A bound on the distance between Loop subdivision surface and its control mesh

By means of direct analysis of the connection between Loop subdivision surface and its control mesh and the computation of the basis functions,we obtain a bound on the distance between Loop subdivision surface patch and its control mesh.The bound can be used to compute the numbers of subdivision for a given tolerance.Finally,two examples are listed in this paper to demonstrate the applications of the bound.     

Improved error estimate for extraordinary Catmull–Clark subdivision surface patches

Based on an optimal estimate of the convergence rate of the second order norm, an improved error estimate for extraordinary Catmull–Clark subdivision surface (CCSS) patches is proposed. If the valence of the extraordinary vertex of an extraordinary CCSS patch is even, a tighter error bound and, consequently, a more precise subdivision depth for a given error tolerance, can be obtained. Furthermore, examples of adaptive subdivision illustrate the practicability of the error estimation approach.     

Subdivision depth computation for n-ary subdivision curves/surfaces

This paper deals with subdivision depth computation technique for n-ary subdivision curves/surfaces. This technique also includes error bound evaluation technique for n-ary subdivision curves/surfaces with their control polygon. Both techniques provide error control tools in subdivision schemes.     

可再生混合高阶指数多项式的插值细分法

通过引入新的形状控制参数,提出一类可以精确插值混合型指数多项式的非静态插值细分法。其基本思想是,通过生成指数多项式空间的指数B样条细分法,得到具有相同空间再生性的插值细分法。与具有相同再生性的其他插值细分法相比,所提细分法具有更小的支撑与更大的自由度。从理论上对细分法的再生性进行了分析,并进一步通过图例分析了初始形状控制参数及自由参数对极限曲线的影响。最后展示了取特殊的初始形状控制参数时,所提细分法对于一些特殊曲线的再生性。     

TSS BVHs: Tetrahedron Swept Sphere BVHs for Ray Tracing Subdivision Surfaces

We present a novel, compact bounding volume hierarchy, TSS BVH, for ray tracing subdivision surfaces computed by the Catmull‐Clark scheme. We use Tetrahedron Swept Sphere (TSS) as a bounding volume to tightly bound limit surfaces of such subdivision surfaces given a user tolerance. Geometric coordinates defining our TSS bounding volumes are implicitly computed from the subdivided mesh via a simple vertex ordering method, and each level of our TSS BVH is associated with a single distance bound, utilizing the Catmull‐Clark scheme. These features result in a linear space complexity as a function of the tree depth, while many prior BVHs have exponential space complexity. We have tested our method against different benchmarks with path tracing and photon mapping. We found that our method achieves up to two orders of magnitude of memory reduction with a high culling ratio over the prior AABB BVH methods, when we represent models with two to four subdivision levels. Overall, our method achieves three times performance improvement thanks to these results. These results are acquired by our theorem that rigorously computes our TSS bounding volumes.     

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.     

基于GPU的视点相关自适应细分

利用GPU的强大浮点数计算能力和并行处理能力,提出一种完全基于GPU的视点相关自适应细分内核进行快速细分计算的方法.在GPU中,依次实现视点相关的面片细分深度值计算、基于基函数表的细分表面顶点求值、细分表面绘制等核心步骤,无须与CPU端系统内存进行几何数据交换.视点相关的自适应细分准则在表面绘制精度保持不变的情况下,有效地降低了细分表面的细分深度和细分的计算量,在此基础上完全基于GPU的细分框架使得曲面细分具有快速高效的特点.该方法还可以在局部重要细节用较大深度值进行实时自适应细分,以逼近极限曲面.     

Feature-Adaptive Rendering of Loop Subdivision Surfaces on Modern GPUs

We present a novel approach for real-time rendering Loop subdivision surfaces on modern graphics hardware. Our algorithm evaluates both positions and normals accurately, thus providing the true Loop subdivision surface. The core idea is to recursively refine irregular patches using a GPU compute kernel. All generated regular patches are then directly evaluated and rendered using tile hardware tessellation unit. Our approach handles triangular control meshes of arbitrary topologies and incorporates common subdivision surface features such as semi-sharp creases and hierarchical edits. While surface rendering is accurate up to machine precision, we also enforce a consistent bitwise evaluation of positions and normals at patch boundaries. This is particularly useful in the context of displacement mapping which strictly requires inatching surface normals. Furthermore, we incorporate efficient level-of-detail rendering where subdivision depth and tessellation density can be adjusted on-the-fly. Overall, our algorithm provides high-quality results at real-time frame rates, thus being ideally suited to interactive rendering applications such as video games or authoring tools.     

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 .     

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     

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.     

Subdivision surfaces for CAD—an overview

Subdivision surfaces refer to a class of modelling schemes that define an object through recursive subdivision starting from an initial control mesh. Similar to B-splines, the final surface is defined by the vertices of the initial control mesh. These surfaces were initially conceived as an extension of splines in modelling objects with a control mesh of arbitrary topology. They exhibit a number of advantages over traditional splines. Today one can find a variety of subdivision schemes for geometric design and graphics applications. This paper provides an overview of subdivision surfaces with a particular emphasis on schemes generalizing splines. Some common issues on subdivision surface modelling are addressed. Several key topics, such as scheme construction, property analysis, parametric evaluation and subdivision surface fitting, are discussed. Some other important topics are also summarized for potential future research and development. Several examples are provided to highlight the modelling capability of subdivision surfaces for CAD applications.     

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。     

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。     

基于任意三角网格的ternary插值细分曲面造型及其分析

针对任意三角网格,提出一种简单有效且局部性更好的带参数的ternary插值曲面细分法,给出并证明了细分法收敛与G1连续的充分条件.在任意给定三角控制网格的条件下,可通过对形状参数的适当选择来实现对插值细分曲面形状的调整.     

曲面三参数binary细分法

在经典四点细分法的基础上,通过在曲线细分过程中引入三个参数,给出一种改进的细分曲线构造的算法,利用生成多项式等方法对细分法的一致收敛性、Ck连续性进行了分析。并把该方法扩展到曲面上,进而提出了曲面三参数binary细分法。在给定初始控制数据的条件下,可以通过对形状参数的适当选择来实现对细分极限曲面形状的调控。数值实验表明该算法较容易控制曲面形状,可方便地应用于工程实际,解决曲线、曲面位置调整和控制问题。     

基于双参数四点细分法的曲面造型

将双参数四点细分曲线方法进行推广，提出了基于双参数四点细分法的曲面造型方法，并对其收敛性进行了分析。该方法通过对两个参数的适当调节能够较容易地控制极限曲面的形状，极限曲面能够达到C4连续，可以应用到对曲面的连续性要求较高的曲面造型中去。在给定初始数据的条件下，可通过对形状参数的适当选择来实现对极限曲面的形状调整和控制，试验表明该算法生成光滑曲面是有效的。     

Construction of minimal subdivision surface with a given boundary

The fascinating characters of minimal surface make it to be widely used in shape design. While the flexibility and high quality of subdivision surface make it a powerful mathematical tool for shape representation. In this paper, we construct minimal subdivision surfaces with given boundaries using the mean curvature flow, a second order geometric partial differential equation. This equation is solved by a finite element method where the finite element space is spanned by the limit functions of an extended Loop’s subdivision scheme proposed by Biermann et al. Using this extended Loop’s subdivision scheme we can treat a surface with boundary, thereby construct the perfect minimal subdivision surfaces with any topology of the control mesh and any shaped boundaries.     

Integration of generalized B-spline functions on Catmull–Clark surfaces at singularities

Subdivision surfaces are a common tool in geometric modelling, especially in computer graphics and computer animation. Nowadays, this concept has become established in engineering too. The focus here is on quadrilateral control grids and generalized B-spline surfaces of Catmull–Clark subdivision type. In the classical theory, a subdivision surface is defined as the limit of the repetitive application of subdivision rules to the control grid. Based on Stam’s idea, the labour-intensive process can be avoided by using a natural parameterization of the limit surface. However, the simplification is not free of defects. At singularities, the smoothness of the classically defined limit surface has been lost. This paper describes how to rescue the parameterization by using a subdivision basis function that is consistent with the classical definition, but is expensive to compute. Based on this, we introduce a characteristic subdivision finite element and use it to discretize integrals on subdivision surfaces. We show that in the integral representation the complicated parameterization reduces to a decisive factor. We compare the natural and the characteristic subdivision finite element approach solving PDEs on surfaces. As model problem we consider the mean curvature flow, whereby the computation is done on the step-by-step changing geometry.