散乱数据点分片二次多项式加权平均插值

将空间散乱数据点划分为三角形网格，在每个给定数据点处构造C^1连续的分片二次多项式曲面片，每个三角形上的曲面片由各个顶点处的C^1连续的分片二次曲面片加权平均确定，整体的C^1曲面由各三角形上的曲面片拼合而成．该方法所构造的曲面函数结构简单、易于计算，具有数据点建议的形状．最后通过实例同其他方法所构造的插值曲面形状进行比较．     

散乱数据点的三次多项式插值

用分片三次多项式曲面对散乱分布数据点插值的方法把给定区域划分成三角形网格，在每个三角形上构造一个三次多项式曲面片，整体的Ｃ１曲面由各三角形上的曲面片拼合而成．讨论了整体Ｃ１曲面需满足的条件组成的方程组的性质，并给出了求解方程组的方法．插值方法的多项式准确集包括所有三次和小于三次的多项式．     

基于径向基函数与B样条的散乱数据拟合方法

针对散乱数据的曲面拟合问题,提出一种径向基函数与B样条插值结合使用的曲面拟合方法.通过分片径向基函数插值,三维散乱点,再从分片插值曲面上获取预先设定好的有序网格点的值,最后利用张量积B样条插值有序网格点,从而得到拟合曲面.该方法较好地解决散乱数据插值和拟合的计算不稳定性问题,最后给出算法实例.     

散乱分布数据曲面重构的光顺-有限元方法

提出了一种基于散乱分布的数据点重构三维曲面的有限元方法.根据最佳逼近与数据光顺理论建立正定的目标泛函,采用有限元最佳拟合使泛函极小化,求得最优解.通过八节点等参数有限元插值计算,重新构造出三维曲面.这种光顺-有限元方法有效地抑制了输入数据上误差噪声的影响,与有限元拟合方法相比,所需的输入数据点少,重构的曲面逼近精度高、光顺性好.数值实验表明,该方法简单,便于应用.     

基于组合方法对图像的双三次多项式拟合

提出了基于图像数据构造拟合曲面的新方法.假设给定的图像数据所对应的原场景曲面能用分片二次多项式曲面表示,原场景曲面称为原曲面.现有方法通常是用图像数据作为插值数据构造原曲面的拟合曲面,而新方法以图像数据生成公式为约束,通过反向采样过程来构造对图像数据拟合的曲面,从而使拟合曲面具有更好的逼近精度.由于图像边缘处的质量对图像的视觉效果起着关键的作用,我们也把图像边缘做为约束条件用于拟合曲面的构造.对于每一个数据点及其邻近区域,新方法以采样公式和图像边缘作为约束条件局部地构造一张二次多项式曲面片,该曲面片具有二次多项式精度.所有的二次多项式曲面片的加权组合形成逼近原曲面的拟合曲面.对算法比较的实验表明,由新方法生成的放大图像具有较高的精度和较好的视觉效果.     

过测地线的优化曲面设计

给定一条曲线,构造以其为测地线的曲面,这是服装鞋帽类产品的设计/制造业中的一个现实课题.已有研究结果是构造出以给定曲线为公共测地线的曲面束,然后用拟合数据点的方法来确定最终曲面.这种通用方法受到曲面参数及其表示方法的影响,且没有对曲面的光顺程度加以考虑.从服装材料的特性和设计思想出发,提出一种利用能量优化来确定最终曲面的新方法.通过改变曲面表示形式和引入能量函数,方便而有效地确定了过给定测地线的一张优化曲面.给出了在插值拟合等约束条件下的相应算法.实例表明,所给算法很好地模拟了成衣的光顺设计与加工,在计算机辅助设计/制造(CAD/CAM)中富有应用价值.     

在三角域上构造三次多项式插值曲面

为满足矿山地形的拟合、水流深度的绘制等很多特殊工程数据量大、有一定的光顺要求但又不需要曲面过于凸起饱满这一需求,提出一种C1连续的三次多项式插值曲面,同时有针对性地提出一种一阶偏导数估计算法.首先将空间散乱数据点投影到平面后进行三角划分;其次针对每个三角形,在其每条边上构造一个C1连续的三次多项式曲面片,由这3个曲面片加权平均形成该三角形的曲面片;最后将所有三角形上的曲面片拼合成整体曲面.为使生成的曲面尽可能地贴近数据点所建议的形状,在曲面求解过程中将数据点分成内部点和边界点分别估计偏导数.实验结果表明,该算法计算量小、具有良好的局部性,并给出了新曲面的效果.     

应用可变形模型进行曲线曲面光顺

根据物理模型的造型技术可以从另一个角度研究曲线曲面的光顺问题.基于能量的可变形曲线曲面可用于光顺一批有误差的散乱数据点.该方法不必要求数据点有序的条件，通过在数据点与曲线和曲面之间连接假想弹簧，可以克服“光顺”和“插值”的矛盾，得到所需形状.文章采用三次B样条几何表达形式，建立了光顺模型，分析了光顺机理，并给出了一个算法和几个应用实例.     

具有多种优点的三角多项式曲线曲面

为了将形状可调性、高阶连续性、自动插值性,以及可以精确表示圆锥曲线曲面等性质融入到一种曲线曲面模型中,构造了一组带2个参数的5次三角多项式调配函数,分析了该调配函数的性质.基于该函数组,分别采用与3次B样条曲线、曲面相同的定义方式,定义了基于4点分段的曲线,并且基于16点分片的曲面,给出了曲线、曲面的性质.曲线、曲面的分段、分片组合结构决定了它们具有B样条方法的局部性.讨论了参数取值的改变对曲线形状的影响;证明了在取特殊参数时曲线可以达到G5或G7的高阶连续性,而且在具有G5连续性时仍然具有形状可调性;通过将2个参数中的一个取为特殊值,即可使曲线、曲面自动插值给定点列、网格点,这种方式不需要反求控制顶点,且插值曲线、曲面中依然存在调整形状的自由度;分别给出了曲线、曲面精确表示椭圆、椭球面的条件.数值实例结果显示了所给曲线曲面表示方法的正确性和有效性.     

集逼近插值于一体的分段3次多项式曲线曲面

为了用一种模型实现逼近与插值的统一,在多项式函数空间上构造了含两组参数的混合函数,并由之定义了基于四点分段的多项式曲线和相应的张量积曲面。当参数取特殊值时,新曲线曲面成为3次均匀B样条曲线曲面。除了继承B样条方法的局部性,自动光滑性等优点之外,新曲线曲面还具有局部形状可调性。限制混合函数中参数的取值范围,可以使新曲线曲面位于控制顶点的凸包内。让混合函数中的一组参数取特定值,可以使新曲线曲面自动插值除边界点以外的控制顶点,且插值曲线曲面的形状依然局部可调。给出了一些曲线曲面图例。     

三角形域上C1连续的四次插值曲面

提出了一种在三角形域上构造C^1曲面的方法，该方法构造的曲面片由4个曲面加权平均产生，在三角形的边界上满足给定的边界曲线和一阶跨界导数．所构造的曲面可看作由一张基本曲面和三张过渡曲面构成．用三条曲线相交于一点且在交点处共面作为约束条件构造基本曲面，在三角形的内部具有较好形状和逼近精度．同边点法相比，文中方法产生的曲面形状更好；且该方法产生的曲面对四次多项式曲面是精确的，因而比Nielson的点边方法具有更高的插值精度．     

Nonuniform B-spline Mesh Fairing Method

A method for fairing a surface composed of a set of discrete data points distributed in anonrectangular topological mesh is presented.All curves are expressed by nonuniform cubic B-splinecurves.The fairing method is minimizing the elastic strain energy of mesh curves and of springs at-tached to the data points.The fairing surface can be generated by interpolating through the meshcurves.The generation and fairing of a ship hull surface is given as an example.     

An Algorithm for Parametric Quadric Patch Construction

An algorithm for constructing arbitrary parametric quadratic quadric triangles in rational Bézier form is presented. The algorithm does not require the knowledge of the underlying quadric, an important property in view of applying this method for the interpolation of triangulated 3D data points. The algorithm consists of four steps starting with the arbitrary choice of the three corner points and corner weights of the patch, by then constructing a certain triangle and a tetrahedron by means of which the remaining inner control points and weights are obtained guaranteeing the resulting patch to lie on a quadric surface.     

Adaptive fairing of digitized point data with discrete curvature

An algorithm for fairing two-dimensional (2D) shape formed by digitised data points is described. The application aims to derive a fair curve from a set of dense and error-filled data points digitised from a complex surface, such that the basic shape information recorded in the original point data is relatively unaffected. The algorithm is an adaptive process in which each cycle consists of several steps. Given a 2D point set, the bad points are identified by analysing the property of their discrete curvatures (D-curvatures) and first-order difference of D-curvatures, in two consecutive fairing stages. The point set is then segmented into single bad point (SBP) segments and multiple bad point (MBP) segments. For each MBP segment, a specially designed energy function is used to identify the bad point to be modified in the current cycle. Each segment is then faired by directly adjusting the geometric position of the worst point. The amount of adjustment in each cycle is kept less than a given shape tolerance. This algorithm is particularly effective in terms of shape preservation when dealing with MBP segments. Case studies are presented that illustrate the efficacy of the developed technique.     

Shape aware normal interpolation for curved surface shading from polyhedral approximation

Independent interpolation of local surface patches and local normal patches is an efficient way for fast rendering of smooth curved surfaces from rough polyhedral meshes. However, the independently interpolating normals may deviate greatly from the analytical normals of local interpolating surfaces, and the normal deviation may cause severe rendering defects when the surface is shaded using the interpolating normals. In this paper we propose two novel normal interpolation schemes along with interpolation of cubic Bézier triangles for rendering curved surfaces from rough triangular meshes. Firstly, the interpolating normal is computed by a Gregory normal patch to each Bézier triangle by a new definition of quadratic normal functions along cubic space curves. Secondly, the interpolating normal is obtained by blending side-vertex normal functions along side-vertex parametric curves of the interpolating Bézier surface. The normal patches by these two methods can not only interpolate given normals at vertices or boundaries of a triangle but also match the shape of the local interpolating surface very well. As a result, more realistic shading results are obtained by either of the two new normal interpolation schemes than by the traditional quadratic normal interpolation method for rendering rough triangular meshes.     

类螺旋特征测点数据的闭曲面建模方法研究

复杂曲面及海量点云测量数据的曲面建模已成为通用CAD/CAM软件的重要功能;然而,对于复杂的闭曲面建模方法,仍然存在许多技术上的难题,至今尚未能很好的解决,比如,基于海量的测量数据,如何进行闭曲面特征点识别,如何进行区域分割与处理,这一切都使得闭曲面建模过程中很难采用已经成熟的自由曲面建模技术和方法.通过研究异步仿形测量原理以及测量数据类型,针对鞋楦测量形成的空间螺旋线数据特征,提出一种闭曲面建模方法.该方法包括如下步骤:首先对测量点数据处理;并以特征螺旋线数据为基础对曲面进行三角分割;最后,以三角Bezier曲面为基础进行曲面构造,并将各曲面进行拼接、裁剪,形成完整的曲面.采用该方法对鞋楦测量数据的建模实例说明,能够有效地对具有空间螺旋线数据特征的闭曲面进行数据处理、曲面重构,提高了产品建模效率.     

B-spline surface fitting by iterative geometric interpolation/approximation algorithms

Recently, the use of $B$-spline curves/surfaces to fit point clouds by iteratively repositioning the $B$-spline’s control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a $B$-spline surface that passes through the data points, whereas approximation generates a $B$-spline surface that passes near the data points, minimizing the deviation of the surface from the data points. For surface interpolation, the data points are assumed to be in grids, whereas for surface approximation the data points are assumed to be randomly distributed. In this paper, an iterative geometric interpolation method, as well as an approximation method, which is based on the framework of the iterative geometric interpolation algorithm, is discussed. These two iterative methods are compared with standard fitting methods using some complex examples, and the advantages and shortcomings of our algorithms are discussed. Furthermore, we introduce two methods to accelerate the iterative geometric interpolation algorithm, as well as a method to impose geometric constraints, such as reflectional symmetry, on the iterative geometric interpolation process, and a novel fairing method for non-uniform complex data points. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms.     

散乱数据曲面造型的凸组合Bezier三角曲面片插值方法

本文讨论了在计算机辅助设计和计算机图形学的散乱数据曲面造型中一种有效的凸组合Ｂｅｚｉｅｒ三角曲面片插值方法。构造曲面的方法是对三角剖分的每一条边,得到一个插值已知条件的５次Ｂｅｚｉｅｒ三角曲面片,它与共此边的相邻５次Ｂｅｚｉｅｒ三角曲面片满足Ｃ＾２连续条件,然后对三角剖分的每一个三角形,将三边对应的Ｂｅｚｉｅｒ三角曲面片作凸组合,使之仍然插值已知条件并满足Ｃ＾２连续条件,从数值例子看,效果是不错的     

Hermite interpolation by triangular cubic patches with small Willmore energy

In this paper, a construction of a cubic Bézier spline surface that interpolates prescribed spatial points and the corresponding normal directions of tangent planes is proposed. Boundary curves of each triangular patch minimize the approximated strain energy. A comparison of optimal boundary curves is given. The interpolant minimizes Willmore energy functional. Some numerical examples and applications of the interpolation scheme are presented: surface approximation, hole filling and condensation of parameters.