Dynamic modeling of butterfly subdivision surfaces

The authors develop integrated techniques that unify physics based modeling with geometric subdivision methodology and present a scheme for dynamic manipulation of the smooth limit surface generated by the (modified) butterfly scheme using physics based “force” tools. This procedure based surface model obtained through butterfly subdivision does not have a closed form analytic formulation (unlike other well known spline based models), and hence poses challenging problems to incorporate mass and damping distributions, internal deformation energy, forces, and other physical quantities required to develop a physics based model. Our primary contributions to computer graphics and geometric modeling include: (1) a new hierarchical formulation for locally parameterizing the butterfly subdivision surface over its initial control polyhedron, (2) formulation of dynamic butterfly subdivision surface as a set of novel finite elements, and (3) approximation of this new type of finite elements by a collection of existing finite elements subject to implicit geometric constraints. Our new physics based model can be sculpted directly by applying synthesized forces and its equilibrium is characterized by the minimum of a deformation energy subject to the imposed constraints. We demonstrate that this novel dynamic framework not only provides a direct and natural means of manipulating geometric shapes, but also facilitates hierarchical shape and nonrigid motion estimation from large range and volumetric data sets using very few degrees of freedom (control vertices that define the initial polyhedron)     

OpenGL与VRML在细分几何造型中的应用

细分曲面的生成被广泛应用于计算机图形研究和几何建模应用。本文以Loop细分模式为例，研究了使用OpenGL与VRML在Windows环境下细分曲面的生成。介绍了细分造型技术的原理和OpenGL与VRML文件的结构，并利用它们强大的功能开发出细分曲面造型的系统。     

基于顶点法向量约束的Catmull-Clark细分插值方法

提出一种基于顶点法向量约束实现插值的两步Catmull-Clark细分方法.第一步,通过改造型Catmull-Clark细分生成新网格.第二步,通过顶点法向量约束对新网格进行调整.两步细分分别运用渐进迭代方法和拉格朗日乘子法,使得极限曲面插值于初始控制顶点和法向量.实验结果证明了该方法可同时实现插值初始控制顶点和法向量,极限曲面具有较好的造型效果.     

实现逼近细分模式的统一分解架构

多边形是计算机图形学的一个普遍的建模原语,为渲染多边形而量身度制的图形硬件也已经成为现实。然而,在实现高度分片逼近光滑曲面时,使用多边形建模存在很多问题。这是因为这样的逼近往往含有数十万的多边形,使得设计者难以自由地控制形状。细分则是解决这个难题的新技术,细分曲面的生成也正被广泛地应用于计算机图形研究和几何建模应用,并将成为下一代几何建模原语。本文研究了使用具有分解因子的统一架构生成以逼近模式为例的多边形网格细分曲面建模,并且实现了基于四边形／三角形混合网格的细分。     

Interpolation over arbitrary topology meshes using a two-phase subdivision scheme

The construction of a smooth surface interpolating a mesh of arbitrary topological type is an important problem in many graphics applications. This paper presents a two-phase process, based on a topological modification of the control mesh and a subsequent Catmull-Clark subdivision, to construct a smooth surface that interpolates some or all of the vertices of a mesh with arbitrary topology. It is also possible to constrain the surface to have specified tangent planes at an arbitrary subset of the vertices to be interpolated. The method has the following features: 1) it is guaranteed to always work and the computation is numerically stable, 2) there is no need to solve a system of linear equations and the whole computation complexity is O(K) where K is the number of the vertices, and 3) each vertex can be associated with a scalar shape handle for local shape control. These features make interpolation using Catmull-Clark surfaces simple and, thus, make the new method itself suitable for interactive free-form shape design.     

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。     

A simple method for interpolating meshes of arbitrary topology by Catmull–Clark surfaces

Interpolating an arbitrary topology mesh by a smooth surface plays important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing Catmull–Clark surface that interpolates a given mesh. The control mesh of the interpolating surface is obtained by one Catmull–Clark subdivision of the given mesh with modified geometric rule. Two methods—push-back operation based method and normal-based method—are presented for the new geometric rule. The interpolation method has the following features: (1) Efficiency: we obtain a generalized cubic B-spline surface to interpolate any given mesh in a robust and simple manner. (2) Simplicity: we use only simple geometric rule to construct control mesh for the interpolating subdivision surface. (3) Locality: the perturbation of a given vertex only influences the surface shape near this vertex. (4) Freedom: for each edge and face, there is one degree of freedom to adjust the shape of the limit surface. These features make interpolation using Catmull–Clark surfaces very simple and thus make the method itself suitable for interactive free-form shape design.     

基于边界采样技术的插值Loop细分方法

本文在分析了传统几何造型的弊端及开曲面造型中光滑边界曲线的插值要求后,针对细分曲面造型方法中较常用的Loop细分,提出了基于边界采样技术的插值细分曲面造型方法。该方法一方面利用了细分曲面造型的优点,如算法简单、可表达任意拓扑结构等;另一方面又满足了工程应用中插值边界曲线的要求。文中详细讨论该算法的步骤,并通过示例验证了该算法的有效性和实用性。     

Adaptive and Feature-Preserving Subdivision for High-Quality Tetrahedral Meshes

We present an adaptive subdivision scheme for unstructured tetrahedral meshes inspired by the       -subdivision scheme for triangular meshes. Existing tetrahedral subdivision schemes do not support adaptive refinement and have traditionally been driven by the need to generate smooth three-dimensional deformations of solids. These schemes use edge bisections to subdivide tetrahedra, which generates octahedra in addition to tetrahedra. To split octahedra into tetrahedra one routinely chooses a direction for the diagonals for the subdivision step. We propose a new topology-based refinement operator that generates only tetrahedra and supports adaptive refinement. Our tetrahedral subdivision algorithm is motivated by the need to have one representation for the modeling, the simulation and the visualization and so to bridge the gap between CAD and CAE. Our subdivision algorithm design emphasizes on geometric quality of the tetrahedral meshes, local and adaptive refinement operations, and preservation of sharp geometric features on the boundary and in the interior of the physical domain.     

Real-time ray casting of algebraic B-spline surfaces

Piecewise algebraic B-spline surfaces (ABS surfaces) are capable of modeling globally smooth shapes of arbitrary topology. These can be potentially applied in geometric modeling, scientific visualization, computer animation and mathematical illustration. However, real-time ray casting the surface is still an obstacle for interactive applications, due to the large amount of numerical root findings of nonlinear polynomial systems that are required. In this paper, we present a GPU-based real-time ray casting method for ABS surfaces. To explore the powerful parallel computing capacity of contemporary GPUs, we adopt iterative numerical root-finding algorithms, e.g., the Newton-Raphson and regula falsi algorithms, rather than recursive ones. To facilitate convergence of the Newton-Raphson or regula falsi algorithm, their initial guesses are determined through rasterization of the isotopic isosurface, and the isosurface is generated based on regular criteria for surface domain subdivision. Meanwhile, polar surfaces are adopted to identify single roots or to isolate different roots, i.e., ray and surface intersections. As an important geometric feature, the silhouette curve is elaborately computed to floating-point accuracy, which can be applied in further anti-aliasing processes. The experimental results show that the proposed method can render thousands of piecewise algebraic surface patches of degrees 6-9 in real time.     

基于曲率流的四边形主导网格的光顺方法

网格模型是计算机图形学和数字几何处理中运用最为广泛的三维几何表达方式.四边网格(以四边形为主的网格)由于其符合人们对几何形状变化的自然感知,在表示三维几何上有其独有的优势,并且可以更为直接地应用在几何造型、细分曲面、建筑设计等方面.文中针对四边形主导网格含有噪声的情况,设计了一种基于表面微分属性的光顺方法,该方法具有易实现、计算效率高的特点.基于曲率流的几何扩散可以有效地保持原网格的几何特征,同时还针对四边形主导网格的T-顶点进行了特殊处理.     

Loop细分格式在基于Java 3D的几何造型系统中的应用

细分方法是一种新的形体表示方法，在计算机图形学和几何造型中有广泛应用。文中研究了Loop细分算法在基于Java 3D的几何造型系统中的应用，该算法可以快速生成具有真实感的任意复杂的形体。     

Globally optimal surface mapping for surfaces with arbitrary topology

Computing smooth and optimal one-to-one maps between surfaces of same topology is a fundamental problem in computer graphics and such a method provides us a ubiquitous tool for geometric modeling and data visualization. Its vast variety of applications includes shape registration/matching, shape blending, material/data transfer, data fusion, information reuse, etc. The mapping quality is typically measured in terms of angular distortions among different shapes. This paper proposes and develops a novel quasi-conformal surface mapping framework to globally minimize the stretching energy inevitably introduced between two different shapes. The existing state-of-the-art inter-surface mapping techniques only afford local optimization either on surface patches via boundary cutting or on the simplified base domain, lacking rigorous mathematical foundation and analysis. We design and articulate an automatic variational algorithm that can reach the global distortion minimum for surface mapping between shapes of arbitrary topology, and our algorithm is sorely founded upon the intrinsic geometry structure of surfaces. To our best knowledge, this is the first attempt towards numerically computing globally optimal maps. Consequently, our mapping framework offers a powerful computational tool for graphics and visualization tasks such as data and texture transfer, shape morphing, and shape matching.     

Interpolatory,solid subdivision of unstructured hexahedral meshes

This paper presents a new, volumetric subdivision scheme for interpolation of arbitrary hexahedral meshes. To date, nearly every existing volumetric subdivision scheme is approximating, i.e., with each application of the subdivision algorithm, the geometry shrinks away from its control mesh. Often, an approximating algorithm is undesirable and inappropriate, producing unsatisfactory results for certain applications in solid modeling and engineering design (e.g., finite element meshing). We address this lack of smooth, interpolatory subdivision algorithms by devising a new scheme founded upon the concept of tri-cubic Lagrange interpolating polynomials. We show that our algorithm is a natural generalization of the butterfly subdivision surface scheme to a tri-variate, volumetric setting.     

自由曲线曲面的任意次非均匀细分

提出一种有效的建模自由曲线曲面的非均匀细分算法。首先在节点插入技术基础上推导出任意次自由曲线的非均匀细分规则,然后把它推广到张量积曲面得到任意次自由曲面的非均匀细分规则,最后对奇异点附近曲面采用类Doo-Sabin和Catmull-Clark的细分规则,从而使该算法可以实现建模任意次具有任意拓扑基网格的非均匀细分曲面。此外,该方法也实现了对传统细分格式的统一,例如,当次数为2并采用均匀节点矢量便转化为Doo-Sabin细分,当次数为3并采用均匀节点矢量便转化为Catmull-Clark细分。     

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.     

自适应细分方法进行曲面造型

充分利用可调控CatmullClark细分规则与均匀的CatmullClark细分规则的优点,提出了自适应细分方法。该方法简单,比传统的单一细分方法有更好的灵活性,通过适当调节控制因子,可使得曲面造型比较灵活。通过分析曲面上点的曲率来控制细分,可以在较低的细分次数下达到良好的曲面造型效果,为曲面造型提供了一个新的方法。     

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.