用逼近型√3细分方法构造闭三角网格的插值曲面

为了避免用逼近型3~(1/2)细分方法构造插值曲面过程中出现的烦琐运算,利用3细分方法极限点计算公式,提出一种用逼近型3~(1/2)细分方法构造闭三角网格插值曲面的方法.给定待插值的闭三角网格,先用一个新的几何规则与原3~(1/2)细分方法的拓扑规则细分一次得到一个初始网格,用3~(1/2)细分方法细分该初始网格得到插值曲面;新几何规则根据极限点公式确定,保证了初始网格的极限曲面插值待插值的三角网格.由于初始网格的顶点仅与待插值顶点2邻域内的点相关,所以插值曲面具有良好的局部性,即改变一个待插值点的位置时,只影响插值曲面在其附近的形状.该方法中只有确定初始网格顶点的几何规则与原3细分方法不同,故易于整合到原有的细分系统中.实验结果表明,该方法具有计算简单、有充分的自由度调整插值曲面的形状等特点,使得利用3~(1/2)细分方法构造三角网格的插值曲面变得极其简单.     

用逼近型3~(1/2)细分方法构造闭三角网格的插值曲面

为了避免用逼近型3~(1/2)细分方法构造插值曲面过程中出现的烦琐运算,利用3细分方法极限点计算公式,提出一种用逼近型3~(1/2)细分方法构造闭三角网格插值曲面的方法.给定待插值的闭三角网格,先用一个新的几何规则与原3~(1/2)细分方法的拓扑规则细分一次得到一个初始网格,用3~(1/2)细分方法细分该初始网格得到插值曲面;新几何规则根据极限点公式确定,保证了初始网格的极限曲面插值待插值的三角网格.由于初始网格的顶点仅与待插值顶点2邻域内的点相关,所以插值曲面具有良好的局部性,即改变一个待插值点的位置时,只影响插值曲面在其附近的形状.该方法中只有确定初始网格顶点的几何规则与原3细分方法不同,故易于整合到原有的细分系统中.实验结果表明,该方法具有计算简单、有充分的自由度调整插值曲面的形状等特点,使得利用3~(1/2)细分方法构造三角网格的插值曲面变得极其简单.     

细分曲面拟合的局部渐进插值方法

逼近型细分方法生成的细分曲面其品质要优于插值型细分方法生成的细分曲面.然而,逼近型细分方法生成的细分曲面不能插值于初始控制网格顶点.为使逼近型细分曲面具有插值能力,一般通过求解全局线性方程组,使其插值于网格顶点.当网格顶点较多时,求解线性方程组的计算量很大,因此,难以处理稠密网格.与此不同,在不直接求解线性方程组的情况下,渐进插值方法通过迭代调整控制网格顶点,最终达到插值的效果.渐进插值方法可以处理稠密的任意拓扑网格,生成插值于初始网格顶点的光滑细分曲面.并且经证明,逼近型细分曲面渐进插值具有局部性质,也就是迭代调整初始网格的若干控制顶点,且保持剩余顶点不变,最终生成的极限细分曲面仍插值于初始网格中被调整的那些顶点.这种局部渐进插值性质给形状控制带来了更多的灵活性,并且使得自适应拟合成为可能.实验结果验证了局部渐进插值的形状控制以及自适应拟合能力.     

基于能量最小化的网格优化算法

对网格优化的方法进行研究,提出一种基于能量最小化的网格优化算法.给定一定数量的三维散乱点数据和一个初始三角网格,使用能量最小化算法对网格顶点位置进行优化,使网格顶点更好地逼近三维散乱点数据;网格也更加逼近实际曲面.实验结果表明,使用该算法能够获得形状良好的网格.     

基于逆3 细分的渐进网格生成算法研究

提出一种基于逆3 细分的渐进网格生成算法,用于解决图形的快速传输和显示问 题。算法的基本思路是：将细密网格通过边折叠操作得到简化网格,以细分极限点逼近原始网 格为准则进行网格调整,采用3 细分得到高密度网格,调整后进行逆3 细分,即逐层次删除 部分顶点,生成用于重构渐进网格模型的基网格,并记录每层删除顶点在采用本层表示时相对 于细分计算位置的几何调整量。3 细分过程中三角片数量增长速度较慢,采用逆3 细分利于 生成多层次的渐进网格,经实例验证,逆3 细分生成渐进网格的效果能满足快速、多分辨率显 示要求。     

蝶形细分面片的光顺

使用蝶形细分法细分一般的初始控制网格得到的细分面片光滑而不光顺 ,面片的视觉效果很差 ,而运用现有的光顺技术 ,又只能直接光顺细分以后的结果 ,其需要保存的数据不仅量大 ,而且会引入误差 .针对这一问题 ,提出了一种新的光顺方法 ,即通过调整初始网格顶点位置来光顺细分以后的结果 .在添加合适的约束后 ,该方法不仅可以在光顺细分面片的同时 ,降低细分面片和三维真实物体表面之间的逼近误差 ,而且由于最终输出的是初始控制网格 ,故需要保存的数据量小 .     

基于逆3~(1/2)细分的渐进网格生成算法研究

提出一种基于逆3~(1/2)细分的渐进网格生成算法,用于解决图形的快速传输和显示问题。算法的基本思路是:将细密网格通过边折叠操作得到简化网格,以细分极限点逼近原始网格为准则进行网格调整,采用3~(1/2)细分得到高密度网格,调整后进行逆3~(1/2)细分,即逐层次删除部分顶点,生成用于重构渐进网格模型的基网格,并记录每层删除顶点在采用本层表示时相对于细分计算位置的几何调整量。3~(1/2)细分过程中三角片数量增长速度较慢,采用逆3~(1/2)细分利于生成多层次的渐进网格,经实例验证,逆3~(1/2)细分生成渐进网格的效果能满足快速、多分辨率显示要求。     

基于C—C细分的四边形网格上插值光滑Bezier曲面的生成

在任意拓扑的四边形网格上构造光滑的曲面是计算机辅助几何设计中的一个重要问题．基于C—C细分,提出一种从四边形网格上生成插值网格顶点的光滑Bezier曲面片的算法．将输入四边形网格作为C—C细分的初始控制网格,在四边形网格的每张面上对应得到一张Bezier曲面,使Bezier曲面片逼近C—C细分极限曲面．曲面片在与奇异顶点相连的边界上G^1连续,其他地方C^2连续．为解决C—C细分的收缩问题,给出了基于误差控制的迭代扩张初始控制网格的方法,使从扩张后网格上生成的曲面插值于初始控制网格的顶点．实验结果表明,该算法效率高,生成的曲面具有较好的连续性,适用于对四边化后的网格模型上重建光滑的曲面．     

基于C-C细分的四边形网格上插值光滑Bézier曲面的生成

在任意拓扑的四边形网格上构造光滑的曲面是计算机辅助几何设计中的一个重要问题.基于C-C细分,提出一种从四边形网格上生成插值网格顶点的光滑Bézier曲面片的算法.将输入四边形网格作为C-C细分的初始控制网格,在四边形网格的每张面上对应得到一张Bézier曲面,使Bézier曲面片逼近C-C细分极限曲面.曲面片在与奇异顶点相连的边界上G1连续,其他地方C2连续.为解决C-C细分的收缩问题,给出了基于误差控制的迭代扩张初始控制网格的方法,使从扩张后网格上生成的曲面插值于初始控制网格的顶点.实验结果表明,该算法效率高,生成的曲面具有较好的连续性,适用于对四边化后的网格模型上重建光滑的曲面.     

隐式曲面重新多边形化

首先用Bloomenthal的多边形化算法生成一个粗糙的初始网格;然后在初始网格上分布若干个新顶点,新顶点可以均匀分布,也可以按曲率分布;再把初始网格上的老顶点和新顶点连接起来,生成一个中间网格,从中间网格上删除初始网格上的老顶点,得到重新多边形化的网格;最后细分这个网格．实验结果表明：该算法可以生成近似等边的、大小由曲率指导的三角网格．     

A framework for quad/triangle subdivision surface fitting: Application to mechanical objects

In this paper we present a new framework for subdivision surface approximation of three‐dimensional models represented by polygonal meshes. Our approach, particularly suited for mechanical or Computer Aided Design (CAD) parts, produces a mixed quadrangle‐triangle control mesh, optimized in terms of face and vertex numbers while remaining independent of the connectivity of the input mesh. Our algorithm begins with a decomposition of the object into surface patches. The main idea is to approximate the region boundaries first and then the interior data. Thus, for each patch, a first step approximates the boundaries with subdivision curves (associated with control polygons) and creates an initial subdivision surface by linking the boundary control points with respect to the lines of curvature of the target surface. Then, a second step optimizes the initial subdivision surface by iteratively moving control points and enriching regions according to the error distribution. The final control mesh defining the whole model is then created assembling every local subdivision control meshes. This control polyhedron is much more compact than the original mesh and visually represents the same shape after several subdivision steps, hence it is particularly suitable for compression and visualization tasks. Experiments conducted on several mechanical models have proven the coherency and the efficiency of our algorithm, compared with existing methods.     

Surface interpolation of meshes by geometric subdivision

Subdivision surfaces are generated by repeated approximation or interpolation from initial control meshes. In this paper, two new non-linear subdivision schemes, face based subdivision scheme and normal based subdivision scheme, are introduced for surface interpolation of triangular meshes. With a given coarse mesh more and more details will be added to the surface when the triangles have been split and refined. Because every intermediate mesh is a piecewise linear approximation to the final surface, the first type of subdivision scheme computes each new vertex as the solution to a least square fitting problem of selected old vertices and their neighboring triangles. Consequently, sharp features as well as smooth regions are generated automatically. For the second type of subdivision, the displacement for every new vertex is computed as a combination of normals at old vertices. By computing the vertex normals adaptively, the limit surface is G¹ smooth. The fairness of the interpolating surface can be improved further by using the neighboring faces. Because the new vertices by either of these two schemes depend on the local geometry, but not the vertex valences, the interpolating surface inherits the shape of the initial control mesh more fairly and naturally. Several examples are also presented to show the efficiency of the new algorithms.     

任意拓扑网的细分改进

在改进任意拓扑网构造光滑表面时,初始控制网格确定的情况下,生成的曲面形状惟一确定,最终的物体造型也随之确定,不具有可调性,因而在曲面细分过程中引入了控制参数和摄动。通过引入控制参数,调节一个参数值,使得所得的细分曲面的表达度可控,可以得到一系列的细分曲面。引入摄动是为了改进了空间位置,允许局部地调控约束曲面的形状。最后给出了曲面设计的实例,表明这种算法简单、有效。     

不规则网格上的曲面设计方法

提出一种在不规则网格上构造曲面的方法.其基本思想是,通过均匀双三次B样条基函数的分解和子基函数的分类,将B样条曲面方法推广到任意四边形网格.给定一个任意四边形控制网格,首先对每个控制点构造一个基函数;所有控制点加权组合形成整体曲面.构造的曲面是分片双三次有理参数多项式曲面.此方法可以看成是均匀B样条曲面构造方法的扩展,如果控制网格是规则四边形网格,那么构造得到的曲面与均匀双三次B样条曲面是一致的.最后,实例证明此方法能够有效地构造曲面.     

Fitting Sharp Features with Loop Subdivision Surfaces

Various methods have been proposed for fitting subdivision surfaces to different forms of shape data (e.g., dense meshes or point clouds), but none of these methods effectively deals with shapes with sharp features, that is, creases, darts and corners. We present an effective method for fitting a Loop subdivision surface to a dense triangle mesh with sharp features. Our contribution is a new exact evaluation scheme for the Loop subdivision with all types of sharp features, which enables us to compute a fitting Loop subdivision surface for shapes with sharp features in an optimization framework. With an initial control mesh obtained from simplifying the input dense mesh using QEM, our fitting algorithm employs an iterative method to solve a nonlinear least squares problem based on the squared distances from the input mesh vertices to the fitting subdivision surface. This optimization framework depends critically on the ability to express these distances as quadratic functions of control mesh vertices using our exact evaluation scheme near sharp features. Experimental results are presented to demonstrate the effectiveness of the method.     

Generalized B-spline subdivision-surface wavelets for geometry compression

We present a new construction of lifted biorthogonal wavelets on surfaces of arbitrary two-manifold topology for compression and multiresolution representation. Our method combines three approaches: subdivision surfaces of arbitrary topology, B-spline wavelets, and the lifting scheme for biorthogonal wavelet construction. The simple building blocks of our wavelet transform are local lifting operations performed on polygonal meshes with subdivision hierarchy. Starting with a coarse, irregular polyhedral base mesh, our transform creates a subdivision hierarchy of meshes converging to a smooth limit surface. At every subdivision level, geometric detail is expanded from wavelet coefficients and added to the surface. We present wavelet constructions for bilinear, bicubic, and biquintic B-spline subdivision. While the bilinear and bicubic constructions perform well in numerical experiments, the biquintic construction turns out to be unstable. For lossless compression, our transform is computed in integer arithmetic, mapping integer coordinates of control points to integer wavelet coefficients. Our approach provides a highly efficient and progressive representation for complex geometries of arbitrary topology.