Elsevier

Computers & Graphics

Volume 41, June 2014, Pages 13-25
Computers & Graphics

Special Section on Uncertainty and Parameter Space Analysis in Visualization
Visualizing the stability of critical points in uncertain scalar fields

https://doi.org/10.1016/j.cag.2014.01.007Get rights and content

Highlights

  • We analyze associated gradient and Hessian matrix fields of scalar ensembles.

  • We derive confidence intervals for gradients and trace and determinant of Hessian.

  • Confidence intervals of gradients show whether critical points exist.

  • Confidence intervals of trace and determinant show type stability of critical points.

Abstract

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.

Introduction

Scalar ensembles consist of several scalar fields, where every field or instance indicates a possible occurrence of the phenomenon represented by the data values. Ensembles are often generated numerically via multiple simulation runs with slightly perturbed input parameter settings. The rationale stems from the observation that the result of every run is affected by a certain degree of uncertainty, for instance, due to model simplifications or approximations inherent to the numerical schemes employed. Generating multiple instances helps predict and quantify the range of outcomes and, thus, allows us to classify features with respect to their stability across instances.

An important class of features in scalar fields is based on level-sets or iso-contours, i.e., the set of all points in the domain where the scalar field takes on a prescribed value, also called an iso-value. The effect of uncertainty on level-sets has been treated in several works [1], [2], or [3], which investigate the positional variations of level-sets due to uncertainty. Such an analysis, however, does not allow making reliable estimates of the possible geometric or topological variations of level-sets.

Recently, Pfaffelmoser et al. [4] have looked into the effect of uncertainty on the variability of gradients in scalar fields. Indicators for the likelihood of geometric changes of level-sets were derived from confidence intervals of the gradient magnitude and orientation, resulting in a stability analysis of both the shape and the slope of level-sets. By using a similar technique to propagate uncertainty for derived quantities in scalar fields that are linear combinations of the input values, and by introducing a method for non-linear combinations, we propose techniques to classify critical points in scalar ensemble fields with respect to different notions of stability. Interesting features often relate to critical points, since these indicate prominent surface components and their topological changes. Depending on the position and type of the critical points, the spatial locations where changes in the surface topology take place and the nature of these changes can be identified: surface components emerge or vanish at minima and maxima, join or split at saddles.

Contribution: We investigate the associated gradient and Hessian matrix fields of the scalar ensemble members to identify the possible locations of the critical points, and assess their stability in type throughout the ensemble. We first summarize ensembles statistically and derive corresponding moments for the gradients. Since critical points occur where the gradients vanish, we use confidence intervals of the gradients to obtain quantities indicating the possibility of a critical point occurring around a given location. We then derive statistical summaries for the trace and determinant of the Hessian matrix, to give insight into the tendency of critical points to behave like minima, maxima, or saddles near a specified location in the ensemble.

The remainder of the paper is as follows: in the next section we review related work. We then introduce methods to analyze critical points in Section 3, which we visualize in Section 4. The proposed approaches are validated in Section 5 and demonstrated on two synthetic and two real world data sets in Section 6. We conclude the paper with a summary of the contributions.

Section snippets

Related work

Uncertainty is a topic relevant to many research domains, and has been classified among the top research areas in visualization. Overviews of uncertainty visualization approaches are given, for instance, by Griethe and Schumann [5], Thomson et al. [6], or Potter et al. [7].

Uncertainty information has often been summarized by quantities such as mean and standard deviation, which have been encoded together with the actual data by means of color maps, opacity, texture, animation, glyphs, etc., by,

Critical points in ensembles

Critical points of scalar fields are those points where the gradient vector vanishes. Several methods can be applied to locate critical points in scalar data sets: finding the crossings of the zero-contours of the x- and y-components of the gradient vector field, or the grid points with non-zero Poincaré indices, etc. The locations of critical points, however, are affected by the uncertainty in the data, which causes variations in the positions and types of critical points throughout the

Visualization

In the following, we present techniques to illustrate the introduced indicators together with the scalar fields of the ensemble. We occasionally display the critical points, even though they are not relevant to computing the indicator functions, in order to contribute to the validation of the proposed techniques. Furthermore, the concurrent visualization allows us to place the indicators and the critical points in space, and observe the possible occurrences of critical points and their type

Validation

In the previous sections we introduced and visualized two types of functions to indicate, at each grid vertex, whether critical points can be assumed to emerge nearby and display a stable behavior. Depending on the indicators, critical points have been classified as more or less stable in location and type.

According to this classification, critical points occurring near grid vertices where positional indicators have positive values are stable, i.e., they are more likely to appear frequently

Further results

We apply the introduced techniques for analysis, visualization, and validation to three other data sets, two synthetic ensembles and another ECMWF ensemble.

The first synthetic data set, of dimensions 100×100, was generated by assigning the three parameters a, b, and c in x4/4y4/4x2y2/2+ax2/2+bxy+cy2/2,(x,y)[2,2]X[2,2], random numbers generated from a multivariate normal distribution with the following mean and covariance matrix: μ=[0.510.5],Σ=[10.50.50.510.50.50.51].

When the

Conclusion

Prominent features display variations across ensembles, potentially changing their location and shape. In this paper, we developed several indicator functions to give insight into the salient features of scalar fields and their stability, by investigating their associated critical points. We summarized ensembles statistically and computed corresponding moments for the associated gradient fields and the determinant and trace of the Hessian matrices. The first were used to derive quantities

Acknowledgments

The work was partly funded by the European Union under the ERC Advanced Grant 291372: Safer-Vis – Uncertainty Visualization for Reliable Data Discovery. We thank Tobias Pfaffelmoser for valuable discussions during the development of the work described here. Access to ECMWF prediction data has been kindly provided in the context of the ECMWF special project “Support Tool for HALO Missions”. We are grateful to the special project members Marc Rautenhaus and Andreas Dörnbrack for providing the

References (37)

  • S. Djurcilov et al.

    Visualizing scalar volumetric data with uncertainty

    Comput Graph

    (2002)
  • B Zehner et al.

    Visualization of gridded scalar data with uncertainty in geosciences

    Computers Geosciences

    (2010)
  • M. Leutbecher et al.

    Ensemble forecasting

    J Comput Phys

    (2008)
  • T. Pfaffelmoser et al.

    Visualizing the variability of gradients in uncertain 2d scalar fields

    Vis Comput Gr, IEEE Trans

    (2013)
  • K. Pothkow et al.

    Positional uncertainty of isocontourscondition analysis and probabilistic measures

    IEEE Trans Vis Comput Graph

    (2011)
  • K. Wu et al.

    A contour tree based visualization for exploring data with uncertainty

    Int J Uncertain Quantif

    (2013)
  • T Pfaffelmoser et al.

    Visualizing the variability of gradients in uncertain 2d scalar fields

    IEEE Transactions on Visualization and Computer Graphics

    (2013)
  • Griethe H, Schumann H. The visualization of uncertain data: methods and problems. In: SimVis; 2006. p....
  • Thomson J, Hetzler E, MacEachren A, Gahegan M, Pavel M. A typology for visualizing uncertainty. In: Proceedings of...
  • Potter K, Rosen P, Johnson CR. From quantification to visualization: a taxonomy of uncertainty visualization...
  • C. Wittenbrink et al.

    Glyphs for visualizing uncertainty in vector fields

    IEEE Trans Vis Comput Graph

    (2002)
  • Rhodes P, Laramee R, Bergeron R, Sparr T. Uncertainty visualization methods in isosurface rendering. In: Eurographics...
  • C. Lundstrom et al.

    Uncertainty visualization in medical volume rendering using probabilistic animation

    IEEE Trans Vis Comput Graph

    (2007)
  • J Sanyal et al.

    Noodles: A tool for visualization of numerical weather model ensemble uncertainty

    IEEE Transactions on Visualization and Computer Graphics

    (2010)
  • A.T. Pang et al.

    Approaches to uncertainty visualization

    Vis Comput

    (1997)
  • G. Grigoryan et al.

    Point-based probabilistic surfaces to show surface uncertainty

    IEEE Trans Vis Comput Graph

    (2004)
  • Brown R. Animated visual vibrations as an uncertainty visualisation technique. In: GRAPHITE '04. New York, NY, USA:...
  • Pfaffelmoser T, Westermann R. Visualization of global correlation structures in uncertain 2D scalar fields. Comput...
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