Computing Morse-Smale Complexes with Accurate Geometry

Area of Science:

• Computational Topology
• Scientific Visualization
• Data Analysis

Background:

• Topological techniques are vital for scientific data analysis and visualization.
• Existing methods for computing structures like Morse-Smale (MS) complexes often yield incorrect connectivity and poor geometry.
• These inaccuracies can invalidate subsequent analyses, even with increased mesh resolution.

Purpose of the Study:

• To introduce novel algorithms for robust and accurate computation of MS complexes.
• To address the limitations of current topological analysis methods regarding connectivity and geometry.
• To provide algorithms that improve with increasing mesh resolution.

Main Methods:

• A randomized algorithm to compute the discrete gradient of a scalar field, designed to converge under refinement.
• A deterministic variant utilizing two ordered function traversals to integrate probabilities and extract accurate geometry and connectivity.
• Extensive empirical studies on synthetic and real-world data to validate algorithm performance.

Main Results:

• The randomized algorithm shows convergence, with average correctness increasing and standard deviation decreasing as mesh resolution rises.
• The deterministic algorithm successfully computes accurate geometry and correct MS complex connectivity.
• Demonstrated advantages over several popular existing approaches in empirical evaluations.

Conclusions:

• The developed algorithms offer significant improvements for computing Morse-Smale complexes.
• These methods provide topologically consistent results with accurate geometry and connectivity.
• The new approaches enhance the reliability of topological data analysis in scientific applications.