this to present a catalog of visualization techniques describing the research that has been done to extend each method to handle uncertainty. Finally we note the responsibility on us all to incorporate any known uncertainty into a visualization, so that integrity of the discipline is maintained.
The goal of visualization is to effectively and accurately communicate data. Visualization research has often overlooked the errors and uncertainty which accompany the scientific process and describe key characteristics used to fully understand the data. The lack of these representations can be attributed, in part, to the inherent difficulty in defining, characterizing, and controlling this uncertainty, and in part, to the difficulty in including additional visual metaphors in a well designed, potent display. However, the exclusion of this information cripples the use of visualization as a decision making tool due to the fact that the display is no longer a true representation of the data. This systematic omission of uncertainty commands fundamental research within the visualization community to address, integrate, and expect uncertainty information. In this chapter, we outline sources and models of uncertainty, give an overview of the state-of-the-art, provide general guidelines, outline small exemplary applications, and finally, discuss open problems in uncertainty visualization.
In many fields of science or engineering, we are confronted with uncertain data. For that reason, the visualization of uncertainty received a lot of attention, especially in recent years. In the majority of cases, Gaussian distributions are used to describe uncertain behavior, because they are able to model many phenomena encountered in science. Therefore, in most applications uncertain data is (or is assumed to be) Gaussian distributed. If such uncertain data is given on fixed positions, the question of interpolation arises for many visualization approaches. In this paper, we analyze the effects of the usual linear interpolation schemes for visualization of Gaussian distributed data. In addition, we demonstrate that methods known in geostatistics and machine learning have favorable properties for visualization purposes in this case.
In scalar fields, critical points (points with vanishing derivatives) are important indicators of the topology of iso-contours. When the data values are affected by uncertainty, the locations and types of critical points vary and can no longer be predicted accurately. In this paper, we derive, from a given uncertain scalar ensemble, measures for the likelihood of the occurrence of critical points, with respect to both the positions and types of the critical points. In an ensemble, every instance is a possible occurrence of the phenomenon represented by the scalar values. We show that, by deriving confidence intervals for the gradient and the determinant and trace of the Hessian matrix in scalar ensembles, domain points can be classified according to whether a critical point can occur at a certain location and a specific type of critical point should be expected there. When the data uncertainty can be described stochastically via Gaussian distributed random variables, we show that even probabilistic measures for these events can be deduced.
An uncertain (scalar, vector, tensor) field is usually perceived as a discrete random field with a priori unknown probability distributions. To compute derived probabilities, e.g. for the occurrence of certain features, an appropriate probabilistic model has to be selected. The majority of previous approaches in uncertainty visualization were restricted to Gaussian fields. In this paper we extend these approaches to nonparametric models, which are much more flexible, as they can represent various types of distributions, including multimodal and skewed ones. We present three examples of nonparametric representations: (a) empirical distributions, (b) histograms and (c) kernel density estimates (KDE). While the first is a direct representation of the ensemble data, the latter two use reconstructed probability density functions of continuous random variables. For KDE we propose an approach to compute valid consistent marginal distributions and to efficiently capture correlations using a principal component transformation. Furthermore, we use automatic bandwidth selection, obtaining a model for probabilistic local feature extraction. The methods are demonstrated by computing probabilities of level crossings, critical points and vortex cores in simulated biofluid dynamics and climate data.
The Reeb graph is a useful tool in visualizing real-valued data obtained from computational simulations of physical processes. We characterize the evolution of the Reeb graph of a time-varying continuous function defined in three-dimensional space. We show how to maintain the Reeb graph over time and compress the entire sequence of Reeb graphs into a single, partially persistent data structure, and augment this data structure with Betti numbers to describe the topology of level sets and with path seeds to assist in the fast extraction of level sets for visualization.
Box plot is a compact representation that encodes the minimum, maximum, mean, median, and quartile information of a distribution. In practice, a single box plot is drawn for each variable of interest. With the advent of more accessible computing power, we are now facing the problem of visualizing data where there is a distribution at each 2D spatial location. Simply extending the box plot technique to distributions over 2D domain is not straightforward. One challenge is reducing the visual clutter if a box plot is drawn over each grid location in the 2D domain. This paper presents and discusses two general approaches, using parametric statistics and shape descriptors, to present 2D distribution data sets. Both approaches provide additional insights compared to the traditional box plot technique.
A computer-assisted linear programming approach is used to study minimum-cardinality decompositions of the cube. A triangulation of the 7-cube into 1493 simplices is given and it is shown that this, and a previously given triangulation of the 6-cube into 308 simplices, are the smallest possible for these dimensions. A characterization is given for the numbers of the various types of simplices used in all minimum-cardinality triangulations of the d-cube for d = 5, 6, 7. It is shown that the minimum of the cardinalities of all corner-slicing triangulations of I7 is 1820. For decompositions of the cube more general than triangulations, it is shown for dimension 5 that the minimum cardinality is 67, and for dimension 6 it is at least 270.
Scientific visualization as currently understood and practiced is still a relatively new discipline. As a result, we visualization researches are not necessarily accustomed to undertaking the sorts of self-examinations that other scientists routinely undergo in relation to their work. Yet if we are to creat a disciplinary culture focused on matters of real scientific importance and committed to real progress, it is essential that we ask ourselves hard questions on an ongoing basis. What are the most important research issues facing us? What underlying assumptions needs to be challenged and perhaps abandoned? What practices need to be reviewed? In this article, I attempt to start a discussion of these issues by proposing a list of top research problems and issues in scientific visualization.
For time-dependent scalar fields, one is often interested in topology changes of contours in time. In this paper, we focus on describing how contours split and merge over a certain time interval. Rather than attempting to describe all individual contour splitting and merging events, we focus on the simpler and therefore more tractable in practice problem: describing and querying the cumulative effect of the splitting and merging events over a user-specified time interval. Using our system one can, for example, find all contours at time t0 that continue to two contours at time t1 without hitting the boundary of the domain. For any such contour, there has to be a bifurcation happening to it somewhere between the two times, but, in addition to that, many other events may possibly happen without changing the cumulative outcome (e.g. merging with several contours born after t0 or splitting off several contours that disappear before t1). Our approach is flexible enough to enable other types of queries, if they can be cast as counting queries for numbers of connected components of intersections of contours with certain simply connected domains. Examples of such queries include finding contours with large life spans, contours avoiding certain subset of the domain over a given time interval or contours that continue to two at a later time and then merge back to one some time later. Experimental results show that our method can handle large 3D (2 space dimensions plus time) and 4D (3D+time) datasets. Both preprocessing and query algorithms can easily be parallelized.
Visualizing confidence intervals for critical point probabilities in 2D scalar field ensembles
- D Vietinghoff
- M Bottinger
- G Scheuermann
- C Heine