带权值的渐进迭代逼近算法及其应用

发布时间：2018-01-28 17:59

本文关键词： 渐进迭代逼近加权渐进迭代逼近插值与逼近加速迭代　出处：《合肥工业大学》2017年硕士论文　论文类型：学位论文

【摘要】：在计算机辅助几何设计与逆向工程中,构造一组满足精度要求的曲线(曲面)来插值或拟合给定的有序点集是一类很重要的课题。反求控制顶点的方法往往因为计算量过大(求解大规模线性方程组)而难以在实际中推广,诸多学者也为此提出了许多不同形式的插值和拟合方法。渐进迭代逼近(the progressive iterative approximation,PIA,又称几何迭代法)的方法以其良好的自适应性和收敛稳定性,受到大多数学者的青睐,该方法通过不断调整与迭代控制顶点,得到一组精度不断提高的曲线(曲面)序列,不仅极大减少了计算量,而且具有明显的几何意义。近年来PIA方法更是在多个领域得到了广泛应用。经典PIA算法虽然能够保证最后得到的极限曲线曲面插值于给定数据点,但是前提是要把所有的数据点都作为每一步迭代的控制顶点。当原始数据规模较大时,经典PIA方法就会出现不够灵活、迭代速度较慢等不足。近年来涌现的一些改进算法有:局部PIA方法、加权PIA方法、Extended PIA方法、最小二乘PIA方法等,这些方法不断地扩大了PIA方法的适用领域、加快了PIA方法的收敛速率、提高了PIA方法的灵活性,同时也丰富了PIA方法的内容。鉴于PIA方法的类型和诸多优点,本文主要做了如下工作:1.PIAWPIA1.研究了PIA方法的发展现状,对带权渐进迭代逼近方法(WPIA)加以改进,即对所有的调整向量取不同权值,并研究其收敛性及迭代效果;2.对局部PIA方法进行了改进,实现了对要调整数据点的加速迭代,同时研究了局部PIA方法和局部代数插值之间的关系。
[Abstract]:In CAD and reverse engineering. It is an important subject to construct a set of curves (surfaces) that satisfy the precision requirement to interpolate or fit a given ordered set of points. It is difficult to be popularized in practice. Many scholars have also proposed many different methods of interpolation and fitting. The progressive iterative approximation. Bia (geometric iterative method) is favored by most scholars because of its good adaptability and convergence stability. This method controls the vertices by constantly adjusting and iterating. A series of curves (surfaces) with increasing accuracy are obtained, which not only greatly reduces the computational complexity. In recent years, the PIA method has been widely used in many fields. Although the classical PIA algorithm can guarantee the final limit curve and surface interpolation to a given data point. But the premise is that all the data points are used as the control vertices of each iteration. When the original data scale is large, the classical PIA method will appear inflexible. Some improved algorithms have emerged in recent years, such as local PIA method, weighted PIA method and extended PIA method, least square PIA method and so on. These methods expand the application field of PIA method, accelerate the convergence rate of PIA method, and improve the flexibility of PIA method. At the same time, it enriches the content of PIA method. In view of the type and many advantages of PIA method, this paper mainly does the following work: 1. PIAWPIA1.The development status of PIA method is studied. The weighted asymptotic iterative approximation method (WPIA) is improved, that is, all the adjustment vectors are given different weights, and their convergence and iterative effect are studied. 2. The local PIA method is improved to realize the accelerated iteration of the data points to be adjusted, and the relationship between the local PIA method and the local algebraic interpolation is studied.
【学位授予单位】：合肥工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP391.7

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