圆曲线拟合的牛顿算法及其应用

Newton Method in the Circular Curve Fitting and Its Application

基于最小二乘准则,圆曲线拟合算法由于目标函数线性化方法及选取拟合参数不同,算法的收敛速度往往存在差异。本文通过将圆曲线参数方程展开至泰勒级数的二阶项,给出了一种基于牛顿法的圆曲线拟合的迭代算法。模拟实验表明:该算法由于每次迭代计算了二阶项,相比圆曲线拟合的经典非线性最小二乘算法具有更快的收敛速度,并且整体表现出观测噪声越小、观测点在圆周上分布越均匀时收敛越快的特点。圆曲线拟合的牛顿算法对于测绘实践中海量数据情况下的非线性曲线和曲面拟合的同类算法,如点云数据的工程建筑、工业设施等拟合算法具有重要的参考和应用价值。

Based on the least-squares rule,the convergence rate of the circular curve fitting algorithm is different due to the different linearization methods of the objective function and the different fitting parameters.In this paper,an iterative algorithm for fitting circular curve based on Newton Method is presented by expanding the parametric equation of circular curve to the second order term of Taylor series..The simulation results show that the proposed algorithm has a faster convergence rate than the classical nonlinear least-squares algorithm fitted by circular curve because it calculates the secondorder terms each iteration,and the algorithm as a whole shows that the smaller the observation noise is and the more uniform the distribution of observation points on the circumference is,the faster the convergence rate is.The Newton algorithm of circular curve fitting has important reference and application value for the similar algorithm of nonlinear curve and surface fitting under the circumstance of mass data in surveying and mapping practice,such as point cloud data fitting method of engineering construction,industrial facilities and so on.