Periodic Global Parameterization

We present a new globally smooth parameterization method for triangulated surfaces of arbitrary topology. Given two orthogonal piecewise linear vector fields defined over the input mesh (typically the estimated principal curvature directions), our method computes two piecewise linear periodic functions, aligned with the input vector fields, by minimizing an objective function. The bivariate function they define is a smooth parameterization almost everywhere on the surface, except in the vicinity of singular vertices, edges and triangles, where the derivatives of the parameterization vanish. We extract a quadrilateral chart layout from the parameterization function and propose an automatic procedure to detect the singularities, and fix them by splitting and re-parameterizing the containing charts. Our method can construct both quasi-conformal (angle preserving) and quasi-isometric (angle and area preserving) parameterizations. The more restrictive class of quasi-isometric parameterizations is constructed at the expense of introducing more singularities. The constructed parameterizations can be used for a variety of geometry processing applications. Since we can align the parameterization with the principal curvature directions, our result is particularly suitable for surface fitting and remeshing

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

Connectivity Control for Quad-Dominant Meshes

abstract: Quad-dominant (QD) meshes, i.e., three-dimensional, 2-manifold polygonal meshes comprising mostly four-sided faces (i.e., quads), are a popular choice for many applications such as polygonal shape modeling, computer animation, base meshes for spline and subdivision surface, simulation, and architectural design. This thesis investigates the topic of connectivity control, i.e., exploring different choices of mesh connectivity to represent the same 3D shape or surface. One key concept of QD mesh connectivity is the distinction between regular and irregular elements: a vertex with valence 4 is regular; otherwise, it is irregular. In a similar sense, a face with four sides is regular; otherwise, it is irregular. For QD meshes, the placement of irregular elements is especially important since it largely determines the achievable geometric quality of the final mesh. Traditionally, the research on QD meshes focuses on the automatic generation of pure quadrilateral or QD meshes from a given surface. Explicit control of the placement of irregular elements can only be achieved indirectly. To fill this gap, in this thesis, we make the following contributions. First, we formulate the theoretical background about the fundamental combinatorial properties of irregular elements in QD meshes. Second, we develop algorithms for the explicit control of irregular elements and the exhaustive enumeration of QD mesh connectivities. Finally, we demonstrate the importance of connectivity control for QD meshes in a wide range of applications.Dissertation/ThesisDoctoral Dissertation Computer Science 201

Quad Meshing

Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semi-regular ones, have advantages for many applications, and significant progress was made in quadrilateral mesh generation and processing during the last several years. In this State of the Art Report, we discuss the advantages and problems of techniques operating on quadrilateral meshes, including surface analysis and mesh quality, simplification, adaptive refinement, alignment with features, parametrization, and remeshing