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

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 (MHs) provide the means to smoothly interpolate data on a manifold and are highly effective, specifically as it relates to geometry representation and editing; MHs form a natural basis for multi-resolution representation (and editing) of complex surfaces and functions defined therein. In this paper, we seek to develop the framework to exploit the benefits of MHs for shape reconstruction. To this end, a highly compressible, multi-resolution shape reconstruction scheme using MHs is developed. The method relies on subdivision basis sets to construct boundary element isogeometric methods for analysis and surface finite elements to construct MHs. This technique is paired with the volumetric source reconstruction method to determine an initial starting point. The examples presented highlight efficacy of the approach in the presence of noisy data, including a significant reduction in the number of degrees of freedom for complex objects, accuracy of reconstruction, and multi-resolution capabilities.