融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法

Nonlinear equality constrained total least squares adjustment combined with orthogonal geometry information and its iterative algorithm

正交距离最小二乘和加权整体最小二乘是解自变量含误差拟合问题的两种独立准则。加权整体最小二乘与正交距离最小二乘不同,它不考虑测量点与拟合点之间的连线垂直于拟合对象的几何信息,不能确保测量点到拟合对象的距离的平方和为极小值。针对该问题,本文将正交几何信息作为约束条件融入加权整体最小二乘,提出一种约束方程带有误差改正数的非线性等式约束整体最小二乘平差法。首先,把加权整体最小二乘平差的函数式看作是非线性方程,连同正交几何约束方程一并线性化,得到线性的平差函数方程;然后,采用拉格朗日乘数法推导其参数估计及精度评定公式,并给出迭代计算算法;最后,以平面直线拟合为例,对本文方法和计算算法进行验证。试验结果表明:①本文方法和算法具有可行性;②与加权最小二乘和加权整体最小二乘相比,本文方法计算的测量点到拟合直线的垂直距离平方和最小;③本文方法计算的测量点到拟合直线的距离与测量点到拟合点的距离相等。

For the problem of fitting the independent variables with error,wo independent approaches including orthogonal distance least squares and weighted total least squares can be employed to solve it.However,the weighted total least square is unlike orthogonal distance least squares.It does not taking into account the orthogonal geometry information,namely,the line segment consisted of observation point and fitted point is vertical to fitted object.Aimed to solve this problem,a nonlinear equality constrained total least squares adjustment model is proposed,in which total least squares is combined with orthogonal geometry information that has been transformed into a nonlinear equality constraints function with unknown corrected errors.After the function model and the nonlinear equality constraints function are linearized,the Lagrange multiplier method is introduced to derive the calculation formula for estimating parameter and assessing accuracy.Two iterative algorithms are given correspondingly.The suggested method and the designed algorithm are tested by the example of fitting a straight-line.The results show that:①the proposed model and the iterative algorithm are feasible;②compared with weighted least squares and weighted total least squares,the sum square of orthogonal distance from the measured point to the fitted line calculated by the new method is the smallest value;③this orthogonal distance is equal to the distance from the measured point to the corrected point calculated by the suggested method,and the other two methods are not like this.