Access Type
Open Access Embargo
Date of Award
January 2025
Degree Type
Dissertation
Degree Name
Ph.D.
Department
Computer Science
First Advisor
zichun zhong
Abstract
\centerline{\bf RESTRICTED DELAUNAY-BASED ANISOTROPIC MESHING WITH RIEMANNIAN METRICS}
{\setlength\baselineskip{0.3in} \begin{center} by\\ \medskip {\bf Sikai Zhong}\\ \medskip {\bf August 2025}\\ \end{center} \Vspc \begin{tabular}{ll} {\bf Advisor:} & Dr. Zichun Zhong \\ {\bf Major:} & Computer Science \\ {\bf Degree:} Doctor of philosophy \end{tabular} }
\bigskip \bigskip
Nowadays, 3D scanners, such as LiDAR (Light Detection and Ranging), have become more affordable and accessible. Capturing high-resolution 3D data is easier than ever before, point clouds, thus, have becomes instrumental in emerging technologies such as 3D manufacturing, animations, autonomous driving and virtual reality (VR), etc. Large-scale and high-resolution raw point clouds are required to represent complicated 3D objects with a lot of features and details. Therefore, it is important to resample the original dense point clouds into the relative sparse point sets with high-quality and high-fidelity. Point clouds lack explicit connection information, which is necessary in many tasks, such as finite element method (FEM) and computational fluid dynamics (CFD). 3D reconstruction is thus required to generate the meshes. Resampling and reconstruction together are commonly referred as remeshing. How to robustly and effectively represent 3D objects with high-fidelity is a challenging research problem. There are mainly three categories regarding remeshing techniques, Centroidal Voronoi diagram (CVT) based methods, particle based methods and geometric embedding based methods. Among these techniques, particle based methods are GPU friendly and easy to implement since it is essentially structurally-aware and physically-aware. Capturing the complicated boundary / surface of 3D objects typically needs intensive linear simplices even with an appropriate anisotropy (defined as Riemannian tensor field). High order surface representations, such as B-spline, implicit functions, and subdivision surfaces, can further reduce the required number of elements. All the existing curved meshing techniques do not essentially consider the Riemannian metrics in arbitrary 3D shape volumes. In short, they are not specifically designed for Riemannian mesh elements followed by the definition of the Riemannian metric field, so that theoretically they could not guarantee a high accuracy and fidelity on both geometric boundary and physical metric simultaneously, which also leads to ``over-smooth'' issue. Riemannian Delaunay Tetrahedralization is a better solution when fidelity is the main concern.
In this thesis, we propose an efficient method that constructs high-quality isotropic / anisotropic hexagonal / quadrilateral resampling results from point clouds. It should be noted that our unified framework can easily be modified and applied to surface / volume meshes. We also propose a practical algorithm to compute curved Riemannian restricted Delaunay tetrahedralization to keep both the geometry and the metric.
The main contributions presented in this thesis are listed below.
\begin{itemize} \item \textit{Unified Straight Restricted Delaunay Triangulation:} We introduce a new unified particle-based formulation for resampling with specific patterns from original point clouds. Given the input point clouds, the proposed $L_{p}$-Gaussian kernel function is defined to simulate the inter-particle energy and force to form the isotropic / adaptive / anisotropic hexagonal and quadrilateral sampling patterns. Then, the particle-based optimization can be easily formulated and computed in parallel scheme with the high-efficiency and the fast convergence, without any control of particle population. Finally, based on the optimized particle distribution, the high-quality surface meshes are reconstructed by computing the restricted Voronoi diagram and its dual mesh with the parallel implementation. The experimental results are demonstrated by using extensive examples and evaluation criteria as well as compared with the state-of-the-art in the point cloud resampling and reconstruction. \item \textit{ Curved Riemannian Restricted Delaunay Tetrahedralization:} We present a new method to compute the high-quality curved anisotropic restricted Delaunay tetrahedralizations (CARDT) for given volumetric shapes endowed with arbitrary Riemannian metric tensor fields. The main idea is to firstly develop an effective meshing technique to generate inversion-avoiding straight-edge coarse anisotropic tetrahedral mesh by computing the isotropic 3-simplices (tetrahedrons) in a high-dimensional (high-d) Euclidean embedding space, where it can retain both the geometric and metric properties from the input Riemannian manifold. After that, we design a simplex-based pointwise optimization strategy to compute the straight simplices into curved simplices in the high-d space to further preserve the Riemannian metrics (geodesics in volume) as well as avoid inversions. We demonstrate that our method is practically useful for generating complicated 3D curved anisotropic volume meshes to well preserve the Riemannian metric fields by evaluating a variety of challenging 3D Riemannian metric tensor fields and real applications in numerical cosmology and geology. Experimental results are compared with the state-of-the-art in anisotropic and curved tetrahedral meshing. \end{itemize}
Recommended Citation
Zhong, Sikai, "Restricted Delaunay-Based Anisotropic Meshing With Riemannian Metrics" (2025). Wayne State University Dissertations. 4295.
https://digitalcommons.wayne.edu/oa_dissertations/4295