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

arXiv: https://arxiv.org/abs/2106.00157 · HTML: https://mlsanity.com/paper/2106.00157/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2021.

## Built on by (most cited first, 14 of 14)

- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows -- An Experimental Study](https://mlsanity.com/paper/2207.14080/) (2022)
- [Principal Geodesic Analysis of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2207.10960/) (2022)
- [Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2307.02509/) (2023)
- [TopoPilot: Reliable Conversational Workflow Automation for Topological Data Analysis and Visualization](https://mlsanity.com/paper/2603.25063/) (2026)
- [Time-varying Extremum Graphs](https://mlsanity.com/paper/2406.17652/) (2024)
- [Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields](https://mlsanity.com/paper/2407.03348/) (2024)
- [Multi-field Visualisation via Trait-induced Merge Trees](https://mlsanity.com/paper/2308.09015/) (2023)
- [Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs](https://mlsanity.com/paper/2508.07637/) (2025)
- [Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport](https://mlsanity.com/paper/2603.15683/) (2026)
- [Dual-Scale Temporal Fusion Reveals Structured Predictability in Subseasonal-to-Seasonal Temperature Prediction](https://mlsanity.com/paper/2605.06911/) (2026)
- [MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport](https://mlsanity.com/paper/2606.08258/) (2026)
- [DebrisTracer: Reliable Tracking in Hypervelocity Impact Fast Imaging](https://mlsanity.com/paper/2607.15986/) (2026)
- [Direct Topology Tracking in Continuous Implicit Models](https://mlsanity.com/paper/2609.12157/) (2026)

## Abstract (arXiv)

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.