二元三方向剖分中B样条的B网结构与递推算法

§1.引言众所周知,de Boor-Con递推公式及微分-差分公式对于一元B样条的理论和应用极为重要。在多元样条中是否存在类似的结果,已成为近年来的研究课题。本文从B网结构出发,讨论三向剖分下不同次数样条空间的B样条之间的递推关系,指出不能简单地把函数形式的de Boor-Con公式搬到这里,然而可以在B网意义下实现递推。与一     

多元线性回归模型参数估计的递推算法及误差分析

讨论新增数据信息对多元线性回归模型的修正原理,给出不断加入新增数据信息的多元线性回归模型参数估计值的一种递推算法.利用影响因子的概念来刻画新增信息对预测误差的影响,并给出了算法和应用实例.     

Pseudoaffinity, de Boor algorithm, and blossoms

In order to ensure existence of a de Boor algorithm (hence of a B-spline basis) in a given spline space with (n+1)-dimensional sections, it is important to be able to generate each spline by restriction to the diagonal of a symmetric function of n variables supposed to be pseudoaffine w.r. to each variable. We proved that a way to obtain these three properties (symmetry, n-pseudoaffinity, diagonal property) is to suppose the existence of blossoms on the set of admissible n-tuples, given that blossoms are defined in a geometric way by means of intersections of osculating flats. In the present paper, we examine the converse: do symmetry, n-pseudoaffinity, and diagonal property imply existence of blossoms?     

Error Correcting Sequence and Projective De Bruijn Graph

Let X be a finite set of q elements, and n, K, d be integers. A subset C ⊂ X ⁿ is an (n, K, d) error-correcting code, if #(C) = K and its minimum distance is d. We define an (n, K, d) error-correcting sequence over X as a periodic sequence {a _i } _i=0,1,... (a _i ∈ X) with period K, such that the set of all consecutive n-tuples of this sequence form an (n, K, d) error-correcting code over X. Under a moderate conjecture on the existence of some type of primitive polynomials, we prove that there is a error correcting sequence, such that its code-set is the q-ary Hamming code with 0 removed, for q > 2 being a prime power. For the case q = 2, under a similar conjecture, we prove that there is a error-correcting sequence, such that its code-set supplemented with 0 is the subset of the binary Hamming code [2 ^m  − 1, 2 ^m  − 1 − m, 3] obtained by requiring one specified coordinate being 0. Received: October 27, 2005. Final Version received: December 31, 2007     

Products of B-patches

Products and tensor products of multivariate polynomials in B-patch form are viewed as linear combinations of higher degree B-patches. Univariate B-spline segments and certain regions of simplex splines are examples of B-patches. A recursive scheme for transforming tensor product B-patch representations into B-patch representations of more variables is presented. The scheme can also be applied for transforming ann-fold product of B-patch expansions into a B-patch expansion of higher degree. Degree raising formulas are obtained as special cases. The scheme calculates the blossom of the (tensor) product surface and generalizes the pyramidal recursive scheme for B-patches.     

基于轮廓关键点的B样条曲线拟合算法

针对逆向工程中的点云切片轮廓数据点列,提出一种基于轮廓关键点的B样条曲线拟合算法.在确保扫描线点列形状保真度的前提下,首先对其进行等距重采样等预处理,并遴选出曲线轮廓关键点,生成初始插值曲线;再利用邻域点比较法求出初始曲线与各采样点间的偏差值,在超过拟合允差处增加新的关键点,并生成新的插值曲线,重复该步骤至拟合曲线满足预定精度要求.实验表明,在对稠密的二维断面数据点进行B样条逼近时,该算法能有效压缩控制顶点数目,并具有较高的计算效率.同时,由于所得控制顶点的分布能准确反映曲线的曲率变化,该方法还可作为误差约束的曲线逼近中的迭代步骤之一.     

Error Analysis of Clenshaw's Algorithm for Evaluating Derivatives of a Polynomial

A forward rounding error analysis is presented for the extended Clenshaw algorithm due to Skrzipek for evaluating the derivatives of a polynomial expanded in terms of orthogonal polynomials. Reformulating in matrix notation the three-term recurrence relation satisfied by orthogonal polynomials facilitates the estimate of the rounding error for the m-th derivative, which is recursively estimated in terms of the one for the (m – 1)-th derivative. The rounding errors in an important case of Chebyshev polynomial are discussed in some detail.This revised version was published online in October 2005 with corrections to the Cover Date.     

Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix

The Lanczos algorithm for tridiagonalizing a symmetric matrixis the basis for several methods for solving sets of linearequations as well as for solving the eigenproblem. These methodsare very useful when the matrix is large and sparse. A completerounding error analysis of the algorithm is presented here,giving among other results an important expression for the lossof orthogonality of the computed vectors. The results here canbe used to analyze the many methods which are basedon the Lanczosalgorithm.     

NURBS曲线曲面拟合数据点的迭代算法

本文推广了文献[1]的结果,将文献[1]中关于B样条曲线曲面拟合数据点的迭代算法推广至有理形式,给出了无需求解方程组反求控制点及权因子即可得到拟合NURBS曲线曲面的迭代方法．该算法和文献[1]的算法本质上是统一的,而后者恰是前者的一种退化形式．文章还给出了收敛性证明以及一些定性分析．文末的数值实例说明该算法简单实用．     

基于离散点和法矢的曲线重构算法

This paper presents a curve reconstruction algorithm based on discrete data points and normal vectors using B-splines.The proposed algorithm has been improved in three steps:parameterization of the discrete data points with tangent vectors,the B-spline knot vector determination by the selected dominant points based on normal vectors,and the determination of the weight to balancing the two errors of the data points and normal vectors in fitting model.Therefore,we transform the B-spline fitting problem into three sub-problems,and can obtain the B-spline curve adaptively.Compared with the usual fitting method which is based on dominant points selected only by data points,the B-spline curves reconstructed by our approach can retain better geometric shape of the original curves when the given data set contains high strength noises.     

关于DC函数最小化邻近点算法的注记

本文证明DC函数最小化问题邻近点算法的一个收敛性定理,并对此问题提出一类非精确邻近点算法．