Persistence Cycles for Visual Exploration of Persistent Homology

Area of Science:

• Topological Data Analysis
• Scientific Visualization
• Computational Topology

Background:

• Persistent homology (PH) is crucial in topological data analysis, with scatter plots being the standard visualization.
• Current scatter plot visualizations for PH data are limited and prone to misinterpretation.
• Effective visualization is key to understanding complex topological features in data.

Purpose of the Study:

• To propose an efficient method for computing persistence cycles, a geometric representation of PH features.
• To demonstrate the utility of persistence cycles in analyzing scalar fields.
• To compare the advantages of this new approach against existing topology-based visualization techniques.

Main Methods:

• Development of an efficient algorithm for computing persistence cycles.
• Implementation of the approach using discrete Morse theory.
• Integration of the method as a new module within the Topology Toolkit.

Main Results:

• The proposed method enables efficient computation of persistence cycles.
• Persistence cycles provide a superior geometric representation for analyzing scalar fields.
• The new implementation demonstrates comparable performance to state-of-the-art methods.

Conclusions:

• Persistence cycles offer a more insightful and less ambiguous visualization of persistent homology data.
• The developed approach enhances the analysis of topological information through improved visualization.
• This work provides a valuable new tool for the field of topology-based data analysis.