Scalable Mesh Refinement for Canonical Polygonal Schemas of Extremely High Genus Shapes

Area of Science:

• Computational geometry
• Computer graphics
• Topology

Background:

• Canonical polygonal schemas are essential for parameterizing shapes with identical topology in graphics and engineering.
• Computing shortest loop systems for these schemas is NP-hard, necessitating alternative approaches.
• Existing greedy algorithms and mesh refinement methods struggle with high-genus shapes, causing memory issues.

Purpose of the Study:

• To investigate local refinement operators for detaching cycles in loop systems.
• To develop scalable and efficient mesh refinement strategies for high-genus shapes.
• To enable the computation of canonical polygonal schemas for complex surfaces.

Main Methods:

• Analysis of various local refinement operators for cycle detachment.
• Development of two novel refinement approaches: one prioritizing mesh reduction, the other balancing complexity and geometric accuracy.
• Experimental validation of the proposed methods on high-genus shapes.

Main Results:

• Identified significant differences in mesh complexity and surface preservation among refinement operators.
• Proposed two novel refinement strategies that overcome scalability limitations of previous methods.
• Demonstrated successful computation of canonical polygonal schemas for extremely high-genus shapes.

Conclusions:

• The proposed refinement strategies are effective and implementable for generating canonical polygonal schemas.
• These methods overcome memory and scalability issues associated with high-genus shapes.
• The new approaches facilitate parameterization of complex surfaces previously intractable.