On symmetrical three-dimensional datum conversion
On symmetrical three-dimensional datum conversion
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DOI：
10.1007/s10291-008-0100-5
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发表时间：
2008
期刊：
影响因子：
4.9
通讯作者：
Y. Felus;R. Burtch
中科院分区：
文献类型：
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作者：
Y. Felus;R. Burtch
三维相似性变换经常用于将基于GPS-WGS 84的坐标转换为使用一组具有两个系统中的坐标值的控制点的本地基准中的坐标。在这种应用中，高斯-马尔可夫（GM）模型通常被用来表示的问题，最小二乘法被用来计算的数学模型内的参数。然而，高斯-马尔可夫模型认为数据矩阵（A）中的源坐标是固定的或无误差的;这是一个不精确的假设，因为这些坐标也是测量的量，并且包括随机误差。变量误差（EIV）模型假设数学模型中的所有变量都被随机误差污染。该模型可以使用由Golub和货车Loan在1980年引入的相对较新的总体最小二乘（TLS）估计技术来求解。本文分析了EIV模型中的相似性变换问题，并提出了一种新的变换参数获取算法。证明了即使在EIV模型下，也可以采用闭合形式的Procrustes方法来获得旋转矩阵和平移参数。变换尺度可以通过求解适当的二次方程来计算。最后通过算例和实际算例对该算法进行了验证，并与EIV模型和GM模型进行了比较。
A 3-D similarity transformation is frequently used to convert GPS-WGS84-based coordinates to those in a local datum using a set of control points with coordinate values in both systems. In this application, the Gauss-Markov (GM) model is often employed to represent the problem, and a least-squares approach is used to compute the parameters within the mathematical model. However, the Gauss–Markov model considers the source coordinates in the data matrix (A) as fixed or error-free; this is an imprecise assumption since these coordinates are also measured quantities and include random errors. The errors-in-variables (EIV) model assumes that all the variables in the mathematical model are contaminated by random errors. This model may be solved using the relatively new total least-squares (TLS) estimation technique, introduced in 1980 by Golub and Van Loan. In this paper, the similarity transformation problem is analyzed with respect to the EIV model, and a novel algorithm is described to obtain the transformation parameters. It is proved that even with the EIV model, a closed form Procrustes approach can be employed to obtain the rotation matrix and translation parameters. The transformation scale may be calculated by solving the proper quadratic equation. A numerical example and a practical case study are presented to test this new algorithm and compare the EIV and the GM models.