Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

TL;DR

This survey tackles the challenge of comparing scalar fields through topology-based descriptors by organizing descriptors into set-based, graph-based, and complex-based families and by classifying applications into single-field, time-varying, and ensemble contexts. It synthesizes a wide range of comparative measures (distances and kernels) and analyzes them against properties like metricity, stability, discriminativity, and computational cost, linking theory to visualization tasks. The three core contributions are (i) a taxonomy of methods aligned to visualization tasks, (ii) a compilation of desirable properties for comparative measures, and (iii) a critical analysis of strengths, limitations, and open problems with guidance for future work. The work aims to bridge theory and practice, informing the design of robust, scalable topological comparisons that can be embedded in visualization tools and workflows with real-world impact across science and engineering.

Abstract

In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse-Smale complexes play an essential role in capturing the shape of scalar field data. We present a state-of-the-art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time-varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.

Figures (19)

• Figure 1: Morse functions with (a) a 1-dimensional and (b) a 2-dimensional domain, respectively.
• Figure 2: (a) The graph of $f: \mathbb{M}{\hbox{$\mathbb{M}$}} \to \mathbb{R}{\hbox{$\mathbb{R}$}}$, where each point $p_i = (x_i, c_i)$ for $c_i = f(x_i)$; together with (b) the 0-dimensional barcode and (c) 0-dimensional persistence diagram of $f$ based on its sublevel set filtration.
• Figure 3: Rotating a persistence diagram in (a) to create a functional representation -- a persistence landscape in (b).
• Figure 4: (a) The graph of a 1-dimensional Morse function $f$ restricted to an interval, $f: \mathbb{M}{\hbox{$\mathbb{M}$}} \to \mathbb{R}{\hbox{$\mathbb{R}$}}$; (b) the merge tree of $f$ shown abstractly, where branches are colored based on its branch decomposition; (c) the graph of $f$ is colored based on the branch decomposition in (b).
• Figure 5: (a) A height function $f: \mathbb{M}{\hbox{$\mathbb{M}$}} \to \mathbb{R}{\hbox{$\mathbb{R}$}}$ defined on a double torus, (b) its Reeb graph embedded in the domain $\mathbb{M}{\hbox{$\mathbb{M}$}}$, and (c) its Reeb graph shown in an abstract view. If the Reeb graph in (c) is further equipped with a function $l_f$ defined on its vertices, where $l_f$ is the restriction of $f$ to $V$, then we obtain a labeled Reeb graph.

Theorems & Definitions (26)

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