形状可调的Loop 细分曲面渐进插值方法

针对Loop 细分无法调整形状与不能插值的问题,提出了一种形状可调的Loop 细分 曲面渐进插值方法。首先给出了一个既能对细分网格顶点统一调整又便于引入权因子实现细分曲 面形状可调的等价Loop 细分模板。其次,通过渐进迭代调整初始控制网格顶点生成新网格,运 用本文的两步Loop 细分方法对新网格进行细分,得到插值于初始控制顶点的形状可调的Loop 细分曲面。最后,证明了该方法的收敛性,并给出实例验证了该方法的有效性。     

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.     

Error bounds for Loop subdivision surfaces

In this paper, we obtain the error bounds on the distance between a Loop subdivision surface and its control mesh. Both local and global bounds are derived by means of computing and analysing the control meshes with two rounds of refinement directly. The bounds can be expressed with the maximum edge length of all triangles in the initial control mesh. Our results can be used as posterior estimates and also can be used to predict the subdivision depth for any given tolerance.     

Estimating subdivision depth of Catmull-Clark surfaces

In this paper, both general and exponential bounds of the distance between a uniform Catmull-Clark surface and its control polyhedron are derived. The exponential bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Based on the exponential bound, we can predict the depth of subdivision within a user-specified error tolerance. This is quite useful and important for pre-computing the subdivision depth of subdivision surfaces in many engineering applications such as surface/surface intersection,mesh generation, numerical control machining and surface rendering.     

Interpolation by geometric algorithm

We present a novel geometric algorithm to construct a smooth surface that interpolates a triangular or a quadrilateral mesh of arbitrary topological type formed by n vertices. Although our method can be applied to B-spline surfaces and subdivision surfaces of all kinds, we illustrate our algorithm focusing on Loop subdivision surfaces as most of the meshes are in triangular form. We start our algorithm by assuming that the given triangular mesh is a control net of a Loop subdivision surface. The control points are iteratively updated globally by a simple local point-surface distance computation and an offsetting procedure without solving a linear system. The complexity of our algorithm is O(mn) where n is the number of vertices and m is the number of iterations. The number of iterations m depends on the fineness of the mesh and accuracy required.     

Conjugate-Gradient Progressive-Iterative Approximation for Loop and Catmull-Clark Subdivision Surface Interpolation

Loop and Catmull-Clark are the most famous approximation subdivision schemes, but their limit surfaces do not interpolate the vertices of the given mesh. Progressive-iterative approximation (PIA) is an efficient method for data interpolation and has a wide range of applications in many fields such as subdivision surface fitting, parametric curve and surface fitting among others. However, the convergence rate of classical PIA is slow. In this paper, we present a new and fast PIA format for constructing interpolation subdivision surface that interpolates the vertices of a mesh with arbitrary topology. The proposed method, named Conjugate-Gradient Progressive-Iterative Approximation (CG-PIA), is based on the Conjugate-Gradient Iterative algorithm and the Progressive Iterative Approximation (PIA) algorithm. The method is presented using Loop and Catmull-Clark subdivision surfaces. CG-PIA preserves the features of the classical PIA method, such as the advantages of both the local and global scheme and resemblance with the given mesh. Moreover, CG-PIA has the following features. 1) It has a faster convergence rate compared with the classical PIA and W-PIA. 2) CG-PIA avoids the selection of weights compared with W-PIA. 3) CG-PIA does not need to modify the subdivision schemes compared with other methods with fairness measure. Numerous examples for Loop and Catmull-Clark subdivision surfaces are provided in this paper to demonstrate the efficiency and effectiveness of CG-PIA.     

曲线插值约束的LOOP细分曲面

提出基于Loop细分方法的曲线插值方法,不需要修改细分规则,只需以插值曲线的控制多边形为中心多边形,向其两侧构造对称三角网格带,该对称三角网格带将收敛于插值曲线。因此,包含有该三角网格带的多面体网格的极限曲面将经过插值曲线。若要插值多条相交曲线只需在交点处构造全对称三角网格。运用该方法可在三角网格生成的细分曲面中插值多达六条的相交曲线。     

散乱数据点的细分曲面重建算法及实现

提出一种对海量散乱数据根据给定精度拟合出无需裁剪和拼接的、反映细节特征的、分片光滑的细分曲面算法．该算法的核心是基于细分的局部特性，通过对有特征的细分控制网格极限位置分析，按照拟合曲面与数据点的距离误差最小原则，对细分曲面控制网格循环进行调整、优化、特征识别、白适应细分等过程，使得细分曲面不断地逼近原始数据．实例表明：该算法不仅具有高效性、稳定性，同时构造出的细分曲面还较好地反映了原始数据的细节特征。     

一种带噪声的密集三角网格细分曲面拟合算法

实现了一个从带噪声的密集三角形拟合出带尖锐特征的细分曲面拟合系统.该系统包括了一种改进的基于图像双边滤波器的网格噪声去除方法,模型的尖锐特征提取以及保持尖锐特征的网格简化和拓扑优化.为了处理局部细节特征和模型数据量问题,提出了自适应细分方法,并将根据给定精度估计最少细分深度引入到细分曲面拟合系统中,使得拟合得到的细分曲面模型具有良好的细节特征和数据量小等特点.大量3D模型实验结果和实际工程应用结果表明了该细分曲面拟合系统的有效性.     

基于二面角逆插值Loop细分的渐进传输方法

随着虚拟现实、增强现实等领域快速发展,渐进传输获得了良好的用户体验。为 了三角网格在移动终端的快速传输和显示,提出了一种基于二面角逆插值 Loop 细分(DRILS)的 渐进传输算法。主要通过对原始三角网格进行基于二面角插值 Loop 细分(DILS)和插值 Loop 细 分(ILS)进行预处理,在局部特征精确保持的同时获得具备细分连通性的精网格。在渐进传输的 过程中通过对该精网格迭代操作 3 个步骤,即奇偶顶点划分、预测偏移量、更新三角网格。由 于采用 DILS 与 ILS 结合获取精网格,在渐进传输的过程中保持了精确的局部特征,同时也加 快了渐进传输的速度。实验对比表明,该算法精确、高效,适应于移动终端设备的显示传输及 存储。     

RGB Subdivision

We introduce the RGB subdivision: an adaptive subdivision scheme for triangle meshes, which is based on the iterative application of local refinement and coarsening operators, and generates the same limit surface of the Loop subdivision, independently on the order of application of local operators. Our scheme supports dynamic selective refinement, as in Continuous Level Of Detail models, and it generates conforming meshes at all intermediate steps. The RGB subdivision is encoded in a standard topological data structure, extended with few attributes, which can be used directly for further processing. We present an interactive tool that permits to start from a base mesh and use RGB subdivision to dynamically adjust its level of detail.     

一种Loop细分小波的边界处理方法

细分小波近年来发展迅速,在计算机图形显示、渐进网格传输和网格多分辨率编辑等领域获得了广泛的应用。Bertram提出的Loop细分小波是基于提升格式的双正交细分小波的典型范例,它所针对的对象均为网格的内部顶点。目前尚未发现相关文献提及细分小波对于边界的处理。该文在Loop细分小波算法的基础上,给出了一种Loop细分小波边界处理的方法,经验证效果令人满意。     

Optimization methods for scattered data approximation with subdivision surfaces

We present a method for scattered data approximation with subdivision surfaces which actually uses the true representation of the limit surface as a linear combination of smooth basis functions associated with the control vertices. A robust and fast algorithm for exact closest point search on Loop surfaces which combines Newton iteration and non-linear minimization is used for parameterizing the samples. Based on this we perform unconditionally convergent parameter correction to optimize the approximation with respect to the L² metric, and thus we make a well-established scattered data fitting technique which has been available before only for B-spline surfaces, applicable to subdivision surfaces. We also adapt the recently discovered local second order squared distance function approximant to the parameter correction setup. Further we exploit the fact that the control mesh of a subdivision surface can have arbitrary connectivity to reduce the L^∞ error up to a certain user-defined tolerance by adaptively restructuring the control mesh. Combining the presented algorithms we describe a complete procedure which is able to produce high-quality approximations of complex, detailed models.     

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 .     

基于加权渐进插值的Loop 细分曲面等距逼近

等距曲面在CAD/CAM 领域有着重要的作用,由于细分曲面没有整体解 析表达式,使得计算细分曲面等距比参数曲面更加困难。针对目前已有的两种等距面逼近算 法进行了改进,利用加权渐进插值技术避免了传统细分等距逼近算法产生网格偏移的问题。 此外,提出了针对边界等距处理方案,使得等距后的细分曲面在内部和边界都均匀等距。该 方法无需求解线性方程组,具有全局和局部特性,能够处理闭网格和开网格,为Loop 细分 曲面数控加工奠定了良好的基础算法。最后给出的实例验证了算法的有效性。     

渐进插值的LOOP曲面细分*

目前很多细分方法都存在不能用同一种方法处理封闭网格和开放网格的问题。对此,一种新的基于插值技术的LOOP曲面细分方法,其主要思想就是给定一个初始三角网格M,反复生成新的顶点,新顶点是通过其相邻顶点的约束求解得到的,从而构造一个新的控制网格M,在取极限的情况下,可以证明插值过程是收敛的;因为生成新顶点使用的是与其相连顶点的约束求解得到的,本质上是一种局部方法,所以,该方法很容易定义。它在本地方法和全局方法中都有优势,能处理任意顶点数量和任意拓扑结构的网格,从而产生一个光滑的曲面并忠实于给定曲面的形状,其控制     

基于逆 Loop 细分的半正则网格简化算法

摘 要：三维网格简化是在保留目标物体几何形状信息的前提下尽量减小精细化三维模型 中的点数和面数的一种操作,对提高三维网格数据的存取和网络传输速度、编辑和渲染效率具 有十分重要的作用。针对大多网格简化算法在简化过程中未考虑网格拓扑结构与视觉质量的问 题,提出了一种基于逆 Loop 细分的半正则网格简化算法。首先根据邻域质心偏移量进行特征 点检测,随后随机选取种子三角形,以边扩展方式获取正则区域并执行逆 Loop 细分进行简化。 最后,以向内分割方式进行边缘拼接,获取最终的简化模型。与经典算法在公开数据集上进行 实验对比,结果表明,该算法能够在简化的同时有效地保持网格特征,尽可能保留与原始网格 一致的规则的拓扑结构,并且在视觉质量上优于边折叠以及聚类简化算法。     

基于形状控制的细分曲面的局部渐进插值方法

提出一种基于形状控制的 Catmull-Clark 细分曲面构造方法,实现局部插值任意拓扑的四边形网格顶点。首先该方法利用渐进迭代逼近方法的局部性质,在初始网格中选取若干控制顶点进行迭代调整,保持其他顶点不变,使得最终生成的极限细分曲面插值于初始网格中的被调整点;其次该方法的 Catmull-Clark 细分的形状控制建立在两步细分的基础上,第一步通过对初始网格应用改造的 Catmull-Clark 细分产生新的网格,第二步对新网格应用 Catmull-Clark 细分生成极限曲面,改造的 Catmull-Clark 细分为每个网格面加入参数值,这些参数值为控制局部插值曲面的形状提供了自由度。证明了基于形状控制的 Catmull-Clark 细分局部渐进插值方法的收敛性。实验结果验证了该方法可同时实现局部插值和形状控制。     

Loop subdivision surfaces interpolating -spline curves

This paper presents a novel method for defining a Loop subdivision surface interpolating a set of popularly-used cubic B-spline curves. Although any curve on a Loop surface corresponding to a regular edge path is usually a piecewise quartic polynomial curve, it is found that the curve can be reduced to a single cubic B-spline curve under certain constraints of the local control vertices. Given a set of cubic B-spline curves, it is therefore possible to define a Loop surface interpolating the input curves by enforcing the interpolation constraints. In order to produce a surface of local or global fair effect, an energy-based optimization scheme is used to update the control vertices of the Loop surface subjecting to curve interpolation constraints, and the resulting surface will exactly interpolate the given curves. In addition to curve interpolation, other linear constraints can also be conveniently incorporated. Because both Loop subdivision surfaces and cubic B-spline curves are popularly used in engineering applications, the curve interpolation method proposed in this paper offers an attractive and essential modeling tool for computer-aided design.     

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.