A Partition Based Method for Spectrum-Preserving Mesh Simplification.

Area of Science:

• Computer Graphics
• Computational Geometry
• Applied Mathematics

Background:

• Existing mesh simplification methods often neglect spectral properties of differential operators, crucial for geometry processing applications.
• Ignoring spectral properties during mesh coarsening can lead to inaccuracies in subsequent computations.
• The spectrum of the Laplace-Beltrami operator is fundamental for many geometry processing tasks.

Purpose of the Study:

• To develop a mesh simplification technique that preserves the spectral properties of the cotangent Laplace-Beltrami operator.
• To address the limitations of current methods that prioritize visual appearance over spectral fidelity.
• To enable accurate geometric computations on simplified 3D triangular meshes.

Main Methods:

• The proposed method employs edge collapses for mesh simplification.
• It focuses on preserving the eigenvalues of the Laplace-Beltrami operator.
• The technique is partition-based, simplifying individual patches of the input mesh.

Main Results:

• The method effectively preserves the spectral properties of the Laplace-Beltrami operator.
• Evaluation using functional maps and quantitative norms confirms the maintenance of eigenvalues and eigenvectors.
• Spectrum preservation is demonstrated to be at least as effective as existing spectral coarsening techniques.

Conclusions:

• The novel partition-based edge collapse method successfully preserves the spectral properties of the Laplace-Beltrami operator.
• This approach ensures the integrity of geometric computations on simplified meshes.
• The method offers a viable alternative to existing spectral coarsening techniques with comparable or superior performance.