Laplace-Beltrami based Multi-Resolution Shape Reconstruction on Subdivision Surfaces

TL;DR

This paper introduces a multi-resolution shape reconstruction framework using Laplace-Beltrami eigenfunctions and subdivision basis sets, enabling efficient and accurate surface modeling even with noisy data.

Contribution

It develops a novel multi-resolution shape reconstruction method leveraging manifold harmonics and subdivision surfaces, improving efficiency and robustness over existing techniques.

Findings

Effective in noisy data scenarios

Reduces degrees of freedom significantly

Achieves accurate multi-resolution surface reconstructions

Abstract

The eigenfunctions of the Laplace-Beltrami operator have widespread applications in a number of disciplines of engineering, computer vision/graphics, machine learning, etc. These eigenfunctions or manifold harmonics, provide the means to smoothly interpolate data on a manifold. They are highly effective, specifically as it relates to geometry representation and editing; manifold harmonics form a natural basis for multi-resolution representation (and editing) of complex surfaces and functioned defined therein. In this paper, we seek to develop the framework to exploit the benefits of manifold harmonics for shape reconstruction. To this end, we develop a highly compressible, multi-resolution shape reconstruction scheme using manifold harmonics. The method relies on subdivision basis sets to construct both boundary element isogeometric methods for analysis and surface finite elements to construct manifold harmonics. We pair this technique with the volumetric source reconstruction method to determine an initial starting point. Examples presented highlight efficacy of the approach in the presence of noisy data, including significant reduction in the number of degrees of freedom for complex objects, the accuracy of reconstruction, and multi-resolution capabilities.