任意三角形网格的基于二元四次箱样条分片C1曲面

提出一种以任意三角剖分为控制网格的二元箱样条曲面算法.二元三方向剖分是方向最少的三角剖分,建立在其上的二元三向四次箱样条在CAGD等领域有着广泛的应用.其规范的箱样条曲面计算仅适用于控制点的价数均为6的网格.从规范的算法出发,提出了一种任意价数控制网格的曲面计算算法,并对算法的连续性等进行了详细的分析.生成的曲面具有保凸性,且是分片C1连续的.该算法可进行3D离散点全局或局部插值,并可应用于3D曲面重构等领域.     

基于C-B样条的Catmull-Clark细分曲面

为了解决 Catum ull- Clark细分曲面在工程上难以推广的问题 ,给出了一种基于 C- B样条的 Catumull-Clark细分曲面的算法 .C- B样条曲线是 B样条曲线的拓广 ,但它们的形状依赖于参数 α.由于新的曲面细分方法充分利用 C- B样条能够精确表示圆、椭圆等规则形体的特性 ,因而使通过此方法生成的细分曲面 ,除了在奇异点处能保持二阶导数连续外 ,还能够像 C- B样条曲线、曲面一样 ,精确地表示圆柱等常规曲面、统一工程曲面等的造型 ;同时它仍然保持细分曲面的造型特点 ,即能够解决 NU RBS曲面难以处理的任意拓扑结构的造型问题 ,另外 ,还可依赖控制参数 α的调节作用来增加造型的自由度 ,而且当 α→ 0时 ,它们就退化成 Catm ul- Clark细分曲面 .在工程图形上的应用实例表明 ,这种算法简单、有效 .     

针对密集点云的快速曲面重建算法

为了能够快速地从高密度散乱点云生成三角形网格曲面,提出一种针对散乱点云的曲面重建算法.首先通过逐层外扩建立原始点云的近似网格曲面,然后对近似网格曲面进行二次剖分生成最终的精确曲面;为了能够处理噪声点云,在剖分过程中所有网格曲面顶点都通过层次B样条进行了优化.相比于其他曲面重建方法,该算法剖分速度快,且能够保证点云到所生成的三角网格曲面的距离小于预先设定容限.实验结果表明,文中算法能够有效地实现高密度散乱点云的三角剖分,且其剖分速度较已有算法有大幅提高.     

NURBS曲面的有限元网格三角剖分

主要介绍一种ＮＵＲＢＳ曲面的有限元网格三角剖分算法。首先讨论ＮＵＲＢＳ曲面的离散算法，接着在此基础上，提出了利用网格前沿技术剖分ＮＵＲＢＳ曲面的算法，并且网格单元和结点同时生成     

NURBS曲面的有限元网格三角划分

主要介绍一种ＮＵＲＢＳ曲面的有限元网格三角剖分算法，首先讨论ＮＵＲＢＳ曲面的离散算法，接着在此基础上，提出了利用网格前沿技术剖分ＮＵＲＢＳ曲面的算法，并且网格单元和结点同时生成。     

任意三角形网格的基于二元四次箱样条分片C1曲面

提出一种以任意三角剖分为控制网格的二元箱样条曲面算法.二元三方向剖分是方向最少的三角剖分,建立在其上的二元三向四次箱样条在CAGD等领域有着广泛的应用.其规范的箱样条曲面计算仅适用于控制点的价数均为6的网格.从规范的算法出发,提出了一种任意价数控制网格的曲面计算算法,并对算法的连续性等进行了详细的分析.生成的曲面具有保凸性,且是分片C¹连续的.该算法可进行3D离散点全局或局部插值,并可应用于3D曲面重构等领域.     

参数曲面的有限元网格化

基于推进波前法,本文提出了一种针对三维Trimmed参数曲面的有限元网格剖分方法,首先对曲面参数空间进行剖分,利用结点密度函数（与曲率有关）和生成内部结点的公式,使网格单元和结点能同时生成,然后把网格反映射到曲面上,从而实现参数曲面的三角形网格剖分。     

Bézier三角曲面片生成显示与求交算法

本文在证明了Bézier三角曲面片的中点部分网格收敛性质的基础上,通过中点剖分算法给出了Bézier三角曲面片的生成显示算法与求交算法。     

基于黎曼度量的复杂参数曲面有限元网格生成方法

给出了三维空间的黎曼度量和曲面自身的黎曼度量相结合的三维复杂参数曲面自适应网格生成的改进波前推进算法.详细阐述了曲面参数域上任意一点的黎曼度量的计算和插值方法;采用可细化的栅格作为背景网格,在降低了程序实现的难度的同时提高了网格生成的速度;提出按层推进和按最短边推进相结合的方法,在保证边界网格质量的同时,提高曲面内部网格的质量.三维自适应黎曼度量的引入,提高了算法剖分复杂曲面的自适应性.算例表明,该算法对复杂曲面能够生成高质量的网格,而且整个算法具有很好的时间特性和可靠性.     

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.     

Progressive Interpolation based on Catmull‐Clark Subdivision Surfaces

We introduce a scheme for constructing a Catmull‐Clark subdivision surface that interpolates the vertices of a quadrilateral mesh with arbitrary topology. The basic idea here is to progressively modify the vertices of an original mesh to generate a new control mesh whose limit surface interpolates all vertices in the original mesh. The scheme is applicable to meshes with any size and any topology, and it has the advantages of both a local scheme and a global scheme.     

A Colour Interpolation Scheme for Topologically Unrestricted Gradient Meshes

Gradient meshes are a 2D vector graphics primitive where colour is interpolated between mesh vertices. The current implementations of gradient meshes are restricted to rectangular mesh topology. Our new interpolation method relaxes this restriction by supporting arbitrary manifold topology of the input gradient mesh. Our method is based on the Catmull‐Clark subdivision scheme, which is well‐known to support arbitrary mesh topology in 3D. We adapt this scheme to support gradient mesh colour interpolation, adding extensions to handle interpolation of colours of the control points, interpolation only inside the given colour space and emulation of gradient constraints seen in related closed‐form solutions. These extensions make subdivision a viable option for interpolating arbitrary‐topology gradient meshes for 2D vector graphics.     

Efficient wavelet construction with Catmull–Clark subdivision

This paper presents an efficient biorthogonal wavelet construction with the generalized Catmull–Clark subdivision based on the lifting scheme. The subdivision wavelet construction scheme is applicable to all variants of Catmull–Clark subdivision, so it is more universal than the previous wavelet construction for the generalized bicubic B-spline subdivision. Because the analysis and synthesis algorithms of the wavelets are composed of a series of local and in-place lifting operations, they can be performed in linear time. The experiments have demonstrated the stability of the proposed wavelet analysis based on the ordinary Catmull–Clark subdivision. Moreover, the resulting Catmull–Clark subdivision wavelets have better fitting quality than the generalized bicubic B-spline subdivision wavelets at a similar computation cost.     

Implicit Hierarchical Quad‐Dominant Meshes

We present a method for producing quad‐dominant subdivided meshes, which supports both adaptive refinement and adaptive coarsening. A hierarchical structure is stored implicitly in a standard half‐edge data structure, while allowing us to efficiently navigate through the different level of subdivision. Subdivided meshes contain a majority of quad elements and a moderate amount of triangles and pentagons in the regions of transition across different levels of detail. Topological LOD editing is controlled with local conforming operators, which support both mesh refinement and mesh coarsening. We show two possible applications of this method: we define an adaptive subdivision surface scheme that is topologically and geometrically consistent with the Catmull–Clark subdivision; and we present a remeshing method that produces semi‐regular adaptive meshes.     

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.     

A subdivision algorithm for axisymmetric sections

In the generation of curved surfaces through a subdivision process, Sabin and Doo applied and extended Chaikin's algorithm to three dimensions by using linear combinations of the vertices of a polyhedron. A similar smoothing subdivision algorithm was brought out by Catmull and Clark. This paper describes an alternative algorithm which uses a similar approach but applies to sections of axisymmetric objects. It shows that axisymmetric free-formed surfaces can be generated easily and effeciently.     

Subdivision Schemes for Quadrilateral Meshes with the Least Polar Artifact in Extraordinary Regions

This paper presents subdivision schemes with subdivision stencils near an extraordinary vertex that are free from or with substantially reduced polar artifact in extraordinary regions while maintaining the best possible bounded curvature at extraordinary positions. The subdivision stencils are firstly constructed to meet tangent plane continuity with bounded curvature at extraordinary positions. They are further optimized towards curvature continuity at an extraordinary position with additional measures for removing or for minimizing the polar artifact in extraordinary regions. The polar artifact for subdivision stencils of lower valences is removed by applying an additional constraint to the subdominant eigenvalue to be the same as that of subdivision at regular vertices, while the polar artifact for subdivision stencils of higher valances is substantially reduced by introducing an additional thin‐plate energy function and a penalty function for maintaining the uniformity and regularity of the characteristic map. A new tuned subdivision scheme is introduced by replacing subdivision stencils of Catmull‐Clark subdivision with that from this paper for extraordinary vertices of valences up to nine. We also compare the refined meshes and limit surface quality of the resulting subdivision scheme with that of Catmull‐Clark subdivision and other tuned subdivision schemes. The results show that subdivision stencils from our method produce well behaved subdivision meshes with the least polar artifact while maintaining satisfactory limit surface quality.     

Ternary subdivision for quadrilateral meshes

A well-documented problem of Catmull and Clark subdivision surfaces is that, in the neighborhood of extraordinary points, the curvature is unbounded and fluctuates. In fact, since one of the eigenvalues that determines elliptic shape is too small, the limit surface can have a saddle point when the designer's input mesh suggests a convex shape. Here, we replace, near the extraordinary point, Catmull–Clark subdivision by another set of rules based on refining each bi-cubic B-spline into nine. This provides many localized degrees of freedom for special rules that need not reach out to neighbor vertices in order to tune the behavior. In this paper, we provide a strategy for setting such degrees of freedom and exhibit tuned ternary quad subdivision that yields surfaces with bounded curvature, nonnegative weights and full contribution of elliptic and hyperbolic shape components.     

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.     

Smooth spline surface generation over meshes of irregular topology

An efficient method for generating a smooth spline surface over an irregular mesh is presented in this paper. Similar to the methods proposed by [1, 2, 3, 4], this method generates a generalised bi-quadratic B-spline surface and achieves C ¹ smoothness. However, the rules to construct the control points for the proposed spline surfaces are much simpler and easier to follow. The construction process consists of two steps: subdividing the initial mesh once using the Catmull–Clark [5] subdivision rules and generating a collection of smoothly connected surface patches using the resultant mesh. As most of the final mesh is quadrilateral apart from the neighbourhood of the extraordinary points, most of the surface patches are regular quadratic B-splines. The neighbourhood of the extraordinary points is covered by quadratic Zheng–Ball patches [6].