A local tangential lifting differential method for triangular meshes

In this note we present a local tangential lifting (LTL) algorithm to compute differential quantities for triangular meshes obtained from regular surfaces. First, we introduce a new notation of the local tangential polygon and lift functions and vector fields on a triangular mesh to the local tangential polygon. Then we use the centroid weights proposed by Chen and Wu [4] to define the discrete gradient of a function on a triangular mesh. We also use our new method to define the discrete Laplacian operator acting on functions on triangular meshes. Higher order differential operators can also be computed successively. Our approach is conceptually simple and easy to compute. Indeed, our LTL method also provides a unified algorithm to estimate the shape operator and curvatures of a triangular mesh and derivatives of functions and vector fields. We also compare three different methods : our method, the least square method and Akima’s method to compute the gradients of functions.     

A new intrinsic numerical method for PDE on surfaces

In this note, we introduce a simple, effective numerical method, the local tangential lifting method, for solving partial differential equations for scalar- and vector-valued data defined on surfaces. Even though we follow the traditional way to approximate the regular surfaces under consideration by triangular meshes, the key idea of our algorithm is to develop an intrinsic and unified way to compute directly the partial derivatives of functions defined on triangular meshes. We present examples in computer graphics and image processing applications.     

Perfect Laplacians for Polygon Meshes

A discrete Laplace‐Beltrami operator is called perfect if it possesses all the important properties of its smooth counterpart. It is known which triangle meshes admit perfect Laplace operators and how to fix any other mesh by changing the combinatorics. We extend the characterization of meshes that admit perfect Laplacians to general polygon meshes. More importantly, we provide an algorithm that computes a perfect Laplace operator for any polygon mesh without changing the combinatorics, although, possibly changing the embedding. We evaluate this algorithm and demonstrate it at applications.     

Spectral Processing of Tangential Vector Fields

We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier‐type representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge–Laplace operator that fits conceptually to the prominent cotan discretization of the Laplace–Beltrami operator. Based on the Fourier representation, we introduce schemes for spectral analysis, filtering and compression of tangential vector fields. Moreover, we introduce a spline‐type editor for modelling of tangential vector fields with interpolation constraints for the field itself and its divergence and curl. Using the spectral representation, we propose a numerical scheme that allows for real‐time modelling of tangential vector fields.     

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.     

Efficient Coding of Nontriangular Mesh Connectivity

We describe an efficient algorithm for coding the connectivity information of general polygon meshes. In contrast to most existing algorithms which are suitable only for triangular meshes, and pay a penalty for treatment of nontriangular faces, this algorithm codes the connectivity information in a direct manner. Our treatment of the special case of triangular meshes is shown to be equivalent to the Edgebreaker algorithm. Using our methods, any triangle mesh may be coded in no more than 2 bits/triangle (approximately 4 bits/vertex), a quadrilateral mesh in no more than 3.5 bits/quad (approximately 3.5 bits/vertex), and the most common case of a quad mesh with few triangles in no more than 4 bits/polygon.     

A computational approach to joint line detection on triangular meshes

In this paper, we present formulae for evaluating differential quantities at vertices of triangular meshes that may approximate potential piecewise smooth surfaces with discontinuous normals or discontinuous curvatures at the joint lines. We also define the C ¹ and C ² discontinuity measures for surface meshes using changing rates of one-sided curvatures or changing rates of curvatures across mesh edges. The curvatures are computed discretely as of local interpolating surfaces that lie within a tolerance to the mesh. Together with proper estimation of local shape parameters, the obtained discontinuity measures own properties like sensitivity to salient joint lines and being scale invariant. A simple algorithm is finally developed for detection of C ¹ or C ² discontinuity joint lines on triangular meshes with even highly non-uniform triangulations. Several examples are provided to demonstrate the effectiveness of the proposed method.     

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。     

Mesh Parameterization with Generalized Discrete Conformal Maps

We introduce a new method to compute conformal parameterizations using a recent definition of discrete conformity, and establish a discrete version of the Riemann mapping theorem. Our algorithm can parameterize triangular, quadrangular and digital meshes. It can also be adapted to preserve metric properties. To demonstrate the efficiency of our method, many examples are shown in the experiment section.     

Polygon Laplacian Made Simple

The discrete Laplace-Beltrami operator for surface meshes is a fundamental building block for many (if not most) geometry processing algorithms. While Laplacians on triangle meshes have been researched intensively, yielding the cotangent discretization as the de-facto standard, the case of general polygon meshes has received much less attention. We present a discretization of the Laplace operator which is consistent with its expression as the composition of divergence and gradient operators, and is applicable to general polygon meshes, including meshes with non-convex, and even non-planar, faces. By virtually inserting a carefully placed point we implicitly refine each polygon into a triangle fan, but then hide the refinement within the matrix assembly. The resulting operator generalizes the cotangent Laplacian, inherits its advantages, and is empirically shown to be on par or even better than the recent polygon Laplacian of Alexa and Wardetzky [AW11] — while being simpler to compute.     

Efficiently computing exact geodesic loops within finite steps

Closed geodesics, or geodesic loops, are crucial to the study of differential topology and differential geometry. Although the existence and properties of closed geodesics on smooth surfaces have been widely studied in mathematics community, relatively little progress has been made on how to compute them on polygonal surfaces. Most existing algorithms simply consider the mesh as a graph and so the resultant loops are restricted only on mesh edges, which are far from the actual geodesics. This paper is the first to prove the existence and uniqueness of geodesic loop restricted on a closed face sequence; it contributes also with an efficient algorithm to iteratively evolve an initial closed path on a given mesh into an exact geodesic loop within finite steps. Our proposed algorithm takes only an O(k) space complexity and an O(mk) time complexity (experimentally), where m is the number of vertices in the region bounded by the initial loop and the resultant geodesic loop, and k is the average number of edges in the edge sequences that the evolving loop passes through. In contrast to the existing geodesic curvature flow methods which compute an approximate geodesic loop within a predefined threshold, our method is exact and can apply directly to triangular meshes without needing to solve any differential equation with a numerical solver; it can run at interactive speed, e.g., in the order of milliseconds, for a mesh with around 50K vertices, and hence, significantly outperforms existing algorithms. Actually, our algorithm could run at interactive speed even for larger meshes. Besides the complexity of the input mesh, the geometric shape could also affect the number of evolving steps, i.e., the performance. We motivate our algorithm with an interactive shape segmentation example shown later in the paper.     

Constrained interpolation (remap) of divergence-free fields

A novel constrained interpolation algorithm for remapping of solenoidal face finite element vector fields is presented. The algorithm is based on explicit recovery, postprocessing and interpolation of a potential for the original vector field and a subsequent application of a curl operator to obtain the desired divergence-free finite element field on the new mesh.The use of interpolation instead of advection in the remap process offers valuable computational advantages. Old and new meshes are neither required to have the same connectivity, nor to be close to each other. Slope limiting and upwinding, which can be sensitive to grid structure, are avoided and replaced by local optimization to control energy of the remapped field.The new method is validated using a suite of cyclic remap problems on random and tensor product mesh sequences. A comparison with a local remapper based on a constrained transport advection algorithm is also included.     

Winslow Smoothing on Two-Dimensional Unstructured Meshes

The Winslow equations from structured elliptic grid generation are adapted to smoothing of two-dimensional unstructured meshes using a finite difference approach. We use a local mapping from a uniform N-valent logical mesh to a local physical subdomain. Taylor Series expansions are then applied to compute the derivatives which appear in the Winslow equations. The resulting algorithm for Winslow smoothing on unstructured triangular and quadrilateral meshes gives generally superior qualilty than traditional Laplacian smoothing, while retaining the resistance to mesh folding on structured quadrilateral meshes.     

Optimized triangle mesh reconstruction from unstructured points

A variety of approaches have been proposed for polygon mesh reconstruction from a set of unstructured sample points. Suffering from severe aliases at sharp features and having a large number of unnecessary faces, most resulting meshes need to be optimized using input sample points in a postprocess. In this paper, we propose a fast algorithm to reconstruct high-quality meshes from sample data. The core of our proposed algorithm is a new mesh evaluation criterion which takes full advantage of the relation between the sample points and the reconstructed mesh. Based on our proposed evaluation criterion, we develop necessary operations to efficiently incorporate the functions of data preprocessing, isosurface polygonization, mesh optimization and mesh simplification into one simple algorithm, which can generate high-quality meshes from unstructured point clouds with time and space efficiency. Published online: 28 January 2003 Correspondence to: Y.-J. Liu     

GEOMPACK — a software package for the generation of meshes using geometric algorithms

We describe the mathematical software package GEOMPACK, which contains standard Fortran 77 routines for the generation of two-dimensional triangular and three-dimensional tetrahedral finite element meshes using efficient geometric algorithms. This package results from our research into mesh generation and geometric algorithms. It contains routines for constructing two- and three-dimensional Delaunay triangulations, decomposing a general polygonal region into simple or convex polygons, constructing the visibility polygon of a simple polygon from a viewpoint, and other geometric algorithms, from which our mesh generation method is built and others can be implemented. Our method generates meshes in polygonal or polyhedral regions specified by their boundary representation and possible interfaces between subregions.     

径向基函数及移动网格在Euler方程数值计算中的应用

基于二维Euler方程,在利用弹簧技术的移动非结构三角形网格上给出了一种基于紧支径向基函数重构的ENO型有限体积格式,方法的主要思想是先对每一个三角形单元构造插值径向基函数,而在计算交界面的流通量采用两点高斯积分公式以保证格式的整体精度,时间离散采用三阶TVD Runge-Kutta方法。最后用该格式对一些典型算例进行了数值模拟,结果表明该方法计算速度快,对间断有很好的分辨能力。     

基于模版的三角网格拓扑压缩

提出一种基于面的高效三角网格拓扑压缩算法.该算法是单分辨率无损压缩算法,是对Edgebreaker算法的改进:在网格遍历部分,通过自适应网格遍历方法使非常影响压缩比的分割图形操作尽可能少;在熵编码部分,为网格遍历后得到的每个操作符各设计一个模版,根据模版确定该操作符的二进制表示,然后采用自适应算术编码方法压缩该二进制表示得到最后的压缩结果.与网格拓扑压缩领域中基于面的最好的算法得到的压缩比相比较,该算法得到的压缩比有很大提高.     

基于可变模版的三角网格拓扑压缩

针对三角网格模型的拓扑信息。提出了一种高效压缩方法.不同于以往的单纯利用算术编码或霍夫曼鳊码对遍历三角网格生成的拓扑流进行编码压缩,根据三角网格模型(特别是规则三角网格模型)的特点,自适应地提高编码过程中对当前编码字符发生的预测准确率,实现对三角网格模型的拓扑信息的高效压缩.算法首先遍历三角网格模型,得到操作符序列;然后对得到的操作符序列的每个操作符作模版可变的自适应算术编码.在编码过程中,根据当前编码字符的前一个操作符、三角网格模型的特点以及网格遍历方法为当前编码操作符计算一个模版,在这个模版中,预测准确率高的操作符用较短的二进制串表示.根据当前编码操作符的可变模版,可以得到该操作符的二进制表示,并对这个二进制表示的每个比特作自适应算术编码.该方法是针对流形三角网格模型的拓扑信息作单分辨率的基于面的无损压缩,可以得到很好的三角网格拓扑信息的压缩结果,其压缩比甚至比拓扑压缩领域压缩比方面最好的TG算法的压缩比还要好.     

基于上下文算术编码的非三角网格拓扑压缩

网格拓扑压缩方法是计算机图形学的基础算法。该文方法是单分辨率,主要针对非三角网格模型的拓扑信息作无损压缩。算法首先遍历网格的所有多边形得到操作系列;然后对操作系列作霍夫曼编码;再对霍夫曼编码结果作基于上下文长度可变的算术编码得到最后的压缩结果。相比于对非三角网格拓扑信息作压缩的压缩比很高的算法,该算法得到的压缩结果更好。此算法的另一个突出优点是在解码时间和空间上有了改进——新算法可以在接收一个多边形的编码后立即完成解码并抛弃这个编码,从而使得该算法特别适用于在线传输和解码的实时与交互应用场合。此外,该算法还可以处理有空洞和柄（handle）的模型。