低次非均匀三角Bézier曲面的最小二乘渐进迭代逼近性

The Least Square Progressive Iterative Approximation Property of Low Degree Non-Uniform Triangular Bézier Surfaces

渐进迭代逼近(简称PIA)是一种直观有效的数据拟合方法.经典的PIA方法要求曲面控制顶点的个数等于拟合数据点的个数,并不适用于大量数据的拟合.为了改造经典PIA方法,特别研究了使用最频繁的三角曲面用PIA来生成的算法,并重点考虑实际中最常用的低次情形.证明了低次(n=2,3,4)非均匀三角Bézier曲面具有最小二乘渐进迭代逼近(简称LSPIA)性质,并且迭代得到的三角Bézier曲面序列的极限就是数据点的最小二乘拟合.同时,还提供了如何选择合适的权值使得迭代拥有最快收敛速度的方法.实例验证了最小二乘PIA方法的有效性.

Progressive-iterative approximation(PIA) is an intuitive and effective method for data fitting. Classical PIA method requires that the number of control points is equal to the number of the data points. It is not suitable for fitting mass data. In order to improve the classical PIA method, the algorithm for fitting data points with triangular surfaces based on PIA method is studied, especially for the low-degree case usually used in practice. It is proved that the quadratic, cubic and quartic non-uniform triangular Bézier surfaces have the property of progressive-iterative approximation for least square fitting(LSPIA). And the limit of the sequence of triangular Bézier surfaces obtained by iteration is just the least square fitting of the data points. Meanwhile, a method is provided to show how to choose the value of the weight so that the iteration has the fastest convergence speed. A numerical example is presented to validate the effectiveness of the LSPIA method.