Probabilistic Inclusion Depth for Fuzzy Contour Ensemble Visualization

TL;DR

This work introduces Probabilistic Inclusion Depth (PID) to extend depth-based contour analysis to ensembles containing fuzzy or probabilistic masks, enabling uncertainty-aware visualization of scalar-field ensembles. PID uses a probabilistic inclusion operator and a mean-based PID-mean approximation to achieve linear-complexity depth computation, with GPU-accelerated parallelization for large 3D datasets. The method is validated through ranking-consistency tests and scalability analyses, and demonstrated on real-world 3D ensembles including medical soft masks and dynamic smoke plumes. The results show robust, threshold-free depth measures that preserve nuanced uncertainty information and offer efficient, scalable contour boxplots for complex ensembles.

Abstract

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.

Figures (10)

• Figure 1: Visualization of probabilistic inclusion operator between two 2D Gaussian distributions. The fixed blue Gaussian $u$ is shown with covariance ellipses at $25\%$ and $95\%$ quantiles, while the red Gaussian $v$ moves along the horizontal axis. The heatmap represents the element-wise penalty term $u(x)(1-v(x))$ of $u\subset_{\!p}v$.
• Figure 2: Probabilistic inclusion between three fuzzy (soft) disks. Here, the blue disk $u_1$ moves horizontally while the red ($u_2$) and orange ($u_3$) disks are fixed. Density values of each disk are 1 inside given radii (3, 2, and 4) and decay normally distributed, with fixed variance 0.8. The computation formulas for $\mathrm{IN}_{\mathrm{in}}^{\mathrm{p}}$ and $\mathrm{IN}_{\mathrm{out}}^{\mathrm{p}}$ are given in \ref{['eq:pid']}. Depending on which of the two is minimal for $u_1$, the overlaid heatmap represents the respective element-wise penalty term for the probabilistic inclusion depth.
• Figure 3: The left panel shows two highly similar Gaussian-blurred disk ensembles, $u_1$ (blue) and $u_2$ (red), with reference contours at the 25% and 95% quantiles. The heatmap depicts the element-wise penalty term $\mathrm{IN}_{\mathrm{in}}^{\mathrm{p}}(u_1(x))$. The middle and right panels show binary masks at two threshold levels (the 25th percentile and the density value 1), resulting in different eID values. Hatched regions indicate support overlap.
• Figure 4: Comparisons of PID and eID on an ensemble of weather simulations. The ranking differences are shown for (a) between eID ranked binary contours of 5620 m and 5640 m, and (b) PID ranks of fuzzy contours centered at 5620 m and 5640 m. Probabilistic maps of the member with the highest PID rank difference is shown in (c), and (d) shows contours of the top 5 members with highest eID rank differences.
• Figure 5: Ranking consistency comparison between PID and probabilistic IoU after removing the 10 lowest-ranked members.