Approximation of Surfaces by Normal Meshes

Citation

Friedel, Ilja Heinrich (2005) Approximation of Surfaces by Normal Meshes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/FVSZ-GF75. https://resolver.caltech.edu/CaltechETD:etd-05242005-164959

Abstract

This thesis introduces a novel geometry processing pipeline based on unconstrained spherical parameterization and normal remeshing. We claim three main contributions: First we show how to increase the stability of Normal Mesh construction, while speeding it up by decomposing the process into two stages: parameterization and remeshing. We show that the remeshing step can be seen as resampling under a small perturbation of the given parameterization. Based on this observation we describe a novel algorithm for efficient and stable (interpolatory) normal mesh construction via parameterization perturbation. Our second contribution is the introduction of Variational Normal Meshes. We describe a novel algorithm for encoding these meshes, and use our implementation to argue that variational normal meshes have a higher approximation quality than interpolating normal meshes, as expected. In particular we demonstrate that interpolating normal meshes have about 60 percent higher Hausdorff approximation error for the same number of vertices than our novel variational normal meshes. We also show that variational normal meshes have less aliasing artifacts than interpolatory normal meshes. The third contribution is on creating parameterizations for unstructured genus zero meshes. Previous approaches could only avoid collapses by introducing artificial constraints or continuous reprojections, which are avoided by our method. The key idea is to define upper bound energies that are still good approximations. We achieve this by dividing classical planar triangle energies by the minimum distance to the sphere center. We prove that these simple modifaction provides the desired upper bounds and are good approximations in the finite element sense. We have implemented all algorithms and provide example results and statistical data supporting our theoretical observations.

Thesis Files