﻿ 一种基于Partial EIV模型的圆曲线拟合解法
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 大地测量与地球动力学  2018, Vol. 38 Issue (11): 1191-1195  DOI: 10.14075/j.jgg.2018.11.018

引用本文

QIU Dechao, LU Tieding, DENG Xiaoyuan. A Circular Curve Fitting Solution Based on Partial EIV Model[J]. Journal of Geodesy and Geodynamics, 2018, 38(11): 1191-1195.

Foundation support

National Natural Science Foundation of China, No. 41374007, 41464001; Jiangxi Province Science and Technology Landing Project, No. KJLD12077;Jiangxi Provincial Department of Education Science and Technology Project, No. GJJ13457; Natural Science Foundation of Jiangxi Province, No. 2017BAB203032; National Key Research and Development Program of China, No. 2016YFB0501405, 2016YFB0502601-04.

第一作者简介

QIU Dechao, postgraduate, majors in modern geodetic data processing, E-mail: 765730354@qq.com.

文章历史

1. 东华理工大学测绘工程学院，南昌市广兰大道418号, 330013;
2. 流域生态与地理环境监测国家测绘地理信息局重点实验室，南昌市广兰大道418号, 33001;
3. 3 云浮市国土资源和城乡规划管理局，广东省云浮市云城区府前路11号，527300;
4. 浙江省地理信息中心，杭州市保俶北路83号，310012

1 拟合模型构建 1.1 圆曲线一般方程形式

 ${x^2} - 2ax + {y^2} - 2by + c = 0$ (1)

1.2 圆曲线参数方程的EIV模型

 $\left\{ \begin{array}{l} x = a + r\cos \theta \\ y = b + r\sin \theta \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} \hat x = \hat a + \hat r\cos \hat \theta \\ \hat y = \hat b + \hat r\sin \hat \theta \end{array} \right.$ (3)

3) 将步骤2)计算的 L(i)C(i)代入式(13)计算${{\mathit{\boldsymbol{\hat t}}}^{\left( {i + 1} \right)}}$

4) 把上一步求得的${{\mathit{\boldsymbol{\hat t}}}^{\left( {i + 1} \right)}}$重新构造成改正后的系数矩阵${{\mathit{\boldsymbol{\hat A}}}^{\left( {i + 1} \right)}}$

5) 根据式(14)的迭代公式计算新的参数估值${{\mathit{\boldsymbol{\hat \xi}}}^{\left( {i + 1} \right)}}$

6) 重复步骤2)~5)，直到前后两次计算的未知参数估值${{\mathit{\boldsymbol{\hat \xi}}}^{\left( {i } \right)}}$满足‖${{\mathit{\boldsymbol{\hat \xi}}}^{\left( {i + 1} \right)}}$-${{\mathit{\boldsymbol{\hat \xi}}}^{\left( {i } \right)}}$‖＜ε且‖${{\mathit{\boldsymbol{\hat t}}}^{\left( {i + 1} \right)}}$-${{\mathit{\boldsymbol{\hat t}}}^{\left( {i } \right)}}$‖＜δ时，迭代终止，输出所求参数的最终估值。

7) 将输出的${{\mathit{\boldsymbol{\hat \xi}}}^{\left( {i } \right)}}$前3个参数加上初始值$\left( {{{\hat a}^0}, {{\hat b}^0}} \right)$r0，得到最终的圆心坐标和半径。

3 圆曲线拟合的精度评定

 $d_i^2 = {\sum\limits_{i = 1}^n {\left[ {\sqrt {{{\left( {{x_i} - a} \right)}^2} + {{\left( {{y_i} - b} \right)}^2}} - r} \right]} ^2}$ (15)

4 算例 4.1 实例数据

4.2 模拟数据

 ${\left( {x + 1} \right)^2} + {\left( {y + 2} \right)^2} = {3^2}$ (16)

5 结语

1) 不同形式构建的圆曲线拟合模型均可用于求解拟合参数。以一般方程构建的最小二乘法求解简单，但拟合精度较低。

2) 以圆曲线参数形式为基础构建的拟合模型，当观测坐标有误差导致系数矩阵有误差时，需采用总体最小二乘算法求解。

3) 本文给出的基于Partial EIV模型的总体最小二乘算法适用于圆曲线拟合，为圆拟合提供了一种新方法，且在拟合精度上具有一定优势。

4) 本文方法较EIV模型的总体最小二乘法，具有系数矩阵改正量少的优点。

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A Circular Curve Fitting Solution Based on Partial EIV Model
QIU Dechao1,2,3     LU Tieding1,2     DENG Xiaoyuan4
1. Faculty of Geomatics, East China University of Technology, 418 Guanglan Road, Nanchang 330013, China;
2. Key Laboratory of Watershed Ecology and Geographical Environment Monitoring, NASMG, 418 Guanglan Road, Nanchang 330013, China;
3. Yunfu City Land and Resources Planning Administration, 11 Fuqian Road, Yunfu 527300, China;
4. Geomatics Center of Zhejiang Province, 83 North-Baochu Road, Hangzhou 310012, China
Abstract: Aiming at the problem of circular curve fitting, based on the parametric equation of circular curve, firstly this paper establishes the EIV model of circular curve fitting, and transforms the model into a more reasonable Partial EIV model for the characteristics of the coefficient matrix. Then, the formula is transformed into the least squares form, and the two-step iterative method is used to solve parameters of the model, ensuring that the corrections of the same elements in the coefficient matrix are same, and the corrections of the constant elements equals zero. Finally, combing case data shows the feasibility of this algorithm, and fitting accuracy is relatively superior.
Key words: circle fitting; parametric equation; total least squares; Partial EIV model