Time-varying Extremum Graphs

TL;DR

This paper introduces the time-varying extremum graph (TVEG), a dynamic extension of the extremum graph built atop the Morse-Smale framework to analyze time-varying scalar fields. TVEG combines per-time-step extremum graphs $\mathcal{G}^t$ with temporal arcs $A^t$ that connect maxima across consecutive time steps, enabling tracking of feature evolution and topological events such as generation, deletion, merge, and split. Temporal correspondence is formulated as a constrained optimization balancing topological persistence $\mathcal{P}$, value difference $\mathcal{J}$, spatial distance $\mathcal{D}$, and neighborhood similarity $\mathcal{N}$, subject to avoidance of $z$-shaped configurations; the computation proceeds in two steps: (i) construct and simplify each $\mathcal{G}^t$ from the MS complex (via existing methods or MS3D/TTK) with a persistence threshold, and (ii) compute $A^t$ with TemporalArcs, yielding a globally coherent $\mathcal{G}^*$. The authors demonstrate TVEG on moving Gaussians, viscous fingers, and a 3D von Kármán vortex street, showing its ability to reveal complex topological dynamics and offering visualization-friendly space-time representations and query-based exploration. Compared to Lifted Wasserstein Matcher, TVEG provides richer tracking by incorporating local neighborhood and function-value criteria, leading to more interpretable tracks and robust event identification, with scalable performance across datasets. The work enables new ways to visualize, query, and analyze time-varying topological features in scientific data, with potential extensions to multi-way merges/splits and additional node/arc attributes.

Abstract

We introduce time-varying extremum graph (TVEG), a topological structure to support visualization and analysis of a time-varying scalar field. The extremum graph is a substructure of the Morse-Smale complex. It captures the adjacency relationship between cells in the Morse decomposition of a scalar field. We define the TVEG as a time-varying extension of the extremum graph and demonstrate how it captures salient feature tracks within a dynamic scalar field. We formulate the construction of the TVEG as an optimization problem and describe an algorithm for computing the graph. We also demonstrate the capabilities of \TVEG towards identification and exploration of topological events such as deletion, generation, split, and merge within a dynamic scalar field via comprehensive case studies including a viscous fingers and a 3D von Kármán vortex street dataset.

Figures (11)

• Figure 3: tveg nodes and arcs. A subset of a tveg between two time steps $t$ and $t+1$. Sets $\{m_1^t,\ldots,m_5^t\}$ and $\{s_1^t,s_2^t\}$ consists of maxima and $(n-1)$-saddles from the extremum graph at time step $t$. Sets $\{m_1^{t+1},m_2^{t+1},m_3^{t+1}\}$ and $\{s_1^{t+1},s_2^{t+1}\}$ are maxima and saddles of the extremum graph at time step $t+1$. Arcs of the extremum graph are shown in blue, temporal arcs as green dashed edges, and connections to nodes outside the figure are shown as black dashed edges. Maxima $m_2^t$ and $m_3^t$ do not have a suitable temporal correspondence and hence die at time step $t$ whereas $m_2^{t+1}$ is born. Maxima $m_4^t$ and $m_5^t$ merge into $m_3^{t+1}$.
• Figure 4: Dynamics in a synthetic sum of Gaussians dataset Gauss8. The data is visualized by displaying the intersection of descending 3-manifolds of maxima with volume enclosed by the isosurface at scalar value 21. The resulting blobs merge over time to form larger components and subsequently split into multiple components. The merge and split behavior is also observable from the extremum graphs.
• Figure 5: Temporal tracks from the tveg of Gauss8. The 3D domain is scaled along the z-axis and stacked to visualize the tracks over time. (top-right) A view of the stacked domains from the top, along the y-axis, shows the tracks over all time steps. A subset of tveg tracks (Y) that exhibits symmetry along time is highlighted in blue. (middle) Track A is a subset of a longer track, consisting of time steps 9-11 and includes merge and split events. Inset depicts the blobs in the corresponding time steps 9, 10, and 11. Track B (red), a subset of Y, is selected to showcase the structural similarity between extremum graphs (inset) sampled at time steps 15, 16, and 17.
• Figure 6: Tracks obtained by (a) tveg and (b-d) lwm for the Gauss8 dataset. All maxima in the first time step are shown as red spheres. The postprocessing step is applied with two different distance thresholds, $0.08 \cdot diag$ and $0.15 \cdot diag$. Here, $diag$ is the length of the long diagonal of the data domain. Individual lwm track components are shown in distinct color and the arcs inserted by the postprocessing step are shown in black.
• Figure 7: Visualizing the viscous fingers dataset using the extremum graph. \ref{['fig_ext_ms_data']} Fingers are visualized as intersection of descending 3-manifolds of salt concentration maxima and volume enclosed by isosurface at scalar value 35. Integral lines between each max-saddle arc in the extremum graph represent the skeletal structure of the fingers. \ref{['fig_ext_graph_data']} All max-saddle arcs in the extremum graph are displayed using straight edges. \ref{['fig_ext_3dgraph']} The extremum graph with maxima (red) and saddles (green).