复数域与实数域最小二乘平差的等价性研究

Research on Equivalence between Complex-Valued and Real-Valued Least Squares Adjustment Method

针对等精度独立观测条件下的同一平差问题,从理论上证明复数域与实数域最小二乘平差结果的等价性,并指出复平差方法在函数模型表达上的简便特点。基于常用的图像几何校正多项式模型构造复数域多项式模型,并设计一阶和二阶多项式图像校正算例。结果表明,两种方法的平差结果完全相同,但复数域平差方法的参数维数和法矩阵阶数仅为实数域平差方法的一半。

Firstly,the inverse formula of complex matrix is provided in this paper,on which the complex-valued least squares adjustment method（CLS）for complex-valued linear model is presented in detail.Then,based on the equivalent adjustment model with independent observations weighted equally,it is proven theoretically that the CLS method is equivalent to the real-valued least squares method（LS）.However,the CLS method is greatly superior to the LS method in efficiency.Lastly,both real-valued and complex-valued polynomial models for image geometric correction are employed to verify the validity of the CLS method.Results show that CLS and LS method have no difference in parameter and in accuracy estimation,and the CLS method is very helpful in simplifying the adjustment model and in improving calculation efficiency,because the number of complex-valued parameters is only equal to the half of the number of real-valued parameters.