Research On Gradient-Domain Based Volumetric Parameterization

With the rapid development of computer graphics and3D scanning oriented hardware technology,3D geometrical models are widely used in various applications. While triangular surface meshes become the main form of3D geometrical data, tetrahedral meshes are getting more and more attention and being applied in diverse fields, such as physical simulation and finite element calculation, which lead to various tetrahedral geometry processing algorithms. Among them, tetrahedral meshes parameterization is one of the fundamental problems in computer graphics. It can be viewed as a one-to-one mapping from the volumetric mesh to a regular parameter domain, which is essential for many applications such as shape matching and analysis, tetrahedral remeshing and volume texture synthesis. Ideally, the mapping should be locally rigid. However for general volumetric meshes, there doesn’t exit such mappings, even conformal mappings. Thus the distortions will unavoidably be introduced during the parameterization. Therefore how to find low distortion while guaranteeing the mapping one-to-one has become a hot and challenging research in numerical geometry and computer graphics fields.This thesis focuses on the volumetric parameterization of tetrahedral meshes based on the information of gradient domain, and proposes a way to prevent the inverse of tetrahedron to ensure the validity of our volumetric parameterization algorithm.The specific research in this paper has the following details:1. We introduce a gradient-domain based algorithm framework for volumetric parameterization. We use the gradient information to describe the distortion metric and optimize the distortion energy to reconstruct the new mesh under the given boundary conditions. In the framework, we implement three types of volumetric parameterization for the different application requirements:Dirichlet energy based, the as-rigid-as-possible energy based and the as-conformal-as-possible energy based.2. We propose a new way to prevent the inverse of tetrahedrons to ensure the locally injection with the idea of interior penalty function method. The method can handle large bodies of meshes with extreme positional constraints.