Manifold-Constrained Geometric Optimization via Local Parameterizations.

Area of Science:

• Computer Graphics
• Computational Geometry
• Optimization

Background:

• Geometric optimization problems often involve complex, nonlinear, and non-convex manifold constraints.
• Existing methods struggle with constraint adherence and achieving high-quality optimization outcomes.

Purpose of the Study:

• To present a novel divide-and-conquer methodology for manifold-constrained geometric optimization problems.
• To overcome the limitations of existing methods in handling nonlinear and non-convex constraints.

Main Methods:

• Decomposing meshes into developable patches with disc topology.
• Utilizing local parameterizations to transform nonlinear constraints into linear ones.
• Flattening patches, optimizing vertices with linear constraints, and re-projecting onto the manifold surface.

Main Results:

• The proposed method demonstrates applicability and robustness across diverse geometric optimization tasks.
• Experimental results indicate superior performance compared to existing approaches.
• Achieved low isometric distortion during patch flattening.

Conclusions:

• The novel divide-and-conquer methodology offers a significant improvement for manifold-constrained geometric optimization.
• The approach effectively decouples optimization from hard constraints, enhancing quality and reliability.
• This method provides a robust solution for complex geometric optimization challenges.