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

A Constrained Resampling Strategy for Mesh Improvement

In many geometry processing applications, it is required to improve an initial mesh in terms of multiple quality objectives. Despite the availability of several mesh generation algorithms with provable guarantees, such generated meshes may only satisfy a subset of the objectives. The conflicting nature of such objectives makes it challenging to establish similar guarantees for each combination, e.g., angle bounds and vertex count. In this paper, we describe a versatile strategy for mesh improvement by interpreting quality objectives as spatial constraints on resampling and develop a toolbox of local operators to improve the mesh while preserving desirable properties. Our strategy judiciously combines smoothing and transformation techniques allowing increased flexibility to practically achieve multiple objectives simultaneously.  We apply our strategy to both planar and surface meshes demonstrating how to simplify Delaunay meshes while preserving element quality, eliminate all obtuse angles in a complex mesh, and maximize the shortest edge length in a Voronoi tessellation far better than the state-of-the-art

Error-Bounded and Feature Preserving Surface Remeshing with Minimal Angle Improvement

The typical goal of surface remeshing consists in finding a mesh that is (1) geometrically faithful to the original geometry, (2) as coarse as possible to obtain a low-complexity representation and (3) free of bad elements that would hamper the desired application. In this paper, we design an algorithm to address all three optimization goals simultaneously. The user specifies desired bounds on approximation error {\delta}, minimal interior angle {\theta} and maximum mesh complexity N (number of vertices). Since such a desired mesh might not even exist, our optimization framework treats only the approximation error bound {\delta} as a hard constraint and the other two criteria as optimization goals. More specifically, we iteratively perform carefully prioritized local operators, whenever they do not violate the approximation error bound and improve the mesh otherwise. In this way our optimization framework greedily searches for the coarsest mesh with minimal interior angle above {\theta} and approximation error bounded by {\delta}. Fast runtime is enabled by a local approximation error estimation, while implicit feature preservation is obtained by specifically designed vertex relocation operators. Experiments show that our approach delivers high-quality meshes with implicitly preserved features and better balances between geometric fidelity, mesh complexity and element quality than the state-of-the-art.Comment: 14 pages, 20 figures. Submitted to IEEE Transactions on Visualization and Computer Graphic

Efficient L 0 resampling of point sets

Abstract(#br)The point data captured by laser scanners or consumer depth cameras are often contaminated with severe noises and outliers. In this paper, we propose a resampling method in an L 0 minimization framework to process such low quality data. Our framework can produce a set of clean, uniformly distributed, geometry-maintaining and feature-preserving oriented points. The L 0 norm improves the robustness to noises (outliers) and the ability to keep sharp features, but introduces a significant efficiency degradation. To further improve the efficiency of our L 0 point set resampling, we propose two accelerating algorithms including optimization-based local half-sampling and interleaved regularization. As demonstrated by the experimental results, the accelerated method is about an order of magnitude faster than the original, while achieves state-of-the-art point set consolidation performance

Scene Reconstruction from Multi-Scale Input Data

Geometry acquisition of real-world objects by means of 3D scanning or stereo reconstruction constitutes a very important and challenging problem in computer vision. 3D scanners and stereo algorithms usually provide geometry from one viewpoint only, and several of the these scans need to be merged into one consistent representation. Scanner data generally has lower noise levels than stereo methods and the scanning scenario is more controlled. In image-based stereo approaches, the aim is to reconstruct the 3D surface of an object solely from multiple photos of the object. In many cases, the stereo geometry is contaminated with noise and outliers, and exhibits large variations in scale. Approaches that fuse such data into one consistent surface must be resilient to such imperfections. In this thesis, we take a closer look at geometry reconstruction using both scanner data and the more challenging image-based scene reconstruction approaches. In particular, this work focuses on the uncontrolled setting where the input images are not constrained, may be taken with different camera models, under different lighting and weather conditions, and from vastly different points of view. A typical dataset contains many views that observe the scene from an overview perspective, and relatively few views capture small details of the geometry. What results from these datasets are surface samples of the scene with vastly different resolution. As we will show in this thesis, the multi-resolution, or, "multi-scale" nature of the input is a relevant aspect for surface reconstruction, which has rarely been considered in literature yet. Integrating scale as additional information in the reconstruction process can make a substantial difference in surface quality. We develop and study two different approaches for surface reconstruction that are able to cope with the challenges resulting from uncontrolled images. The first approach implements surface reconstruction by fusion of depth maps using a multi-scale hierarchical signed distance function. The hierarchical representation allows fusion of multi-resolution depth maps without mixing geometric information at incompatible scales, which preserves detail in high-resolution regions. An incomplete octree is constructed by incrementally adding triangulated depth maps to the hierarchy, which leads to scattered samples of the multi-resolution signed distance function. A continuous representation of the scattered data is defined by constructing a tetrahedral complex, and a final, highly-adaptive surface is extracted by applying the Marching Tetrahedra algorithm. A second, point-based approach is based on a more abstract, multi-scale implicit function defined as a sum of basis functions. Each input sample contributes a single basis function which is parameterized solely by the sample's attributes, effectively yielding a parameter-free method. Because the scale of each sample controls the size of the basis function, the method automatically adapts to data redundancy for noise reduction and is highly resilient to the quality-degrading effects of low-resolution samples, thus favoring high-resolution surfaces. Furthermore, we present a robust, image-based reconstruction system for surface modeling: MVE, the Multi-View Environment. The implementation provides all steps involved in the pipeline: Calibration and registration of the input images, dense geometry reconstruction by means of stereo, a surface reconstruction step and post-processing, such as remeshing and texturing. In contrast to other software solutions for image-based reconstruction, MVE handles large, uncontrolled, multi-scale datasets as well as input from more controlled capture scenarios. The reason lies in the particular choice of the multi-view stereo and surface reconstruction algorithms. The resulting surfaces are represented using a triangular mesh, which is a piecewise linear approximation to the real surface. The individual triangles are often so small that they barely contribute any geometric information and can be ill-shaped, which can cause numerical problems. A surface remeshing approach is introduced which changes the surface discretization such that more favorable triangles are created. It distributes the vertices of the mesh according to a density function, which is derived from the curvature of the geometry. Such a mesh is better suited for further processing and has reduced storage requirements. We thoroughly compare the developed methods against the state-of-the art and also perform a qualitative evaluation of the two surface reconstruction methods on a wide range of datasets with different properties. The usefulness of the remeshing approach is demonstrated on both scanner and multi-view stereo data

Diskrete Spin-Geometrie für Flächen

This thesis proposes a discrete framework for spin geometry of surfaces. Specifically, we discretize the basic notions in spin geometry, such as the spin structure, spin connection and Dirac operator. In this framework, two types of Dirac operators are closely related as in smooth case. Moreover, they both induce the discrete conformal immersion with prescribed mean curvature half-density.In dieser Arbeit wird ein diskreter Zugang zur Spin-Geometrie vorgestellt. Insbesondere diskretisieren wir die grundlegende Begriffe, wie zum Beispiel die Spin-Struktur, den Spin-Zusammenhang und den Dirac Operator. In diesem Rahmen sind zwei Varianten für den Dirac Operator eng verwandt wie in der glatten Theorie. Darüber hinaus induzieren beide die diskret-konforme Immersion mit vorgeschriebener Halbdichte der mittleren Krümmung

Robust and parallel mesh reconstruction from unoriented noisy points.

Sheung, Hoi.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (p. 65-70).Abstract also in Chinese.Abstract --- p.vAcknowledgements --- p.ixList of Figures --- p.xiiiList of Tables --- p.xvChapter 1 --- Introduction --- p.1Chapter 1.1 --- Main Contributions --- p.3Chapter 1.2 --- Outline --- p.3Chapter 2 --- Related Work --- p.5Chapter 2.1 --- Volumetric reconstruction --- p.5Chapter 2.2 --- Combinatorial approaches --- p.6Chapter 2.3 --- Robust statistics in surface reconstruction --- p.6Chapter 2.4 --- Down-sampling of massive points --- p.7Chapter 2.5 --- Streaming and parallel computing --- p.7Chapter 3 --- Robust Normal Estimation and Point Projection --- p.9Chapter 3.1 --- Robust Estimator --- p.9Chapter 3.2 --- Mean Shift Method --- p.11Chapter 3.3 --- Normal Estimation and Projection --- p.11Chapter 3.4 --- Moving Least Squares Surfaces --- p.14Chapter 3.4.1 --- Step 1: local reference domain --- p.14Chapter 3.4.2 --- Step 2: local bivariate polynomial --- p.14Chapter 3.4.3 --- Simpler Implementation --- p.15Chapter 3.5 --- Robust Moving Least Squares by Forward Search --- p.16Chapter 3.6 --- Comparison with RMLS --- p.17Chapter 3.7 --- K-Nearest Neighborhoods --- p.18Chapter 3.7.1 --- Octree --- p.18Chapter 3.7.2 --- Kd-Tree --- p.19Chapter 3.7.3 --- Other Techniques --- p.19Chapter 3.8 --- Principal Component Analysis --- p.19Chapter 3.9 --- Polynomial Fitting --- p.21Chapter 3.10 --- Highly Parallel Implementation --- p.22Chapter 4 --- Error Controlled Subsampling --- p.23Chapter 4.1 --- Centroidal Voronoi Diagram --- p.23Chapter 4.2 --- Energy Function --- p.24Chapter 4.2.1 --- Distance Energy --- p.24Chapter 4.2.2 --- Shape Prior Energy --- p.24Chapter 4.2.3 --- Global Energy --- p.25Chapter 4.3 --- Lloyd´ةs Algorithm --- p.26Chapter 4.4 --- Clustering Optimization and Subsampling --- p.27Chapter 5 --- Mesh Generation --- p.29Chapter 5.1 --- Tight Cocone Triangulation --- p.29Chapter 5.2 --- Clustering Based Local Triangulation --- p.30Chapter 5.2.1 --- Initial Surface Reconstruction --- p.30Chapter 5.2.2 --- Cleaning Process --- p.32Chapter 5.2.3 --- Comparisons --- p.33Chapter 5.3 --- Computing Dual Graph --- p.34Chapter 6 --- Results and Discussion --- p.37Chapter 6.1 --- Results of Mesh Reconstruction form Noisy Point Cloud --- p.37Chapter 6.2 --- Results of Clustering Based Local Triangulation --- p.47Chapter 7 --- Conclusions --- p.55Chapter 7.1 --- Key Contributions --- p.55Chapter 7.2 --- Factors Affecting Our Algorithm --- p.55Chapter 7.3 --- Future Work --- p.56Chapter A --- Building Neighborhood Table --- p.59Chapter A.l --- Building Neighborhood Table in Streaming --- p.59Chapter B --- Publications --- p.63Bibliography --- p.6