﻿ 加权总体最小二乘和RBF神经网络的三维坐标转换
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 大地测量与地球动力学  2023, Vol. 43 Issue (1): 29-33  DOI: 10.14075/j.jgg.2023.01.006

### 引用本文

ZHAO Hui, GUO Chunxi, MENG Jingjuan, et al. Three-Dimensional Coordinate Transformation Combined with Weighted Total Least Squares Method and RBF Neural Network[J]. Journal of Geodesy and Geodynamics, 2023, 43(1): 29-33.

### Foundation support

The Joint Funds of State Key Laboratory of Geo-Information Engineering and Key Laboratory of Surveying and Mapping Science and Geospatial Information Technology of MNR, No.20210106; The Founds of Beijing Key Laboratory of Urben Spatial Information Engineering, No.20220116.

### Corresponding author

MENG Jingjuan, engineer, majors in geodetic data processing, E-mail: jingjuanmeng@126.com.

### 第一作者简介

ZHAO Hui, engineer, majors in geodetic data processing, E-mail: zhaohuiln@163.com.

### 文章历史

1. 自然资源部大地测量数据处理中心，西安市友谊东路334号，710054

1 三维坐标转换的加权总体最小二乘法WTLS 1.1 三维坐标转换模型

 $\begin{gathered} {\left[\begin{array}{c} X^{\mathrm{T}} \\ Y^{\mathrm{T}} \\ Z^{\mathrm{T}} \end{array}\right]=\left[\begin{array}{c} \Delta X_0 \\ \Delta Y_0 \\ \Delta Z_0 \end{array}\right]+} \\ (1+m)\left[\begin{array}{ccc} 1 & \varepsilon_Z & -\varepsilon_Y \\ -\varepsilon_Z & 1 & \varepsilon_X \\ \varepsilon_Y & -\varepsilon_X & 1 \end{array}\right]\left[\begin{array}{l} X^{\mathrm{S}} \\ Y^{\mathrm{S}} \\ Z^{\mathrm{S}} \end{array}\right] \end{gathered}$ (1)

a1=m+1、a2=a1εXa3=a1εYa4=a1εZ，则式(1)可改写为：

 $\begin{matrix} \overset{{\boldsymbol{y}}}{\mathop{\left[ \begin{matrix} X_{1}^{y} \\ Y_{1}^{\text{T}} \\ Z_{1}^{\text{T}} \\ \vdots \\ X_{n}^{\text{T}} \\ Y_{n}^{\text{T}} \\ Z_{n}^{\text{T}} \\ \end{matrix} \right]}}\, = \\ \overset{{\boldsymbol{A}}}{\mathop{\left[ \begin{matrix} 1 & 0 & 0 & X_{1}^{\text{S}} & 0 & -Z_{1}^{\text{S}} & Y_{1}^{\text{S}} \\ 0 & 1 & 0 & Y_{1}^{\text{S}} & Z_{1}^{\text{S}} & 0 & -X_{1}^{\text{S}} \\ 0 & 0 & 1 & Z_{1}^{\text{S}} & -Y_{1}^{\text{S}} & X_{1}^{\text{S}} & 0 \\ {} & {} & {} & {} & \vdots & {} & {} \\ 1 & 0 & 0 & X_{n}^{\text{S}} & 0 & -Z_{n}^{\text{S}} & Y_{n}^{\text{S}} \\ 0 & 1 & 0 & Y_{n}^{\text{S}} & Z_{n}^{\text{S}} & 0 & -X_{n}^{\text{S}} \\ 0 & 0 & 1 & Z_{n}^{\text{S}} & -Y_{n}^{\text{S}} & X_{n}^{\text{S}} & 0 \\ \end{matrix} \right]}}\, \overset{{\boldsymbol{x}}}{\mathop{\left[ \begin{matrix} \Delta {{X}_{0}} \\ \Delta {{Y}_{0}} \\ \Delta {{Z}_{0}} \\ {{a}_{1}} \\ {{a}_{2}} \\ {{a}_{3}} \\ {{a}_{4}} \\ \end{matrix} \right]}}\, \\ \end{matrix}$ (2)

 $\boldsymbol{x}_{\mathrm{LS}}=\boldsymbol{x}_0+\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{Q}_y^{-1}\right)^{-1} \boldsymbol{A}^{\mathrm{T}} \boldsymbol{Q}_y^{-1}\left(\boldsymbol{y}-\boldsymbol{A} \boldsymbol{x}_0\right)$ (3)

1.2 Partial EIV模型的加权总体最小二乘法

 $\left\{\begin{array}{l} \boldsymbol{y}-\boldsymbol{e}_y=\left(\boldsymbol{x}^{\mathrm{T}} \otimes \boldsymbol{I}_n\right)(\boldsymbol{h}+\boldsymbol{B} \overline{\boldsymbol{a}}) \\ \boldsymbol{a}-\boldsymbol{e}_a=\overline{\boldsymbol{a}} \\ \operatorname{vec}(\boldsymbol{A})=\boldsymbol{h}+\boldsymbol{B} \boldsymbol{a} \end{array}\right.$ (4)

 $\boldsymbol{e}=\left[\begin{array}{l} \boldsymbol{e}_y \\ \boldsymbol{e}_a \end{array}\right] \sim N\left(\left[\begin{array}{l} 0 \\ 0 \end{array}\right], \sigma_0^2\left[\begin{array}{cc} \boldsymbol{Q}_y & 0 \\ 0 & \boldsymbol{Q}_a \end{array}\right]\right)$ (5)

$\boldsymbol{Q}=\left[\begin{array}{cc} \boldsymbol{Q}_y & 0 \\ 0 & \boldsymbol{Q}_a \end{array}\right]$，则Partial EIV模型的参数估计准则为eTQ－1e=min。迭代步骤如下[7-8]

1) 设置初值：x(0)=xLSea(0)=0。

2) 求参数更新值: $\boldsymbol{A}_{(i)}=\operatorname{ivec}\left(\boldsymbol{h}+\boldsymbol{B} \boldsymbol{a}_{(i)}\right), \boldsymbol{C}_{(i)}$$=\left[\begin{array}{cc} \boldsymbol{A}_{(i)} & \left(\boldsymbol{x}_{(i)}^{\mathrm{T}} \otimes \boldsymbol{I}_n\right) \boldsymbol{B} \\ \boldsymbol{O} & \boldsymbol{I}_t \end{array}\right], \boldsymbol{L}_{(i)}=\left[\begin{array}{c} \boldsymbol{y}-\boldsymbol{A}_{(i)} \boldsymbol{x}_{(i)} \\ \boldsymbol{a}-\boldsymbol{a}_{(i)} \end{array}\right],$$ \left[\begin{array}{ll} \delta \boldsymbol{x} & \delta \boldsymbol{a} \end{array}\right]_{(i+1)}^{\mathrm{T}}=\left(\boldsymbol{C}_{(i)}^{\mathrm{T}} \boldsymbol{Q}^{-1} \boldsymbol{C}_{(i)}\right)^{-1} \boldsymbol{C}_{(i)}^{\mathrm{T}} \boldsymbol{Q}^{-1} \boldsymbol{L}_{(i)}$。式中，ivec为矩阵列向量化vec的逆运算。

3) 更新参数: $\left[\begin{array}{l} \boldsymbol{x}_{(i+1)} \\ \boldsymbol{a}_{(i+1)} \end{array}\right]=\left[\begin{array}{l} \boldsymbol{x}_{(i)} \\ \boldsymbol{a}_{(i)} \end{array}\right]+\left[\begin{array}{l} \delta \boldsymbol{x} \\ \delta \boldsymbol{a} \end{array}\right]_{(i+1)},$重复步骤2)，直到$\left\|\left[\begin{array}{ll} \delta \boldsymbol{x} & \delta \boldsymbol{a} \end{array}\right]_{(i+1)}^{\mathrm{T}}\right\|<\boldsymbol{\varepsilon}_0$时终止迭代。

 $\boldsymbol{h}=\left[\begin{array}{c} \boldsymbol{h}_1 \\ \boldsymbol{h}_2 \\ \vdots \\ \boldsymbol{h}_7 \end{array}\right], \boldsymbol{B}=\left[\begin{array}{c} \boldsymbol{B}_1 \\ \boldsymbol{B}_2 \\ \vdots \\ \boldsymbol{B}_7 \end{array}\right]$ (6)

2 顾及源坐标系坐标误差的RBF组合解法 2.1 RBF神经网络

RBF神经网络是一种3层前馈局部逼近网络，将输入层数据非线性变化到高维空间的隐含层中，实现低维度空间内线性不可分问题在高维度空间内的线性可分，再将隐含层线性变换到输出层中[10]

 $\varphi_j(\boldsymbol{X})=\exp \left(-\frac{\left\|\boldsymbol{X}-\mu_j\right\|^2}{2 \sigma_j^2}\right)$ (7)

 $y_k=\sum\limits_{j=1}^m w_{k j} \varphi_j(\bf{X})$ (8)

2.2 组合解法的建立

 图 1 组合方法坐标转换流程 Fig. 1 Coordinate transformation process of composite method

1) 根据§1.1中公式，利用重合点坐标计算七参数加权最小二乘解，并将其作为初值；

2) 根据§1.2中迭代过程计算Partial EIV模型的加权总体最小二乘解；

3) 利用步骤2)计算出的源坐标系坐标改正数训练RBF神经网络；

4) 基于RBF神经网络计算的待转换点坐标改正数，利用Partial EIV模型的加权总体最小二乘法求出的七参数进行转换。

3 实验分析

 图 2 源坐标改正数 Fig. 2 Correction of source coordinate

 图 3 PWTLS+RBF组合法转换残差 Fig. 3 Residuals of coordinate transformation method combining PWTLS and RBF

4 结语

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Three-Dimensional Coordinate Transformation Combined with Weighted Total Least Squares Method and RBF Neural Network
ZHAO Hui1     GUO Chunxi1     MENG Jingjuan1     GENG Xiaoyan1     WANG Wenchao1
1. Geodetic Data Processing Center, MNR, 334 East-Youyi Road, Xi'an 710054, China
Abstract: We analyze the application and influence of weighted total least squares method in partial errors-in-variables weight total least squares model (PWTLS), weighted total least squares method(WTLS) and least squares method(LS) in parameters calculation of the three-dimensional coordinate transformation model. Then, we deduce a coordinate transformation method that combines PWTLS and RBF Neural Network. The results show that, when both constant elements and repeated elements exist in the design matrix of coordinate transformation model, unit weighted error and precision of inner coincidence calculated by PWTLS algorithm are better than LS algorithm; in addition, correction of source coordinate calculated by PWTLS is more reasonable than WTLS. The combination method of PWTLS and RBF realizes the practicability of the parameters calculated by PWTLS, which remarkably improves the accuracy of coordinate transformation in practical application.
Key words: coordinate transformations; WTLS; partial errors-in-variables model; RBF neural network