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.

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