Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Berlin, Germany.
IEEE Trans Vis Comput Graph. 2011 Oct;17(10):1393-406. doi: 10.1109/TVCG.2010.247.
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.
不确定性在科学、工程和医学中无处不在。从不确定的数据中得出结论是正常情况，而不是例外。虽然统计图形学领域已经成熟，但只有少数 2D 和 3D 可视化和特征提取方法考虑了不确定性。我们提出了基于标准概率论和统计学的等轮廓不确定等价物的数学公式，并将其应用于交互式可视化方法中。作为输入数据，我们考虑离散化的不确定标量场，并将其建模为随机场。为了创建适合可视化的连续表示，我们引入了插值概率密度函数。此外，我们引入数值条件作为基于特征的可视化的通用手段。条件数——在等轮廓问题中可能发散——描述了输入数据中的误差在特征计算中是如何放大的。我们展示了等轮廓的平均数值条件如何有助于选择对应于稳健等轮廓的阈值。此外，我们引入了等轮廓密度和水平穿越概率场；这两个用于不确定等轮廓空间分布的度量直接基于输入数据的概率模型。最后，我们自适应交互式可视化方法来评估和显示这些度量，并将其应用于 2D 和 3D 数据集。
