Large-scale Geometric Data Decomposition, Processing and Structured Mesh Generation

Mesh generation is a fundamental and critical problem in geometric data modeling and processing. In most scientific and engineering tasks that involve numerical computations and simulations on 2D/3D regions or on curved geometric objects, discretizing or approximating the geometric data using a polygonal or polyhedral meshes is always the first step of the procedure. The quality of this tessellation often dictates the subsequent computation accuracy, efficiency, and numerical stability. When compared with unstructured meshes, the structured meshes are favored in many scientific/engineering tasks due to their good properties. However, generating high-quality structured mesh remains challenging, especially for complex or large-scale geometric data. In industrial Computer-aided Design/Engineering (CAD/CAE) pipelines, the geometry processing to create a desirable structural mesh of the complex model is the most costly step. This step is semi-manual, and often takes up to several weeks to finish. Several technical challenges remains unsolved in existing structured mesh generation techniques. This dissertation studies the effective generation of structural mesh on large and complex geometric data. We study a general geometric computation paradigm to solve this problem via model partitioning and divide-and-conquer. To apply effective divide-and-conquer, we study two key technical components: the shape decomposition in the divide stage, and the structured meshing in the conquer stage. We test our algorithm on vairous data set, the results demonstrate the efficiency and effectiveness of our framework. The comparisons also show our algorithm outperforms existing partitioning methods in final meshing quality. We also show our pipeline scales up efficiently on HPC environment

ℓ_1-Based Construction of Polycube Maps from Complex Shapes

Polycube maps of triangle meshes have proved useful in a wide range of applications, including texture mapping and hexahedral mesh generation. However, constructing either fully automatically or with limited user control a low-distortion polycube from a detailed surface remains challenging in practice. We propose a variational method for deforming an input triangle mesh into a polycube shape through minimization of the ℓ_1-norm of the mesh normals, regularized via an as-rigid-as-possible volumetric distortion energy. Unlike previous work, our approach makes no assumption on the orientation, or on the presence of features in the input model. User-guided control over the resulting polycube map is also offered to increase design flexibility. We demonstrate the robustness, efficiency, and controllability of our method on a variety of examples, and explore applications in hexahedral remeshing and quadrangulation

Heterogeneous volumetric data mapping and its medical applications

With the advance of data acquisition techniques, massive solid geometries are being collected routinely in scientific tasks, these complex and unstructured data need to be effectively correlated for various processing and analysis. Volumetric mapping solves bijective low-distortion correspondence between/among 3D geometric data, and can serve as an important preprocessing step in many tasks in compute-aided design and analysis, industrial manufacturing, medical image analysis, to name a few. This dissertation studied two important volumetric mapping problems: the mapping of heterogeneous volumes (with nonuniform inner structures/layers) and the mapping of sequential dynamic volumes. To effectively handle heterogeneous volumes, first, we studied the feature-aligned harmonic volumetric mapping. Compared to previous harmonic mapping, it supports the point, curve, and iso-surface alignment, which are important low-dimensional structures in heterogeneous volumetric data. Second, we proposed a biharmonic model for volumetric mapping. Unlike the conventional harmonic volumetric mapping that only supports positional continuity on the boundary, this new model allows us to have higher order continuity $C^1$ along the boundary surface. This suggests a potential model to solve the volumetric mapping of complex and big geometries through divide-and-conquer. We also studied the medical applications of our volumetric mapping in lung tumor respiratory motion modeling. We were building an effective digital platform for lung tumor radiotherapy based on effective volumetric CT/MRI image matching and analysis. We developed and integrated in this platform a set of geometric/image processing techniques including advanced image segmentation, finite element meshing, volumetric registration and interpolation. The lung organ/tumor and surrounding tissues are treated as a heterogeneous region and a dynamic 4D registration framework is developed for lung tumor motion modeling and tracking. Compared to the previous 3D pairwise registration, our new 4D parameterization model leads to a significantly improved registration accuracy. The constructed deforming model can hence approximate the deformation of the tissues and tumor

Geometric modeling and optimization over regular domains for graphics and visual computing

The effective construction of parametric representation of complicated geometric objects can facilitate many design, analysis, and simulation tasks in Computer-Aided Design (CAD), Computer-Aided Manufacturing (CAM), and Computer-Aided Engineering (CAE). Given a 3D shape, the procedure of finding such a parametric representation upon a canonical domain is called geometric parameterization. Regular geometric regions, such as polycubes and spheres, are desirable domains for parameterization. Parametric representations defined upon regular geometric domains have many desirable mathematical properties and can facilitate or simplify various surface/solid modeling and processing computation. This dissertation studies the construction of parameterization on regular geometric domains and explores their applications in shape modeling and computer-aided design. Specifically, we studies (1) the surface parameterization on the spherical domain for closed genus-zero surfaces; (2) the surface parameterization on the polycube domain for general closed surfaces; and (3) the volumetric parameterization for 3D-manifolds embedded in 3D Euclidean space. We propose novel computational models to solve these geometric problems. Our computational models reduce to nonlinear optimizations with various geometric constraints. Hence, we also need to explore effective optimization algorithms. The main contributions of this dissertation are three-folded. (1) We developed an effective progressive spherical parameterization algorithm, with an efficient nonlinear optimization scheme subject to the spherical constraint. Compared with the state-of-the-art spherical mapping algorithms, our algorithm demonstrates the advantages of great efficiency, lower distortion, and guaranteed bijectiveness, and we show its applications in spherical harmonic decomposition and shape analysis. (2) We propose a first topology-preserving polycube domain optimization algorithm that simultaneously optimizes polycube domain together with the parameterization to balance the mapping distortion and domain simplicity. We develop effective nonlinear geometric optimization algorithms dealing with variables with and without derivatives. This polycube parameterization algorithm can benefit the regular quadrilateral mesh generation and cross-surface parameterization. (3) We develop a novel quaternion-based optimization framework for 3D frame field construction and volumetric parameterization computation. We demonstrate our constructed 3D frame field has better smoothness, compared with state-of-the-art algorithms, and is effective in guiding low-distortion volumetric parameterization and high-quality hexahedral mesh generation

Surface parameterization over regular domains

Surface parameterization has been widely studied and it has been playing a critical role in many geometric processing tasks in graphics, computer-aided design, visualization, vision, physical simulation and etc. Regular domains, such as polycubes, are favored due to their structural regularity and geometric simplicity. This thesis focuses on studying the surface parameterization over regular domains, i.e. polycubes, and develops effective computation algorithms. Firstly, the motivation for surface parameterization and polycube mapping is introduced. Secondly, we briefly review existing surface parameterization techniques, especially for extensively studied parameterization algorithms for topological disk surfaces and parameterizations over regular domains for closed surfaces. Then we propose a polycube parameterization algorithm for closed surfaces with general topology. We develop an efficient optimization framework to minimize the angle and area distortion of the mapping. Its applications on surface meshing, inter-shape morphing and volumetric polycube mapping are also discussed