Adaptive Sampling of 3D Spatial Correlations for Focus+Context Visualization

TL;DR

This work tackles the challenge of visualizing spatial correlations in large 3D ensembles under memory and time constraints. It introduces adaptive sampling via Bayesian Optimal Sampling (BOS) to estimate region-to-region correlation maxima and embeds these estimates in two chord diagrams (context and focus) using hierarchical edge bundling and a space-filling curve layout, complemented by GPU-accelerated mutual information computations. The approach supports both single-field correlation analysis and inter-variable comparisons, with a mean-tree aggregation strategy for memory-limited cases and an interactive focus refinement workflow. Evaluations on synthetic data and two large weather ensembles demonstrate faster convergence of BOS versus random sampling, scalable performance on GPUs, and effective visualization of multi-scale correlation structures, enabling interactive exploration of 1000-member ensembles.

Abstract

Visualizing spatial correlations in 3D ensembles is challenging due to the vast amounts of information that need to be conveyed. Memory and time constraints make it unfeasible to pre-compute and store the correlations between all pairs of domain points. We propose the embedding of adaptive correlation sampling into chord diagrams with hierarchical edge bundling to alleviate these constraints. Entities representing spatial regions are arranged along the circular chord layout via a space-filling curve, and Bayesian optimal sampling is used to efficiently estimate the maximum occurring correlation between any two points from different regions. Hierarchical edge bundling reduces visual clutter and emphasizes the major correlation structures. By selecting an edge, the user triggers a focus diagram in which only the two regions connected via this edge are refined and arranged in a specific way in a second chord layout. For visualizing correlations between two different variables, which are not symmetric anymore, we switch to showing a full correlation matrix. This avoids drawing the same edges twice with different correlation values. We introduce GPU implementations of both linear and non-linear correlation measures to further reduce the time that is required to generate the context and focus views, and to even enable the analysis of correlations in a 1000-member ensemble.

Figures (13)

• Figure 1: Bottom left: Context view of a large weather forecast ensemble Necker2020 shows mutual information (MI) maxima between regions in a 3D space partition, laid out on a chord along a space filling curve and visualized via edge bundling. The ensemble spread is encoded in the outer shell. Maxima are estimated via Bayesian optimal sampling (BOS) for all 3828 pairs of regions, each comprising $32 \times 32 \times 20$ grid points, in less than 16 seconds. Selected regions (red and blue triangle) are shown in a 3D view (top left). Middle: Focus diagram shows refined MI estimates - computed on the fly using our GPU implementation in less than 4 seconds - between selected regions. Right: Focus view on regions selected in the first focus diagram.
• Figure 2: Locations of the first 100 samples picked by different sampling schemes for finding the maximum in a 2D field, here the probability density function of a 2D multivariate normal distribution.
• Figure 3: Left: Subdivision of a $250 \times 352 \times 20$ simulation grid into $8 \times 11 \times 1$ bricks of size $32 \times 32 \times 20$. The bricks are traversed using a Z-order curve. Right: Nodes in the chord diagram. The leaves ordered by the Z-order curve are linearized in the outermost circle. Inner octree nodes used for hierarchical edge bundling (they are hidden during rendering). Nodes with the same parent are assigned the same color.
• Figure 4: Context chord diagram using MI as dependence measure for a large simulation ensemble Necker2020. Left: Spatial correlations in the temperature field tk. Middle: Sub-selection of correlations with MI $\ge 0.62$. Right: Sub-selection of correlations with brick distance $\ge 570$km (measured from the brick centers) and MI $\ge 0.62$. Ensemble spread $\sigma$ of tk is shown on the outer ring.
• Figure 5: Recursive linear layout of bricks along a z-curve. Two bricks are selected (red and blue), and laid out on the respective upper and lower half-circle in the top left focus view. Entities in each brick are linearized via a z-curve. In further refinements of the focus view (indicated by blue arrows), the picked regions are again laid out along z-curves.