Conforming Morse-Smale Complexes.

Area of Science:

• Topological Data Analysis
• Computational Geometry
• Scientific Visualization

Background:

• Morse-Smale (MS) complexes are valuable for feature-driven data analysis but are limited by geometric embedding quality and reliance solely on scalar fields.
• Existing methods struggle to incorporate application-specific features or prior domain knowledge into MS complex construction.

Purpose of the Study:

• To develop a new combinatorial technique for computing MS complexes that conform to both scalar fields and prior domain segmentation.
• To enable the incorporation of application-dependent features and expert knowledge into topological data representations.
• To introduce edit operations for MS complexes, allowing direct user modification while preserving topological robustness.

Main Methods:

• A novel combinatorial approach to compute MS complexes constrained by a prior domain segmentation.
• Integration of streamline integration for basin/mountain computation to generate segmentation input.
• Development of edit operations for direct manipulation of MS complex features.

Main Results:

• The proposed method successfully constrains MS complex computation, ensuring segmentation boundaries are captured as separatrices.
• Demonstrated utility in applications involving numerical integration for feature extraction and expert-driven data refinement.
• The resulting MS complexes maintain geometric accuracy and topological robustness after user modifications.

Conclusions:

• The new technique significantly enhances the applicability of MS complexes by allowing integration of prior segmentation and expert knowledge.
• This approach provides a versatile tool for accurate feature representation and interactive data exploration.
• It bridges the gap between purely data-driven topology and application-specific feature requirements.