Probabilistic Inclusion Depth for Fuzzy Contour Ensemble Visualization

2025-12-17

AI Summary

Visualizing scalar field ensembles faces challenges in unifying fuzzy (e.g., soft) and binary contours, and efficiently computing contour boxplots for probabilistic masks, heterogeneous grids, or large-scale 3D data. Method: We propose the Probabilistic Inclusion Depth (PID) model, which formulates isosurface extraction as a probabilistic fuzzy decision process—establishing the first general data depth framework supporting fuzzy contours. We introduce the probabilistic inclusion operator ⊂ₚ, define the mean-probability contour approximation, and design a GPU-parallel algorithm enabling order-of-magnitude speedup. Results: Experiments demonstrate that PID robustly generates contour boxplots for probabilistic masks, multi-type grids, and large-scale 3D ensembles—outperforming state-of-the-art methods on both synthetic and real-world datasets—and has been validated by domain experts.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data visualization

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.

Problem

• Proposes PID for visualizing fuzzy contour ensembles
• Extends contour extraction to incorporate probabilistic sensitivity information
• Enables efficient computation for large 3D ensembles on GPU

Innovation

• Probabilistic inclusion operator for fuzzy contour ensembles
• Efficient GPU parallel algorithm for fast computation
• Mean probabilistic contour approximation to reduce complexity