We propose Probabilistic Inclusion Depth (PID) for the ensemble visualization of scalar fields. By introducing a probabilistic inclusion operator $\subset_{\!p}$, our method is a general data depth model supporting ensembles of fuzzy contours, such as soft masks from modern segmentation methods, and conventional ensembles of binary contours. We also advocate to extend contour extraction in scalar field ensembles to become a fuzzy decision by considering the probabilistic distribution of an isovalue to encode the sensitivity information. To reduce the complexity of the data depth computation, an efficient approximation using the mean probabilistic contour is devised. Furthermore, an order of magnitude reduction in computational time is achieved with an efficient parallel algorithm on the GPU. Our new method enables the computation of contour boxplots for ensembles of probabilistic masks, ensembles defined on various types of grids, and large 3D ensembles that are not studied by existing methods. The effectiveness of our method is evaluated with numerical comparisons to existing techniques on synthetic datasets, through examples of real-world ensemble datasets, and expert feedback.

翻译：我们提出概率包含深度（PID）用于标量场集合的可视化。通过引入概率包含算子$\subset_{\!p}$，我们的方法是一种通用的数据深度模型，支持模糊轮廓集合（例如现代分割方法产生的软掩码）以及传统的二值轮廓集合。我们还主张将标量场集合中的轮廓提取扩展为模糊决策，通过考虑等值面的概率分布来编码敏感性信息。为降低数据深度计算的复杂度，我们设计了一种使用平均概率轮廓的高效近似方法。此外，通过GPU上的高效并行算法，实现了计算时间数量级的缩减。我们的新方法能够计算概率掩码集合、定义在不同类型网格上的集合以及现有方法未研究的大型三维集合的轮廓箱线图。我们通过合成数据集上与现有技术的数值比较、真实世界集合数据集的示例以及专家反馈，评估了本方法的有效性。