Laplace-Beltrami based multi-resolution shape reconstruction on subdivision surfaces

Area of Science:

• Mathematics
• Computer Science
• Engineering

Background:

• Eigenfunctions of the Laplace-Beltrami operator, known as manifold harmonics (MHs), are crucial for data interpolation and geometry representation on manifolds.
• MHs offer a natural basis for multi-resolution analysis and editing of complex surfaces and functions.

Purpose of the Study:

• To develop a framework for shape reconstruction leveraging the benefits of manifold harmonics.
• To create a compressible, multi-resolution scheme for shape reconstruction using MHs.

Main Methods:

• The developed method utilizes subdivision basis sets for boundary element isogeometric analysis.
• Surface finite elements are employed for constructing manifold harmonics.
• A volumetric source reconstruction method is integrated for initial starting point determination.

Main Results:

• The proposed technique demonstrates efficacy in reconstructing shapes from noisy data.
• Significant reduction in degrees of freedom for complex objects was achieved.
• High accuracy and effective multi-resolution capabilities were highlighted in presented examples.

Conclusions:

• The developed MHs-based framework provides an efficient and accurate approach to multi-resolution shape reconstruction.
• The method is robust in the presence of noise and offers significant advantages for complex geometric data.