Point-based manifold harmonics.

Area of Science:

• Computer Graphics
• Geometric Processing
• Numerical Analysis

Background:

• Spectral analysis on manifold surfaces requires orthogonal bases.
• Existing discrete Laplace-Beltrami Operators (LBOs) for point clouds lack guaranteed symmetrizability, hindering orthogonal basis construction.
• Point-Based Manifold Harmonic Bases (PB-MHB) are crucial for spectral analysis on sampled surfaces.

Purpose of the Study:

• To develop a novel algorithm for constructing orthogonal PB-MHB on point-sampled manifold surfaces.
• To address the limitation of non-symmetrizable discrete LBOs in existing methods.
• To provide a robust framework for spectral geometric analysis and processing tasks.

Main Methods:

• A new point-wise discrete Laplace-Beltrami Operator (LBO) is proposed, guaranteeing symmetrizability.
• The convergence of the new discrete LBO is mathematically proven.
• The eigenproblem of the symmetrizable LBO is solved to define orthogonal bases over point clouds.

Main Results:

• The proposed discrete LBO is proven to be symmetrizable and convergent.
• The new operator demonstrates superior convergence compared to other symmetrizable discrete Laplacians, like the graph Laplacian.
• Orthogonal bases are successfully generated for spectral analysis on point-sampled surfaces.

Conclusions:

• The developed algorithm effectively generates orthogonal PB-MHB by utilizing a novel, symmetrizable discrete LBO.
• The proposed method offers improved convergence and reliability for spectral analysis on point clouds.
• This work provides a foundation for advanced spectral geometric processing techniques on manifold surfaces.