Topological Feature Search in Time-Varying Multifield Data

TL;DR

This work tackles the challenge of identifying topological features in time-varying multifield data, where single-scalar analyses can miss critical cross-field structures. It introduces fiber-component distributions over the multifield range and defines metric-based distances, including a singular-values–weighted variant, to compare consecutive time steps with theoretical metric guarantees. The approach is implemented in a JCN/VTK workflow and validated on synthetic and real datasets (e.g., nuclear scission in plutonium and fermium densities, Pt-CO orbital interactions), demonstrating the ability to detect topology-driven changes that may be invisible to individual components. The framework provides a scalable, interpretable tool for rapid topological event detection and lays the groundwork for future Reeb-space distance measures and sub-domain analysis.

Abstract

A wide range of data that appear in scientific experiments and simulations are multivariate or multifield in nature, consisting of multiple scalar fields. Topological feature search of such data aims to reveal important properties useful to the domain scientists. It has been shown in recent works that a single scalar field is insufficient to capture many important topological features in the data, instead one needs to consider topological relationships between multiple scalar fields. In the current paper, we propose a novel method of finding similarity between two multifield data by comparing their respective fiber component distributions. Given a time-varying multifield data, the method computes a metric plot for each pair of histograms at consecutive time stamps to understand the topological changes in the data over time. We validate the method using real and synthetic data. The effectiveness of the proposed method is shown by its ability to capture important topological features that are not always possible to detect using the individual component scalar fields.

Figures (6)

• Figure 1: Figure shows a bivariate synthetic data and corresponding structures to understand its topology. (a) Paraboloid and height field with Jacobi set (red), total $9$ connected components of the Jacobi set are numbered as $1$ to $9$ (b) Singular fiber-components that pass through the Jacobi set points, (c) Reeb Space (JCN) with Jacobi structure (in red). Jacobi structure components that are the projection of the Jacobi set components on the Reeb Space are shown by the corresponding dashed numbers. (d) Histogram with singular values (bins).
• Figure 2: Plots of distance measures between consecutive sites in a series of bivariate (height, paraboloid) fields. (a) Various distance measures show a peak at site 11, indicating a topological change. The proposed metric $d_q^{\mathbb{S}}$ also exhibits a peak, more significant than other distance measures.(b) Root-mean-square plot is not able to capture the topological change. This indicates the need for a topological data structures for multifield data that captures topological changes. (c) Fiber-component distributions for selected sites. Singular values are highlighted in red. Blue nodes indicate regular nodes and the shades of blue indicate the number of nodes in a particular bin (light indicates low). (d) Corresponding Reeb spaces. The height field is mapped to color (blue is low and red is high).
• Figure 3: Plots of the distance measures for the scission data for the plutonium atom. (a) Distance measure between fields at consecutive time steps vs. the time step in the range [$665-699$]. The proposed distance measure $d_q^{\mathbb{S}}$ exhibits a prominent peak between time step $690-692$, which indicates a significant change. (b) Geometry of the plutonium atom at various time steps. The point of scission is between site $690-692$ and can be seen in the geometry.
• Figure 4: Plots of the distance measures for the scission data for the fermium-256 atom. (a) Distance measure between fields at consecutive time steps vs. the time step in the range [$20,39$]. The proposed distance measure $d_q^{\mathbb{S}}$ exhibits a prominent peak at time step $26$, which indicates a significant change. (b) Geometry of the fermium-256 atom at various time steps. The point of scission is at site $26$ and can be seen in the geometry.
• Figure 5: Plots of the distance measures for the orbital density data of Pt-CO bond at different time steps. (a) Distance measure between fields at consecutive time steps vs. the time step in the range [$0,39$]. The plots are for two field values, HOMO and LUMO and the highest peak is obtained at time stamp 21. The proposed distance measure $d_q^{\mathbb{S}}$ exhibits a prominent peak, which indicates a significant change. (b) Pt-CO Bond length vs time. Bond length stabilizes at time step 21. (c) Geometry of the Pt-CO bond creation at various time steps, visualized using the tool Avogadro. Although the bond is visible at time step 13, the bond length is not stable at this site.