Constructing triangular patch by basic approximation operator plus additional interpolation operator

~~Constructing triangular patch by basic approximation operator plus additional interpolation operator1. Barahill, R. E., Birkhoff, G., Gordon, W. J., Smooth interpolation in triangles, J. Approx. Theory, 1973, 8: 114-128. 2. Gregory, J. A., Smooth interpolation without twist constraints, in Computer Aided Geometric Design (eds. Barn-hill, R. E., Riesenfeld, R. R), New York: Academic Press, 1974, 71-88. 3. Charrot, P., Gregory, J. A., A pentagonal surface patch for comput…     

基于最佳平方逼近的B样条曲线降阶

提出了一种基于带约束的最佳平方逼近的B样条曲线降阶的方法.首先讨论了降阶后曲线控制顶点个数以及节点向量的取法、保端点的B样条曲线降阶方法,并把带约束的最佳平方逼近技术引入到B样条曲线的降阶,即误差大的区域施加较大的权函数以降低最大误差.为满足给定误差限制下的降阶,提出了对原曲线插入节点的准则,即对不满足误差限制的区域插入节点.并用实例对新方法和基于扰动约束技术的降阶方法进行了比较.     

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。     

Hermite approximation for free-form deformation of curves and surfaces

Free-Form Deformation Techniques (FFD) are commonly used to generate animations, where a polygonal approximation of the final object suffices for visualization purposes. However, for some CAD/CAM applications, we need an explicit expression of the object, rather than a collection of sampled points. If both object and deformation are polynomial, their composition yields a result that is also polynomial, albeit very high degree, something undesirable in real applications. To solve this problem, we transform each curve or surface composing the object, usually expressed in the Bernstein basis, to a modified Newton form. In this representation, the two-point analogue of Taylor expansions, the composition admits a simple expression in terms of discrete convolutions, and degree reduction corresponding to Hermite approximation is trivial by dropping high-degree coefficients. Furthermore, degree-reduction can be incorporated into the composition. Finally, the deformed curve or surface is converted back to the Bernstein form. This method extends to general non-polynomial deformation, such as bending and twisting, by computing a polynomial approximant of the deformation.     

Efficient selection, storage, and retrieval of irregularly distributed elevation data

The advantages and disadvantages of various digital terrain models are discussed briefly and a new model for triangulating a set of nonuniformly distributed three-dimensional surface observations is described. An algorithm for hierarchical subdivision of a set of data points into nested triangles is proposed. The algorithm selects a subset of the data points which reduce the maximum error between a piecewise linear approximation of the surface using only the selected points and the elevations of the points not selected. The data structure used is such that for any given degree of approximation (in the maximum-error sense) only the necessary points need to be stored. Furthermore, an efficient method is available to approximate the elevation of the surface at any point not included in the original data. The performance of the algorithm is demonstrated experimentally.     

带端点插值条件的Bézier曲线降多阶逼近

研究了两端点具有任意阶插值条件的Bézier曲线降多阶逼近的问题.对于给定的首末端点的各阶插值条件,给出了一种新的一次降多阶逼近算法,应用Chebyshev多项式逼近理论达到了满足端点插值条件下的近似最佳一致逼近.此算法易于实现,误差计算简单,且所得降阶曲线具有很好的逼近效果,结合分割算法,可获得相当高的误差收敛速度.     

Second order approximation of tool envelope surface for 5-axis machining with single point contact

For 5-axis machining with single point contact, this paper proposes a method to calculate second order approximation of the tool envelope surface by using only one tool position. As we known, the true machining errors are deviations between designed surface and tool envelope surface. But it is impossible to determine the whole shape of the tool envelope surface before all tool positions are obtained. Hence, it is difficult to position the tool individually and consider true errors at the same time. Basic Curvature Equations of Locally Tool Positioning (BCELTP) are presented to solve this problem in some degree. By using them under some special conditions, given one tool position, the local shape (second order approximation) of the tool envelope surface can be calculated precisely at the corresponding cutter contact point. These equations make it convenient to adjust the tool position individually until true errors are reduced in some degree. So, they are useful for optimizing tool positions locally. Finally, some examples are given to verify the correctness and practicability of theory.     

Multi-degree reduction of tensor product Bézier surfaces with conditions of corners interpolations

This paper studies the multi-degree reduction of tensor product B(?)zier surfaces with any degree interpolation conditions of four corners, which is urgently to be resolved in many CAD/CAM systems. For the given conditions of corners interpolation, this paper presents one intuitive method of degree reduction of parametric surfaces. Another new approximation algorithm of multi-degree reduction is also presented with the degree elevation of surfaces and the Chebyshev polynomial approximation theory. It obtains the good approximate effect and the boundaries of degree reduced surface can preserve the prescribed continuities. The degree reduction error of the latter algorithm is much smaller than that of the first algorithm. The error bounds of degree reduction of two algorithms are also presented .     

在任意拓扑三角形网格上的光滑样条曲面

介绍了一种在控制三角形网格上创建光滑样条曲面的算法,该控制网格能够刻画具有或没有边界的任意自由曲面.生成的曲面有一个4次参数多项式表示并且被表示成一个切平面连续的三角形Bézier片网.曲面对网格的逼近程度受到一个混合比控制,当混合比为0时,产生的曲面插值网格.该算法是一种局部方法,简单且效率高,适合于外形设计.     

基于四边形网格均值坐标的K-2环网格曲面构造

为在曲面造型中避免产生扭曲、褶皱等现象,将四边形网格中内点的 K-2 环网格作为控制网格, 提出一种简单、灵活的曲面构造方法。给定一个 K-2 环控制网格,构造一个与其具有同样拓扑结构的平面网格。 再将平面网格拓展成空间四边形网格,同时在平面网格内进行采样。然后计算采样点关于空间四边形网格顶点 的四边形网格均值坐标,最后利用四边形网格均值坐标生成曲面,保证曲面上的每一点都满足 C∞ 。在这个过 程中设置了一个全局形状因子 h,用于控制曲面与初始控制网格的逼近程度,通过实例证明,h 越小,曲面越 逼近初始控制网格。     

Wide‐band analysis of wire antenna using MoM combined with best uniform rational approximation and surface–surface configuration

In this article, an efficient method based on the method of moments (MoM) combined with the best uniform rational approximation and surface–surface configuration is presented for the wide‐band analysis of wire antenna. Compared with the asymptotic waveform evaluation technique, the major advantage of the Maehly approximation is that it can be easily implemented into an existing MoM computer code. Compared with the wire‐surface configuration, the surface–surface configuration is easy to ensure the current continuity at the junction and maintain the required good accuracy. Numerical results including the input admittance and return loss show that the CPU time took by Maehly approximation is about 1/10 of the CPU time used by the traditional MoM. Good agreement between the presented method and the exact solution is observed. © 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2010.     

Variational Surface Approximation and Model Selection

We consider the problem of approximating an arbitrary generic surface with a given set of simple surface primitives. In contrast to previous approaches based on variational surface approximation, which are primarily concerned with finding an optimal partitioning of the input geometry, we propose to integrate a model selection step into the algorithm in order to also optimize the type of primitive for each proxy. Our method is a joint global optimization of both the partitioning of the input surface as well as the types and number of used shape proxies. Thus, our method performs an automatic trade‐off between representation complexity and approximation error without relying on a user supplied predetermined number of shape proxies. This way concise surface representations are found that better exploit the full approximative power of the employed primitive types.     

Adaptive reconstruction of bone geometry from serial cross-sections

Many biomedical applications, such as the design of customized orthopaedic implants, require accurate mathematical models of bone geometry. The surface geometry is often generated by fitting closed parametric curves, or contours, to the edge points extracted from a sequence of evenly spaced planar images acquired using computed tomography (CT), magnetic resonance imaging (MRI), or ultrasound imaging. The Bernstein basis function (BBF) network described in this paper is a novel neural network approach to performing functional approximation tasks such as curve and surface fitting. In essence, the BBF architecture is a two-layer basis function network that performs a weighted summation of nonlinear Bernstein polynomials. The weight values generated during network training are equivalent to the control points needed to create a smooth closed Bézier curve in a variety of commercially available computer-aided design software. Modifying the number of basis neurons in the architecture is equivalent to changing the degree of the Bernstein polynomials. An increase in the number of neurons will improve the curve fit, however, too many neurons will diminish the network's ability to generate a smooth approximation of the cross-sectional boundary data. Additional constraints are imposed on the learning algorithm in order to ensure positional and tangential continuity for the closed curve. A simulation study and real world experiment are presented to show the effectiveness of this functional approximation method for reverse engineering bone structures from serial medical imagery.     

Surface Reconstruction from Unorganized Point Data with Quadrics

We present a reverse engineering method for constructing a surface approximation scheme whose input is a set of unorganized noisy points in space and whose output is a set of quadric patches. The local surface properties, necessary for the subsequent segmentation, are estimated directly from the data using a simple and efficient data structure—the neighborhood graph. Our segmentation scheme, based on principal curvatures, constructs initial point subsets, which may be enlarged or further subdivided based on associated approximation error estimates obtained through approximation of the initial segments by quadric surfaces. Our method is highly efficient and produces a high‐quality piecewise quadric surface approximation of engineering objects, which we demonstrate for several simple and complex example data sets.     

Error Estimates for an LDG Method Applied to Signorini Type Problems

In this paper we propose and analyze a Local Discontinuous Galerkin method for an elliptic variational inequality of the first kind that corresponds to a Poisson equation with Signorini type condition on part of the boundary. The method uses piecewise polynomials of degree one for the field variable and of degree zero or one for the approximation of its gradient. We show optimal convergence for the method and illustrate it with some numerical experiments.     

BIAWGN信道中改进的LT码优化方法

在二进制输入加性高斯白噪声信道中传输LT码时,采用高斯近似方法预测置信传播译码算法的误比特率性能不够准确。为此,提出一种改进的高斯近似方法,其中,输入节点度分布采用泊松分布,相应的软信息为高斯混合物,在此基础上给出一种LT码度分布优化方法。仿真结果证明,该方法相比同类方法性能更优越。     

Adaptive detection and approximation of unknown surface discontinuities from scattered data

We consider a method for the detection and approximation of fault lines of a surface, which is known only on a finite number of scattered data. In particular, we present an adaptive approach to detect surface discontinuities, which allows us to give an (accurate) approximation of the detected faults. First, to locate all the fault points, i.e. the nodes on or close to fault lines, we consider a procedure based on a local interpolation scheme involving a cardinal radial basis formula. Second, we find further sets of points generally closer to the faults than the fault points. Finally, after applying a nearest-neighbour searching procedure and a powerful refinement technique, we outline some different approximation methods. Numerical results highlight the efficiency of our approach.     

Reliability analysis of structures using stochastic response surface method and saddlepoint approximation

Stochastic response surface method (SRSM) is a technique used for reliability analysis of complex structural systems having implicit or time consuming limit state functions. The main aspects of the SRSM are the collection of sample points, the approximation of response surface and the estimation of the probability of failure. In this paper, sample points are selected close to the most probable point of failure and the actual limit state surface (LSS). The response surface is fitted using the weighted regression technique, which allows the fitting points to be weighted based on their distance from the LSS. The cumulant generating function (CGF) of the response surface is derived analytically. The saddlepoint approximation (SPA) method is utilized to compute the probability of failure of the structural system. Finally, four numerical examples compare the proposed algorithm with the traditional quadratic polynomial SRSM, Kriging based SRSM and direct MCS.     

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.     

拓扑结构正确的三线性插值曲面的三角片逼近

在等值面的三角片逼近问题中，采样点的选择对于逼近等值面拓扑结构的正确性和逼近的精确性都非常关键．现有的Marching Cubes以及对其进行改进的方法缺乏对原始曲面拓扑结构的考虑，通常选择同类采样点，无法保证逼近等值面具有正确的拓扑结构．为解决上述问题，将Morse理论的基本思想引入到等值面逼近问题中，提出基于拓扑复杂度的等值面逼近的新方法，该方法根据体元内部曲面拓扑复杂度不同，自适应地提取两类等值点作为采样点：临界点和边界等值点．由于临界点是反映曲面拓扑结构的关键点，因此，无论原始曲面的拓扑结构复杂与否，新方法都能保证逼近等值面具有正确的拓扑结构、较高的逼近精度且基本不增加计算量和数据量．用实例对新方法和已有方法的逼近结果做了比较．