Surface Fairing towards Regular Principal Curvature Line Networks

Freeform surfaces whose principal curvature line network is regularly distributed, are essential to many real applications like CAD modeling, architecture design, and industrial fabrication. However, most designed surfaces do not hold this nice property because it is hard to enforce such constraints in the design process. In this paper, we present a novel method for surface fairing which takes a regular distribution of the principal curvature line network on a surface as an objective. Our method first removes the high‐frequency signals from the curvature tensor field of an input freeform surface by a novel rolling guidance tensor filter, which results in a more regular and smooth curvature tensor field, then deforms the input surface to match the smoothed field as much as possible. As an application, we solve the problem of approximating freeform surfaces with regular principal curvature line networks, discretized by quadrilateral meshes. By introducing the circular or conical conditions on the quadrilateral mesh to guarantee the existence of discrete principal curvature line networks, and minimizing the approximate error to the original surface and improving the fairness of the quad mesh, we obtain a regular discrete principal curvature line network that approximates the original surface. We evaluate the efficacy of our method on various freeform surfaces and demonstrate the superiority of the rolling guidance tensor filter over other tensor smoothing techniques. We also utilize our method to generate high‐quality circular/conical meshes for architecture design and cyclide spline surfaces for CAD modeling.     

三维CAD曲面模型距离-曲率形状分布检索算法

产品三维CAD模型检索对实现产品设计信息的有效重用具有重要意义和作用.针对在工程应用中存在大量含有曲面的产品,提出一种基于距离-曲率形状分布的三维CAD曲面模型检索算法.首先在模型表面随机取点,计算出所取点的曲率以及该点到模型质心的距离;然后以模型质心为球心,统计处在不同半径球壳间所取随机点的曲率分布,分别以球壳半径和曲率为坐标轴构建距离-曲率平面网格,并统计每个网格中随机点出现的频次,从而形成距离-曲率矩阵;最后利用特定的规则计算矩阵的相似值,用该值衡量2个模型的相似程度,实现模型的相似性检索.实验结果表明,文中算法的检索性能比传统的形状分布算法有较大提高,尤其适用于自由曲面模型的检索.     

基于CAD模型的3D散乱数据点三角剖分方法

提出了一种针对空间大规模散乱数据点三角剖分的方法。该方法基于可用的CAD模型，采用“分而治之”的思想。对齐测量数据点与CAD模型、记录数据点及在CAD裁剪NURBS曲面实体上投影点。分别对每块实体的参数区域(u,v)相应点2D-Delaunay三角化、根据R2区域的连通结构反构造出3D三角网。进行冗余三角形删除和网格片缝合等优化处理。与其他方法不同的是，它不受测量数据的分布方式和物体曲面形状的拓扑结构限制。实际的算例结果表明，该方法高效且可靠实用。     

基于径向基函数与B样条的散乱数据拟合方法

针对散乱数据的曲面拟合问题,提出一种径向基函数与B样条插值结合使用的曲面拟合方法.通过分片径向基函数插值,三维散乱点,再从分片插值曲面上获取预先设定好的有序网格点的值,最后利用张量积B样条插值有序网格点,从而得到拟合曲面.该方法较好地解决散乱数据插值和拟合的计算不稳定性问题,最后给出算法实例.     

离散插值曲面

散乱插值在实际应用中有重要作用.本文通过Vornonoi图构造了一张具有点插值和法 向插值的曲面.本文的方法优于目前散乱插值中常用的Shepard方法和三角剖分方法.本 文还给出了一种求离散插值曲面的算法,算法的时间复杂性为O(Nlog N),N为散敌点数.     

一种大规模散乱数据自适应压缩与曲面重建方法

针对大规模散乱数据点云,提出了一种基于曲率与距离的三角网格抽样方法。算法既能保证所生成网格曲面中每个三角片具有较好的形状,又能较鲜明地刻画曲面的细节特征。同时还能将原先规模较大的点云压缩到事先可控的数量上,是一种简单高效的自适应压缩和曲面生成方法。     

用参数样条插值挖补方法进行大规模散乱数据曲面造型

利用矩形域中带连续边界条件的多元散乱数据最优插值方法，结合张量积型参数样条插值，从挖补的思想得到启发，提出一种适合大规模散乱数据曲面造型的参数样条插值挖补方法．用该方法构造的参数曲面内部C^m，n连续，挖补的矩形边界分别为C^m-10.和C^0,n-1连续．最后就常见的m=n=2时的双三次样条给出一些数值例子，说明该算法简单易行，效果良好．     

散乱数据可视化研究综述

散乱数据的可视化已广泛应用于众多的科学与工程领域。文中回顾了散乱数据的可视化技术发展的历史。在此基础上，通过对已有的散乱数据插值或拟合的方法进行了分析，对近年来散乱数据可视化研究的热点进行了归纳，对进一步的研究工作进行了展望。     

Natural neighbor extrapolation using ghost points

Among locally supported scattered data schemes, natural neighbor interpolation has some unique features that makes it interesting for a range of applications. However, its restriction to the convex hull of the data sites is a limitation that has not yet been satisfyingly overcome. We use this setting to discuss some aspects of scattered data extrapolation in general, compare existing methods, and propose a framework for the extrapolation of natural neighbor interpolants on the basis of dynamic ghost points.     

An adaptive method for smooth surface approximation to scattered 3D points

The construction of a surface from arbitrarily scattered data is an important problem in many applications. When there are a large number of data points, the surface representations generated by interpolation methods may be inefficient in both storage and computational requirements. This paper describes an adaptive method for smooth surface approximation from scattered 3D points. The approximating surface is represented by a piecewise cubic triangular Bézier surface possessing C¹ continuity. The method begins with a rough surface interpolating only boundary points and, in the successive steps, refines it by adding the maximum error point at a time among the remaining internal points until the desired approximation accuracy is reached. Our method is simple in concept and efficient in computational time, yet realizes efficient data reduction. Some experimental results are given to show that surface representations constructed by our method are compact and faithful to the original data points.     

结合插值细分和径向基函数的3维扫描数据孔洞修补

目的 逆向工程中3维扫描数据通常产生孔洞影响逆向造型精度.针对已有算法补洞会导致的边界突变问题,提出基于插值细分和基于径向基函数的孔洞修复算法。方法 首先,对有噪声孔洞边界进行拉普拉斯平滑预处理;其次,通过快速重心插值细分孔洞;然后,结合孔洞周围曲率信息,利用边界和法线约束点进行隐式曲面求解;最后,利用求得的隐式曲面方程,利用梯度下降法调整孔洞插值点,获得平滑修补孔洞结果。结果 对3维经典造型以及实际机械工件等两类不同的数据进行扫描并进行孔洞修补实验。由于算法针对有噪声孔洞结合了孔洞周围曲率信息并通过插值细分进行约束求解,保证了补洞效果的平滑性。实验结果表明,本文算法使得基于径向基函数隐式曲面对有噪声孔洞的适应性更强,其修补结果更加平滑,符合周围曲率变化,改进了已有孔洞修补的边缘突变和修补痕迹明显问题。结论 本文算法针对基于径向基函数的隐式曲面求解对噪声敏感的局限性,进行平滑预处理,结合孔洞周围曲率,提高了孔洞修补效果。由于基于径向基函数的隐式曲面对光顺的流形曲面模拟较好,所以算法对特征孔洞的修补存在一定的不足,快速重心插值法针对不规则孔洞也有一定的局限性。     

Uniformization and Density Adaptation for Point Cloud Data Via Graph Laplacian

Point cloud data is one of the most common types of input for geometric processing applications. In this paper, we study the point cloud density adaptation problem that underlies many pre‐processing tasks of points data. Specifically, given a (sparse) set of points Q sampling an unknown surface and a target density function, the goal is to adapt Q to match the target distribution. We propose a simple and robust framework that is effective at achieving both local uniformity and precise global density distribution control. Our approach relies on the Gaussian‐weighted graph Laplacian and works purely in the points setting. While it is well known that graph Laplacian is related to mean‐curvature flow and thus has denoising ability, our algorithm uses certain information encoded in the graph Laplacian that is orthogonal to the mean‐curvature flow. Furthermore, by leveraging the natural scale parameter contained in the Gaussian kernel and combining it with a simulated annealing idea, our algorithm moves points in a multi‐scale manner. The resulting algorithm relies much less on the input points to have a good initial distribution (neither uniform nor close to the target density distribution) than many previous refinement‐based methods. We demonstrate the simplicity and effectiveness of our algorithm with point clouds sampled from different underlying surfaces with various geometric and topological properties.     

第四代地理信息系统研究中的尺度转换数字模型

分析了空间插值模型,数字地面模型和它们与地理信息系统集成的研究进展,存在的缺陷和需要解决的理论问题,论述了建立基于曲面论和遥感反演方法数字模型及实现其与地理信息系统有效集成的必要性和可行性。     

Improved shape for multi-surface blends

For many design applications, where multiple primary surface pieces meet, the distribution of curvature is more important than formally achieving exact curvature continuity. For parametric spline surfaces, when constructing a multi-sided surface cap, we demonstrate a strong link between the uniform variation of the re-parameterization between (boundary) data of the joining pieces and a desirable distribution of curvature. We illustrate this interdependence between parameterization quality and surface quality by developing a G¹ bi-quintic surface cap consisting of n pieces that smoothly fills holes in a piecewise bi-cubic tensor-product spline complex. These bi-5 surface caps have arguably better shape than higher-degree, formally curvature continuous alternatives.     

等值线绘制中的多重网格剖分快速搜索算法*

搜索散乱采样数据的网格化过程是等值线生成中的重要步骤,现有矩形网格剖分的网格化方法,对分布不均匀的采样数据搜索效率低,大大影响了网格化的效率。本文提出一种多重网格剖分快速搜索算法,该算法首先将采样数据进行多重网格剖分,建立网格索引;然后通过索引搜索多重网格,合并采样数据;最后利用快速搜索算法得到的采样数据计算出待插值点。实际数据的网格化测试结果表明,该算法有效地提高了散乱采样数据的搜索效率,具有实际应用价值。     

激光体积测量中散乱数据的可视化

在激光体积测量应用中,测量数据往往为三维散乱数据,实现其可视化,还原测量对象的外形与轮廓有着重要的意义.文章使用按距离平方反比加权插值与按方位取点加权插值相结合的方法,实现了测量数据的插值,应用趋势面拟合的方法重构,使曲面更光滑,并提出程序结束判断算法,实际应用表明该方法是可行的.     

Computing area filling contours for surfaces defined by piecewise polynomials

For contour plotting, bivariate polynomials are often used to interpolate between the data points. The polynomials are defined over triangular or rectangular domains and have continuous interfaces resulting from various interpolation schemes. In this paper the principal ideas of the Trip Algorithm are presented, which finds closed polygons for filling the area between contour lines. These polygons consist of points P defining the contours, intersections S with the boundary of the domain, and vertices V of the domain. The points P are computed successively by a nonlinear method that combines extrapolation and the regula falsi in order to adjust the distance between the points to the curvature of the contour. For the extrapolation, derivatives of Lagrange polynomials are used. Empirical parameters for the automatic step size control are given. Once the points S are determined, the method is independent of the type of the interpolation function f(x, y). Two examples for applications of the Trip Algorithm are presented: one from scattered data interpolation, the other from stress analysis by finite element methods.     

基于Voronoi Cells的自然邻点插值法在地形可视化中的应用

本文基于Voronoi Cells的数据结构，实现了二维平面中大规模散乱数据点的非结构网格剖分的相应算法，并在此基础上运用自然邻点插值法构造出了具有良好数学性质的插值基函数，然后将该剖分方法和插值算法应用到GIS地形可视化的实际数据中，并利用OpenGL三维显示技术显示结果，收到了良好的效果。     

一种新的亚取样图象内插方法——基于SHSF－AMS的误差扩散算法

亚取样／内插是一种快速简便有效的图象压缩编码方法，是高分辨率电视压缩编码中的一项重要技术．本文提出了一种新的亚取样图象内插算法即基于尖帽函数型联想记忆系统ＳＨＳＦ－ＡＭＳ的误差扩散算法．计算机模拟结果表明：该内插算法较已知的其他算法具有更好的性能     

Duplex fitting of zero-level and offset surfaces

Offset surfaces play an important role in various CAD/CAM applications. Given a set of oriented points, we propose a hierarchical method in this paper to fit both the zero-level surface and its offset surface with a single implicit function. The implicit function is formed by compactly supported radial basis functions (CSRBFs). Different from other existing methods in literature, our approach reconstructs an implicit function which interpolates or approximates both the zero-level surface and the offset surface of a given point set, simultaneously. Employing locally supported functions leads to an efficient computational procedure, while the coarse-to-fine hierarchy makes our approach insensitive to the density of scattered data and allows us to reconstruct large parts of missing data. The performance of our method is demonstrated by a number of examples and the application of adaptive slicing hollowed models in rapid prototyping.