测绘学报

• 学术论文 •    

病态总体最小二乘问题的广义正则化

葛旭明1,伍吉仓2   

  1. 1. 同济大学
    2. 同济大学测量与国土信息工程系
  • 收稿日期:2011-06-07 修回日期:2011-10-14 出版日期:2012-06-25 发布日期:2012-06-25
  • 通讯作者: 葛旭明

Generalized Regularization to Ill-posed Total Least Squares Problem

,WU Ji-Cang   

  • Received:2011-06-07 Revised:2011-10-14 Online:2012-06-25 Published:2012-06-25

摘要: 总体最小二乘法通过建立合理的“变量中的误差模型”(Error—In—Variables, EIV)得到更加准确的解算结果,在近几年得到了广泛的应用。基于病态最小二乘理论分析可知,系数矩阵在很大程度上均存在一定的病态性,其对求解结果将造成不稳定的影响,正则化方法是处理该类病态性问题的有效工具。总体最小二乘(TLS)算法可以视为一个降正则化的过程,对比最小二乘算法,病态总体最小二乘方法的解受数据误差和观测值误差的影响将更为严重。本文探讨用广义正则化的方法降低病态性对总体最小二乘数值求解的影响,以提高求解结果的稳定性。通过多组算例结果表明:本文采用的广义正则化方法在处理病态总体最小二乘问题上具有明显的优势。

Abstract: By formulating a more reasonable Error—In—Variables (EIV) model, total least squares method has been wildly used in recent years. Based on ill-posed least squares theory, coefficient matrixes are always ill-conditioned, and which will cause the results oscillate. The regularization is a well-known tool for solving these problems. In fact, total least squares method is a deregularizing procedure, so the ill-posed problems will be more serious. That means errors in the data are more likely to affect the total least squares solution than the least squares solution. In this paper, we propose using generalized regularization to solve ill-posed problems in total least squares, so as to improve stability of the results. Finally, numerical experiments are carried out to demonstrate the performance and efficiency of the generalized regularization method which have significant advantages in solving ill-posed problems.