形状可调插值曲线曲面的参数选择

目的 因大多数插值基函数中的参数都是全局参数,从而导致插值曲线曲面的形状无法进行局部调整。另外,当插值曲线曲面形状可调时,也存在如何选择参数才能获得形状较为理想的曲线曲面的问题,为此给出一种无需反求控制顶点、包含局部形状调整参数、具有显式表达式、能重构部分二次曲线曲面的插值曲线曲面构造方法,同时给出易于使用的形状参数确定方案。方法 基于经典3次Hermite插值曲线的Bernstein基函数表达形式,将其中的Bernstein基换成已证明具有全正性的一组三角基函数,根据三角基的端点性质调整曲线表达式以保证其插值性,然后设定插值数据点处的导向量,在其中引入参数,并保证相邻曲线段之间的连续性,得到了一种新的三角基插值曲线。结果 新曲线可以整理成以待插值数据点为控制顶点与一组插值基函数的线性组合形式,插值基表达式简单,插值曲线含一组局部形状调整参数,一个参数的改变只影响一条曲线段的形状,相邻曲线段之间G¹连续,曲线可以重构椭圆。根据不同目标给出了3种用于确定曲线中形状参数的准则,每种准则都提供了可以直接使用的公式。相应的插值曲面具有与插值曲线类似的性质。结论 形状参数选取准则的给出使含参数插值曲线曲面的设计由随意变为确定,这使得采用本文方法更易于得到满意的结果。本文所给插值基函数的构造方法具有一般性,可以采用相同的思路构造其他函数空间上性质类似的插值基。     

散乱数据点多项式插值光顺曲面的构造

为了得到光顺的多项式插值曲面,首先把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,针对各数据点的邻接点个数不同,分别利用弯折能量和拉伸能量建立目标函数,极小化目标函数确定插值曲面的未知量,在保持原有的形状特征的同时构造光顺的分片插值曲面,最后用实例说明了文中方法的有效性．     

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．     

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．     

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.     

基于三次样条插值的亚像素边缘检测方法

为了实现图像边缘的亚像素定位,针对阶跃形式的边缘类型,提出了一种基于三次样条插值的亚像素边缘检测算法,根据样条插值原理,获得一维边缘的连续灰度分布,通过计算边缘点两侧三次样条函数的二阶导数为零点,实现一维边缘点的亚像素定位;利用一维曲面拟合方法,获得边缘投影方向,根据一维曲面投影方向像素灰度不变理论,利用投影公式将数字窗口中所有三维点投影到相同的投影平面,从而可将二维边缘检测问题转化为一维边缘检测,利用一维边缘检测方法获得边缘点位置,结合投影方向,最终实现二维边缘定位。通过与现有亚像素边缘检测方法的比较,可知算法对噪声具有较好的鲁棒性,同时计算时间相对较快,因此在实际应用中具有较好的适用性。     

分形插值曲面

研究一类迭代函数系统和分形插值曲面的构造，得到分形插值曲面面连续的充分必要条件，由此给出了简单的分形超限插值方法和新的网格点插值方法；从而利用网格曲线或网格点上的特征生成连续的分形插值曲面，这种简单有效的方法可广泛地用于分形造型和数据拟合。     

基于三角区域有理函数的图像自适应插值

基于有理函数模型提出了一种新的图像插值算法。此类有理函数基于三角区域构造 并且具有简洁且灵活的表达式,同时含有一个可调节参数,在不改变插值曲面输入数据的前提下, 可以通过调整参数来微调曲面弯曲程度从而达到更加理想的插值效果。首先将图像区域进行三角 剖分,将有理函数模型定义域转化到其特殊域（等腰直角三角形域）,通过区域变换使插值曲面 达到更好的连续性和光滑性,有效提升了插值精度;然后利用一种基于边缘走向的权值确定方法 分别确定各个三角域的权值;最后通过等值线分析将图像划分为不同区域,在平滑区域上随机选 择或者固定参数进行插值即可,在非平滑区域上则进行参数的最优化选取,使当前的插值曲面块 达到最优,进一步提升了插值精度。本文算法在边缘区域和纹理信息保持方面相对于传统插值算 法具有一定的优势,有效地消除了常见的振铃、走样等现象,并且具有良好的视觉效果。     

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.     

基于空间网格表面距离的插值曲面

主要讨论基于空间网格表面距离的插值曲面.定义了一种空间网格表面距离,基于空间网格表 面距离,给出了两种定义在空间网格上的插值曲面：基于网格表面距离的径向基函数插值曲 面和基于网格表面距离的Kriging插值曲面,讨论了曲面的无穷阶光滑性,给出了一种计算网 格表面距离的折线跟踪算法.作为整体曲面,此类插值曲面与网格表面形状较为相符,比较适 合于局部插值与参数估计,如梯度、法方向的估计等计算.     

曲线插值的一种保凸细分方法

为了弥补以四点插值细分方法为代表的线性细分方法在形状控制方面的缺陷,提出一种基于几何的插值型保凸细分方法.细分过程每一步中,每条边所对应的新控制顶点由原控制顶点及其切向共同确定;每点处的切向由其邻近的点所确定,并且随细分过程逐步调整.理论分析表明,该方法的极限曲线是G1连续的保凸曲线.如果所有的初始点取自圆弧段,则极限曲线就是该圆弧段.数值实例表明,采用文中方法得到的曲线较为光顺.     

Non-four-sided patch expressions with control points

This paper introduces new mathematical expressions which describe surface patches with three, five or six sides. Each side of a patch is given as a space curve expressed in Bézier type of second or third degree. Patch shape is modified predictably by moving its control points in the same way as with a four-sided patch. Distribution of cross tangent vectors along a patch boundary is also expressed in the same form with a few control vectors belonging to the boundary concerned. Therefore it can be connected with its surrounding four-sided patches smoothly. A general method of derivation of these equations is explained for further applications.     

Bezier曲面的Hermite方法

基于Kirov定理,研究带有附加导数条件的Bezier曲(线)面。该方法可以在每个型值点再给出导数条件,因此与通常的Bezier曲面拟合相比,有更多的自由度,但其拟合曲面的次数仅比Bezier曲面高一次。这一方法有助于CAGD领域的工程人员采用Bezier技术达到控制所设计曲面形状的目的。     

基于Helmholtz 方程的过渡曲面设计

阐述了二阶和四阶Helmholtz 方程的一类周期边界问题的差分解法及其在 过渡曲面设计中的应用。这类方法不同于传统的PDE 方法中的二阶和四阶的偏微分方程, 比传统的二阶和四阶偏微分方程有了更多的自由项,因此,在曲面设计的时候,就有更多的 形状控制参数可进行调整,文中重点讨论了方程中的系数对曲面形状的影响,并研究了边界 切矢条件对曲面形状的影响及其在曲面形状设计中的应用。设计者只需给出边界曲线和边界 切矢,并通过对它们的控制就可构造和修改曲面形状。     

三次三角域Bézier 曲面的同次扩展

为了在不提升基函数次数的前提下赋予三次三角域Bézier 曲面形状调整的能力, 构造了一组含一个参数的三次双变量基函数,由之定义了由10 个控制顶点确定的三角域曲面 片。新曲面具有角点插值性,在角点处的切平面为由角点和其所在的两条边上与之相邻的两个 顶点确定的平面。改变参数取值,可以调整曲面形状。为了方便应用,给出了曲面片之间的G1 光滑拼接条件及曲面的几何迭代算法,分析了算法的收敛性以及收敛速度与参数取值之间的关 系。图例显示了所给方法的正确性和有效性。     

On the normal vector estimation for point cloud data from smooth surfaces

Reliable estimation of the normal vector at a discrete data point in a scanned cloud data set is essential to the correct implementation of modern CAD/CAM technologies when the continuous CAD model representation is not available. A new method based on fitted directional tangent vectors at the data point has been developed to determine its normal vector. A local Voronoi mesh, based on the 3D Voronoi diagram and the proposed mesh growing heuristic rules, is first created to identify the neighboring points that characterize the local geometry. These local Voronoi mesh neighbors are used to fit a group of quadric curves through which the directional tangent vectors are obtained. The normal vector is then determined by minimizing the variance of the dot products between a normal vector candidate and the associated directional tangent vectors. Implementation results from extensive simulated and practical point cloud data sets have demonstrated that the present method is robust and estimates normal vectors with reliable consistency in comparison with the existing plane fitting, quadric surface fitting, triangle-based area weighted average, and triangle-based angle weighted average methods.     

海量散乱点的曲面重建算法研究

基于海量散乱点的曲面重建在机械产品测量造型、计算机视觉、根据切片数据的医学图像重建等领域中有重要应用.给出了一种以物体表面上不附加任何几何和拓扑信息(包括测点法矢、曲面边界信息)的散乱点集为处理对象,自动生成物体表面的三角网格模型的算法.该算法首先根据测点的邻近测点估算曲面在该测点处的法矢,并采用优化的顺序对法矢方向进行调整以使各测点处的法矢都指向曲面外侧,最后用步进立方体算法输出三角网格模型.采用新的方法计算切平面,不但进一步提高了效率,而且改善了曲面边界及尖锐棱边区域的重建效果.还提出并解决了法矢方向传播中可能出现的局部“孤岛”问题.同时,提出了一种对海量数据进行空间划分的算法,从而大大提高了海量数据的处理效率.应用实例表明,算法效果良好     

On increasing the developability of a trimmed NURBS surface

Developable surfaces are desired in designing products manufactured from planar sheets. Trimmed non-uniform rational B-spline (NURBS) surface patches are widely adopted to represent 3D products in CAD/CAM. This paper presents a new method to increase the developability of an arbitrarily trimmed NURBS surface patch. With this tool, designers can first create and modify the shape of a product without thinking about the developable constraint. When the design is finished, our approach is applied to increase the developability of the designed surface patches. Our method is an optimisation-based approach. After defining a function to identify the developability of a surface patch, the objective function for increasing the developability is derived. During the optimisation, the positions and weights of the free control points are adjusted. When increasing the developability of a given surface patch, its deformation is also minimised and the singular points are avoided. G⁰ continuity is reserved on the boundary curves during the optimisetion, and the method to reserve G¹ continuity across the boundaries is also discussed in this paper. Compared to other existing methods, our approach solves the problem in a novel way that is close to the design convention, and we are dealing with the developability problem of an arbitrarily trimmed NURBS patch.     

Free-form object reconstruction from silhouettes, occluding edges and texture edges: a unified and robust operator based on duality

In this paper, the duality in differential form is developed between a 3D primal surface and its dual manifold formed by the surface's tangent planes, i.e., each tangent plane of the primal surface is represented as a four-dimensional vector which constitutes a point on the dual manifold. The iterated dual theorem shows that each tangent plane of the dual manifold corresponds to a point on the original 3D surface, i.e., the dual of the dual goes back to the primal. This theorem can be directly used to reconstruct 3D surface from image edges by estimating the dual manifold from these edges. In this paper we further develop the work in our original conference papers resulting in the robust differential dual operator. We argue that the operator makes good use of the information available in the image data, by using both points of intensity discontinuity and their edge directions; we provide a simple physical interpretation of what the abstract algorithm is actually estimating and why it makes sense in terms of estimation accuracy; our algorithm operates on all edges in the images, including silhouette edges, self occlusion edges, and texture edges, without distinguishing their types (thus resulting in improved accuracy and handling locally concave surface estimation if texture edges are present); the algorithm automatically handles various degeneracies; and the algorithm incorporates new methodologies for implementing the required operations such as appropriately relating edges in pairs of images, evaluating and using the algorithm's sensitivity to noise to determine the accuracy of an estimated 3D point. Experiments with both synthetic and real images demonstrate that the operator is accurate, robust to degeneracies and noise, and general for reconstructing free-form objects from occluding edges and texture edges detected in calibrated images or video sequences.     

Application of splitting scheme and multigrid method for TV-Stokes denoising

Based on some previous work on the connection between image restoration and fluid dynamics,we apply a two-step algorithm for image denoising.In the first step,using a splitting scheme to study a nonlinear Stokes equation,tangent vectors are obtained.In the second step,an image is restored to fit the constructed tangent directions.We apply a fixed point iteration to solve the total variation-based image denoising problem,and use algebraic multigrid method to solve the corresponding linear equations.Numerical...