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

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.

References in corpus (3)

• Sliced Wasserstein Kernel for Persistence Diagrams
• Edit Distance between Merge Trees
• Kernel method for persistence diagrams via kernel embedding and weight factor

Cited by in corpus (14)

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• A Task-Parallel Approach for Localized Topological Data Structures
• Fast Comparative Analysis of Merge Trees Using Locality Sensitive Hashing
• Jacobi Set Driven Search for Flexible Fiber Surface Extraction
• A General Framework for Augmenting Lossy Compressors with Topological Guarantees
• Taming Horizontal Instability in Merge Trees: On the Computation of a Comprehensive Deformation-based Edit Distance
• Flexible and Probabilistic Topology Tracking with Partial Optimal Transport
• Merge Tree Geodesics and Barycenters with Path Mappings
• Time-varying Extremum Graphs
• Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar Fields
• Bounding the Interleaving Distance for Mapper Graphs with a Loss Function
• A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering