Connectivity Editing for Quad‐Dominant Meshes

We propose a connectivity editing framework for quad‐dominant meshes. In our framework, the user can edit the mesh connectivity to control the location, type, and number of irregular vertices (with more or fewer than four neighbors) and irregular faces (non‐quads). We provide a theoretical analysis of the problem, discuss what edits are possible and impossible, and describe how to implement an editing framework that realizes all possible editing operations. In the results, we show example edits and illustrate the advantages and disadvantages of different strategies for quad‐dominant mesh design.     

边界简化与多目标优化相结合的高质量四边形网格生成

目的 高质量四边形网格生成是计算机辅助设计、等几何分析与图形学领域中一个富有挑战性的重要问题。针对这一问题,提出一种基于边界简化与多目标优化的高质量四边形网格生成新框架。方法 首先针对亏格非零的平面区域,提出一种将多连通区域转化为单连通区域的方法,可生成高质量的插入边界;其次,提出"可简化角度"和"可简化面积比率"两个阈值概念,从顶点夹角和顶点三角形面积入手,将给定的多边形边界简化为粗糙多边形;然后对边界简化得到的粗糙多边形进行子域分解,并确定每个子域内的网格顶点连接信息;最后提出四边形网格的均匀性和正交性度量目标函数,并通过多目标非线性优化技术确定网格内部顶点的几何位置。结果 在同样的离散边界下,本文方法与现有方法所生成的四边网格相比,所生成的四边网格顶点和单元总数目较少,网格单元质量基本类似,计算时间成本大致相同,但奇异点数目可减少70% 80%,衡量网格单元质量的比例雅克比值等相关指标均有所提高。结论 本文所提出的四边形网格生成方法能够有效减少网格中的奇异点数目,并可生成具有良好光滑性、均匀性和正交性的高质量四边形网格,非常适用于工程分析和动画仿真。     

Inspired quadrangulation

This paper presents a new approach for the quadrangulation of triangular surfaces. While previous work focused on fully automatic computations or explicitly involved user control for the integration of subjective decisions, we introduce a new example-based quad-meshing paradigm, in order to easily reproduce the subjective decisions made in the design of reference examples found in a corpus. The algorithm enables to reproduce the subjective aspects of the example (extraordinary vertex layout) while minimizing the induced distortion; allowing users to leverage reference meshes considered of quality for fast prototyping. At the core of our technique, we provide the analytic gradient of as-rigid-as-possible 2D transformations. This expression is implemented in a fast solver to automatically register planar unfoldings of the geometries, yielding low-distortion cross maps of the examples onto the input in 3D. In addition, our technique provides interactive feedback for user modifications of the cross maps computed automatically. Our technique supports localized mesh composition and enables to reproduce meshing styles despite intrinsic reflective symmetry, large variation from isometry or even topological variation. Experiments demonstrate the accuracy of the mimicking process as well as its time efficiency.     

Template-based quadrilateral meshing

Generating quadrilateral meshes is a highly non-trivial task, as design decisions are frequently driven by specific application demands. Automatic techniques can optimize objective quality metrics, such as mesh regularity, orthogonality, alignment and adaptivity; however, they cannot make subjective design decisions. There are a few quad meshing approaches that offer some mechanisms to include the user in the mesh generation process; however, these techniques either require a large amount of user interaction or do not provide necessary or easy to use inputs. Here, we propose a template-based approach for generating quad-only meshes from triangle surfaces. Our approach offers a flexible mechanism to allow external input, through the definition of alignment features that are respected during the mesh generation process. While allowing user inputs to support subjective design decisions, our approach also takes into account objective quality metrics to produce semi-regular, quad-only meshes that align well to desired surface features.     

Filling N-Sided Regions by Quad Meshes for Subdivision Surfaces

Given an n-sided region bounded by a loop of n polylines, we present a general algorithm to fill such a region by a quad mesh suitable for a subdivision scheme. Typically, the approach consists of two phases: the topological phase and the geometrical phase. In the first part, the connectivity of the mesh is based on determining a partitioning of the region into rectangular subregions across which regular grid could be constructed. The geometrical phase generalizes discrete Coon's patches to position the vertices in the 3D space. The generated mesh could be taken as input to any quad-based subdivision scheme, such as that of Catmull–Clark or Doo–Sabin to generate the corresponding limit surface. The goal of the algorithm is to generate smooth meshes with minimum number and less valence of extraordinary vertices deemed undesirable in such subdivision schemes.     

Consistent quadrangulation for shape collections via feature line co-extraction

This paper presents a method that takes a collection of 3D surface shapes, and produces a consistent and individually feature preserving quadrangulation of each shape. By exploring the correspondence among shapes within a collection, we coherently extract a set of representative feature lines as the key characteristics for the given shapes. Then we compute a smooth cross-field interpolating sparsely distributed directional constraints induced from the feature lines and apply the mixed-integer quadrangulation to generate the quad meshes. We develop a greedy algorithm to extract aligned cut graphs across the shape collection so that the meshes can be aligned in a common parametric domain. Computational results demonstrate that our approach not only produces consistent quad meshes across the entire collection with significant geometry variation but also achieves a trade-off between global structural simplicity for the collection and local geometry fidelity for each shape.     

Motorcycle Graphs: Canonical Quad Mesh Partitioning

We describe algorithms for canonically partitioning semi‐regular quadrilateral meshes into structured submeshes, using an adaptation of the geometric motorcycle graph of Eppstein and Erickson to quad meshes. Our partitions may be used to efficiently find isomorphisms between quad meshes. In addition, they may be used as a highly compressed representation of the original mesh. These partitions can be constructed in sublinear time from a list of the extraordinary vertices in a mesh. We also study the problem of further reducing the number of submeshes in our partitions—we prove that optimizing this number is NP‐hard, but it can be efficiently approximated.     

Animation‐Aware Quadrangulation

Geometric meshes that model animated characters must be designed while taking into account the deformations that the shape will undergo during animation. We analyze an input sequence of meshes with point‐to‐point correspondence, and we automatically produce a quadrangular mesh that fits well the input animation. We first analyze the local deformation that the surface undergoes at each point, and we initialize a cross field that remains as aligned as possible to the principal directions of deformation throughout the sequence. We then smooth this cross field based on an energy that uses a weighted combination of the initial field and the local amount of stretch. Finally, we compute a field‐aligned quadrangulation with an off‐the‐shelf method. Our technique is fast and very simple to implement, and it significantly improves the quality of the output quad mesh and its suitability for character animation, compared to creating the quad mesh based on a single pose. We present experimental results and comparisons with a state‐of‐the‐art quadrangulation method, on both sequences from 3D scanning and synthetic sequences obtained by a rough animation of a triangulated model.     

Anisotropic quadrangulation

Quadrangulation methods aim to approximate surfaces by semiregular meshes with as few extraordinary vertices as possible. A number of techniques use the harmonic parameterization to keep quads close to squares, or fit parametrization gradients to align quads to features. Both types of techniques create near-isotropic quads; feature-aligned quadrangulation algorithms reduce the remeshing error by aligning isotropic quads with principal curvature directions. A complementary approach is to allow for anisotropic elements, which are well-known to have significantly better approximation quality.In this work we present a simple and efficient technique to add curvature-dependent anisotropy to harmonic and feature-aligned parameterization and improve the approximation error of the quadrangulations. We use a metric derived from the shape operator which results in a more uniform error distribution, decreasing the error near features.     

A Projective Framework for Polyhedral Mesh Modelling

We present a novel framework for polyhedral mesh editing with face‐based projective maps that preserves planarity by definition. Such meshes are essential in the field of architectural design and rationalization. By using homogeneous coordinates to describe vertices, we can parametrize the entire shape space of planar‐preserving deformations with bilinear equations. The generality of this space allows for polyhedral geometric processing methods to be conducted with ease. We demonstrate its usefulness in planar‐quadrilateral mesh subdivision, a resulting multi‐resolution editing algorithm, and novel shape‐space exploration with prescribed transformations. Furthermore, we show that our shape space is a discretization of a continuous space of conjugate‐preserving projective transformation fields on surfaces. Our shape space directly addresses planar‐quad meshes, on which we put a focus, and we further show that our framework naturally extends to meshes with faces of more than four vertices as well.     

A new interpolation subdivision scheme for triangle/quad mesh

In this paper, we present a new interpolation subdivision scheme for mixed triangle/quad meshes that is C¹ continuous. The new scheme is capable of reproducing the well-known four-point based interpolation subdivision in the quad region but does not reproduce Butterfly subdivision in the triangular part. The new scheme defines rules that produce surfaces both at the regular quad/triangle vertices and isolated, extraordinary points. We demonstrate the visually satisfying of our surfaces through several examples.     

Field‐aligned Quadrangulation for Image Vectorization

Image vectorization is an important yet challenging problem, especially when the input image has rich content. In this paper, we develop a novel method for automatically vectorizing natural images with feature‐aligned quad‐dominant meshes. Inspired by the quadrangulation methods in 3D geometry processing, we propose a new directional field optimization technique by encoding the color gradients, sidestepping the explicit computing of salient image features. We further compute the anisotropic scales of the directional field by accommodating the distance among image features. Our method is fully automatic and efficient, which takes only a few seconds for a 400×400 image on a normal laptop. We demonstrate the effectiveness of the proposed method on various image editing applications.     

Quad/triangle subdivision,nonhomogeneous refinement equation and polynomial reproduction

The quad/triangular subdivision, whose control net and refined meshes consist of both quads and triangles, provides better visual quality of subdivision surfaces. While some theoretical results such as polynomial reproduction and smoothness analysis of quad/triangle schemes have been obtained in the literature, some issues such as the basis functions at quad/triangle vertices and design of interpolatory quad/triangle schemes need further study. In our study of quad/triangle schemes, we observe that a quad/triangle subdivision scheme can be derived from a nonhomogeneous refinement equation. Hence, the basis functions at quad/triangle vertices are shifts of the refinable function associated with a nonhomogeneous refinement equation. In this paper a quad/triangle subdivision surface is expressed analytically as the linear combination of these basis functions and the polynomial reproduction of matrix-valued quad/triangle schemes is studied. The result on polynomial reproduction achieved here is critical for the smoothness analysis and construction of matrix-valued quad/triangle schemes. Several new schemes are also constructed.     

An Interactive Approach to Point Cloud Triangulation

We present an interactive system for the generation of high quality triangle meshes that allows us to handle hybrid geometry (point clouds, polygons,. . .) as input data. In order to be able to robustly process huge data sets, we exploit graphics hardware features like the raster manager and the z-buffer for specific sub-tasks in the overall procedure. By this we significantly accelerate the stitching of mesh patches and obtain an algorithm for sub-sampling the data points in linear time. The target resolution and the triangle alignment in sub-regions of the resulting mesh can be controlled by adjusting the screen resolution and viewing transformation. An intuitive user interface provides a flexible tool for application dependent optimization of the mesh.     

Spectral Geometry Processing with Manifold Harmonics

We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis allowing for such a transform. However, computing even just a few eigenvectors is out of reach for meshes with more than a few thousand vertices, and storing these eigenvectors is prohibitive for large meshes. To overcome these limitations, we propose a band‐by‐band spectrum computation algorithm and an out‐of‐core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. We also propose a limited‐memory filtering algorithm, that does not need to store the eigenvectors. Using this latter algorithm, specific frequency bands can be filtered, without needing to compute the entire spectrum. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering. These technical achievements are supported by a solid yet simple theoretic framework based on Discrete Exterior Calculus (DEC). In particular, the issues of symmetry and discretization of the operator are considered with great care.     

Surface modeling with ternary interpolating subdivision

In this paper, a new interpolatory subdivision scheme, called ternary interpolating subdivision, for quadrilateral meshes with arbitrary topology is presented. It can be used to deal with not only extraordinary faces but also extraordinary vertices in polyhedral meshes of arbitrary topologies. It is shown that the ternary interpolating subdivision can generate a C¹-continuous interpolatory surface. Some applications with open boundaries and curves to be interpolated are also discussed.     

Progressive Lossless Mesh Compression Via Incremental Parametric Refinement

In this paper, we propose a novel progressive lossless mesh compression algorithm based on Incremental Parametric Refinement, where the connectivity is uncontrolled in a first step, yielding visually pleasing meshes at each resolution level while saving connectivity information compared to previous approaches. The algorithm starts with a coarse version of the original mesh, which is further refined by means of a novel refinement scheme. The mesh refinement is driven by a geometric criterion, in spirit with surface reconstruction algorithms, aiming at generating uniform meshes. The vertices coordinates are also quantized and transmitted in a progressive way, following a geometric criterion, efficiently allocating the bit budget. With this assumption, the generated intermediate meshes tend to exhibit a uniform sampling. The potential discrepancy between the resulting connectivity and the original one is corrected at the end of the algorithm. We provide a proof-of-concept implementation, yielding very competitive results compared to previous works in terms of rate/distortion trade-off.     

A Weighted Delaunay Triangulation Framework for Merging Triangulations in a Connectivity Oblivious Fashion

Simplicial meshes are useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, making the simulations unstable. In this paper we propose a framework for updating simplicial meshes that undergo geometric and topological changes. Instead of explicitly maintaining connectivity information, we keep a collection of weights associated with mesh vertices, using a Weighted Delaunay Triangulation (WDT). These weights implicitly define mesh connectivity and allow direct merging of triangulations. We propose two formulations for computing the weights, and two techniques for merging triangulations, and finally illustrate our results with examples in two and three dimensions.     

Automatic construction of quadrilateral network of curves from triangular meshes

The aim of this study is to propose a method for building quadrilateral network of curves automatically from a huge number of triangular meshes. The curve net can be served as the framework of automatic surface reconstruction. The proposed method mainly includes three stages: mesh simplification, quadrangulation and curve net generation. Mesh simplification is employed to reduce the number of meshes in accordance with a quadratic error metric for each vertex. Additional post-processing criteria are also employed to improve the shape of the reduced meshes. For quadrangulation, a front composed of a sequence of edges is introduced. An algorithm is proposed to combine each pair of triangles along the front. A new front is then formed and quadrangulation is continued until all triangles are combined or converted. For curve net generation, each edge of quadrilateral meshes is projected onto the triangular meshes to acquire a set of slicing points first. A constrained curve fitting is then employed to convert all sets of slicing points into B-spline curves, with appropriate continuity conditions across adjacent curves. Several examples have been presented to demonstrate the feasibility of the proposed method and its application in automatic surface reconstruction.