﻿ 病态总体最小二乘靶向奇异值修正法
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 大地测量与地球动力学  2019, Vol. 39 Issue (8): 856-862  DOI: 10.14075/j.jgg.2019.08.017

### 引用本文

WU Guangming, LU Tieding. New Methods of Ill-Posed Total Least-Squares with Targeting Singular Value Corrections[J]. Journal of Geodesy and Geodynamics, 2019, 39(8): 856-862.

### Foundation support

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

### Corresponding author

LU Tieding, PhD, professor, majors in error theory and adjustment, E-mail:tdlu@whu.edu.cn.

### 第一作者简介

WU Guangming, postgraduate, majors in surveying and mapping data processing, E-mail: 821345314@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. 国家自然资源部流域生态与地理环境监测重点实验室, 南昌市广兰大道 418 号, 33001;
3. 江西省数字国土重点实验室, 南昌市广兰大道418号, 330013

1 病态总体最小二乘正则化法

G-M模型是顾及观测向量L的随机误差e，平差模型及最小二乘平差准则为：

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{L}} = \mathit{\boldsymbol{AX}} + \mathit{\boldsymbol{e}},\mathit{\boldsymbol{e}} \sim N\left( {0,\sigma _0^2\mathit{\boldsymbol{I}}} \right)\\ f\left( \mathit{\boldsymbol{e}} \right) = {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{e}} = \min \end{array} \right.$ (1)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{\hat X}} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}}\\ {\mathop{\rm cov}} \left( {\mathit{\boldsymbol{\hat X}}} \right) = \sigma _0^2{\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}}} \right)^{ - 1}} \end{array} \right.$ (2)

 $D\left( {\mathit{\boldsymbol{\hat X}}} \right) = {\rm{tr}}\left[ {{\mathop{\rm cov}} \left( {\mathit{\boldsymbol{\hat X}}} \right)} \right] = \sigma _0^2\sum\limits_{i = 1}^n {\frac{1}{{\mathit{\Lambda }_i^2}}}$ (3)

 $\mathit{\boldsymbol{L}} = \left( {\mathit{\boldsymbol{A}} + {\mathit{\boldsymbol{E}}_A}} \right)\mathit{\boldsymbol{X}} + \mathit{\boldsymbol{e}}$ (4)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{L}} = \left( {\mathit{\boldsymbol{A}} + {\mathit{\boldsymbol{E}}_A}} \right)\mathit{\boldsymbol{X}} + \mathit{\boldsymbol{e}} = \mathit{\boldsymbol{AX}} + \\ \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{I}}&{{\mathit{\boldsymbol{X}}^{\rm{T}}} \otimes \mathit{\boldsymbol{I}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{e}}\\ {{\mathit{\boldsymbol{e}}_A}} \end{array}} \right],{\mathit{\boldsymbol{e}}_A} = {\rm{vec}}\left( {{\mathit{\boldsymbol{E}}_A}} \right)\\ \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{e}}\\ {{\mathit{\boldsymbol{e}}_A}} \end{array}} \right] \sim N\left( {\left[ {\begin{array}{*{20}{c}} 0\\ 0 \end{array}} \right],\sigma _0^2\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{I}}_n}}&0\\ 0&{{\mathit{\boldsymbol{I}}_n} \otimes {\mathit{\boldsymbol{I}}_n}} \end{array}} \right]} \right) \end{array} \right.$ (5)

 $f\left( {{\mathit{\boldsymbol{E}}_A},\mathit{\boldsymbol{e}}} \right) = \mathit{\boldsymbol{e}}_A^{\rm{T}}{\mathit{\boldsymbol{e}}_A} + {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{e}} = \min$ (6)

 $\begin{array}{*{20}{c}} {F\left( {{\mathit{\boldsymbol{E}}_A},\mathit{\boldsymbol{e}}} \right) = }\\ {\mathit{\boldsymbol{e}}_A^{\rm{T}}{\mathit{\boldsymbol{e}}_A} + {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{e}} + 2{\mathit{\boldsymbol{\lambda }}^{\rm{T}}}\left[ {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{e}} - \mathit{\boldsymbol{AX}} - {\mathit{\boldsymbol{E}}_A}\mathit{\boldsymbol{X}}} \right]} \end{array}$ (7)

 ${\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A\hat X}} - {\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}} = \mathit{\boldsymbol{\hat X}}\frac{{{{\left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A\hat X}}} \right)}^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A\hat X}}} \right)}}{{1 + {{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}}}}$ (8)

${\mathit{\boldsymbol{\hat u}}^{\left( k \right)}} = \frac{{{{\left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}} \right)}^{\rm{T}}}\left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}} \right)}}{{1 + {{\mathit{\boldsymbol{\hat X}}}^{\left( k \right){\rm{T}}}}{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}}}$，将得到的迭代式

 ${\mathit{\boldsymbol{\hat X}}^{\left( {k + 1} \right)}} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}}} \right)^{ - 1}}\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}} + {{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}{{\mathit{\boldsymbol{\hat \mu }}}^{\left( k \right)}}} \right)$ (9)

 $\begin{array}{*{20}{c}} {f\left( {{\mathit{\boldsymbol{E}}_A},\mathit{\boldsymbol{e}}} \right) = {\rm{vec}}{{\left( {{\mathit{\boldsymbol{E}}_A}} \right)}^{\rm{T}}}{\rm{vec}}\left( {{\mathit{\boldsymbol{E}}_A}} \right) + }\\ {{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{e}} + \alpha {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{RX}} = \min } \end{array}$ (10)

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat X}}}^{\left( {k + 1} \right)}} = {{\left[ {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}} + \alpha \left( {1 + {{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)T}}{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}} \right)\mathit{\boldsymbol{R}}} \right]}^{ - 1}} \cdot }\\ {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}} + {{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}{{\hat \mu }^{\left( k \right)}}} \right)} \end{array}$ (11)

2 病态总体最小二乘靶向奇异值修正法

 ${\mathit{\boldsymbol{R}}_1} = \sum\limits_{i = j}^n {{\mathit{\boldsymbol{G}}_i}\mathit{\boldsymbol{G}}_i^{\rm{T}}}$ (12)

 $\sum\limits_{i = j}^n {\frac{1}{{{\mathit{\Lambda }_i}}}} \ge 95\% \sum\limits_{i = 1}^n {\frac{1}{{{\mathit{\Lambda }_i}}}}$ (13)

2.1 方法1

 $\begin{array}{*{20}{c}} {F\left( {{\mathit{\boldsymbol{E}}_A},\mathit{\boldsymbol{e}}} \right) = \mathit{\boldsymbol{e}}_A^{\rm{T}}{\mathit{\boldsymbol{e}}_A} + {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{e}} + \alpha {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{RX}} + }\\ {2{\mathit{\boldsymbol{\lambda }}^{\rm{T}}}\left[ {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{e}} - \mathit{\boldsymbol{AX}} - {\mathit{\boldsymbol{E}}_A}\mathit{\boldsymbol{X}}} \right]} \end{array}$ (14)

 $\left\{ \begin{array}{l} \frac{{\partial F}}{{\partial \mathit{\boldsymbol{e}}}} = 2{\mathit{\boldsymbol{e}}^{\rm{T}}} - 2{\mathit{\boldsymbol{\lambda }}^{\rm{T}}} = 0\\ \frac{{\partial F}}{{\partial {\mathit{\boldsymbol{e}}_A}}} = 2\mathit{\boldsymbol{e}}_A^{\rm{T}} + 2{\mathit{\boldsymbol{\lambda }}^{\rm{T}}}\left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}} \right) = 0\\ \frac{{\partial F}}{{\partial \mathit{\boldsymbol{\hat X}}}} = 2\alpha {\mathit{\boldsymbol{X}}^{\rm{T}}}\mathit{\boldsymbol{R}} - 2{\mathit{\boldsymbol{\lambda }}^{\rm{T}}}\left( {\mathit{\boldsymbol{A}} + {\mathit{\boldsymbol{E}}_A}} \right) = 0 \end{array} \right.$ (15)

 $\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A\hat X}} = {\mathit{\boldsymbol{E}}_A}\mathit{\boldsymbol{\hat X}} + \mathit{\boldsymbol{e}} = \lambda \left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}} + 1} \right)$ (16)

 $\mathit{\boldsymbol{e}} = \mathit{\boldsymbol{\lambda }} = \left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A\hat X}}} \right){\left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}} + 1} \right)^{ - 1}}$ (17)

 ${\mathit{\boldsymbol{\hat E}}_A} = \mathit{\boldsymbol{\lambda }}{\mathit{\boldsymbol{\hat X}}^{\rm{T}}} = \left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A\hat X}}} \right){\left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}} + 1} \right)^{ - 1}}{\mathit{\boldsymbol{\hat X}}^{\rm{T}}}$ (18)

 $\mathit{\boldsymbol{\hat A}} = \mathit{\boldsymbol{A}} + {\mathit{\boldsymbol{E}}_A}$ (19)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat X = }}{{\left[ {{{\left( {\mathit{\boldsymbol{A}} + {\mathit{\boldsymbol{E}}_A}} \right)}^{\rm{T}}}\left( {\mathit{\boldsymbol{A}} + {\mathit{\boldsymbol{E}}_A}} \right) + \alpha \mathit{\boldsymbol{R}}} \right]}^{ - 1}}{{\left( {\mathit{\boldsymbol{A}} + {\mathit{\boldsymbol{E}}_A}} \right)}^{\rm{T}}}\mathit{\boldsymbol{L}}}\\ { = {{\left( {{{\mathit{\boldsymbol{\hat A}}}^{\rm{T}}}\mathit{\boldsymbol{\hat A}} + \alpha \mathit{\boldsymbol{R}}} \right)}^{ - 1}}{{\mathit{\boldsymbol{\hat A}}}^{\rm{T}}}\mathit{\boldsymbol{L}}} \end{array}$ (20)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{\hat E}}_A^{\left( k \right)} = \left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}} \right){\left( {{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right){\rm{T}}}}{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}} + 1} \right)^{ - 1}}{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right){\rm{T}}}}\\ {{\mathit{\boldsymbol{\hat X}}}^{\left( {k + 1} \right)}} = {\left( {{{\mathit{\boldsymbol{\hat A}}}^{\left( k \right){\rm{T}}}}{{\mathit{\boldsymbol{\hat A}}}^{\left( k \right)}} + \alpha \mathit{\boldsymbol{R}}} \right)^{ - 1}}{{\mathit{\boldsymbol{\hat A}}}^{\left( k \right){\rm{T}}}}\mathit{\boldsymbol{L}} \end{array} \right.$ (21)

$\left\| {{{\mathit{\boldsymbol{\hat X}}}^{\left( {k + 1} \right)}} - {{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}} \right\| < \varepsilon$时，迭代终止。本文病态TLS靶向奇异值修正法是将式(21)中R替换成R1，靶向矩阵R1可以靶向修正较小奇异值。${\mathit{\boldsymbol{\hat A}}}$可以求得，进而靶向矩阵R1可按顾勇为等[9]的方法求出，这样在迭代过程中R1随着${\mathit{\boldsymbol{\hat A}}}$的变化而变化，从而使靶向修正法矩阵${\mathit{\boldsymbol{\hat A}}}^{\rm{T}}{\mathit{\boldsymbol{\hat A}}}$奇异值变得合理。具体解算步骤如下：1)采用最小二乘估计求出迭代初值${\mathit{\boldsymbol{\hat X}}}_0$；2)根据式(21)求出系数矩阵的误差阵EA，并得出改正的系数矩阵${\mathit{\boldsymbol{\hat A}}}$；3)用改正的系数矩阵${\mathit{\boldsymbol{\hat A}}}$求出新的法矩阵，并根据式(12)求出靶向矩阵R1；4)根据求得的R1用L-曲线法[1]求出相应的正则化参数α；5)利用式(21)求出参数估值，迭代得出最优估值。

2.2 方法2

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{L}} = \mathit{\boldsymbol{\tilde AX}} + \mathit{\boldsymbol{e}}\\ \mathit{\boldsymbol{A}} = \mathit{\boldsymbol{A}} - {\mathit{\boldsymbol{E}}_A} \end{array} \right.$ (22)

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{L}}^{\rm{T}}} = {\mathit{\boldsymbol{X}}^{\rm{T}}}{{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}} + {\mathit{\boldsymbol{e}}^{\rm{T}}}\\ {\mathit{\boldsymbol{A}}^{\rm{T}}} = {{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}} - \mathit{\boldsymbol{E}}_A^{\rm{T}} \end{array} \right.$ (23)

 $\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{L}}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{A}}^{\rm{T}}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{X}}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{I}}_n}} \end{array}} \right]{\mathit{\boldsymbol{\tilde A}}^{\rm{T}}} + \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{e}}^{\rm{T}}}}\\ { - \mathit{\boldsymbol{E}}_A^{\rm{T}}} \end{array}} \right]$ (24)

 $f\left( {\mathit{\boldsymbol{e}},{\mathit{\boldsymbol{E}}_A}} \right) = {\left\| {\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{e}}^{\rm{T}}}}\\ { - \mathit{\boldsymbol{E}}_A^{\rm{T}}} \end{array}} \right]} \right\|_F} = \min$ (25)

 ${\left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{I}}_n}} \end{array}} \right]^{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{L}}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{A}}^{\rm{T}}}} \end{array}} \right] = {\left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{I}}_n}} \end{array}} \right]^{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{I}}_n}} \end{array}} \right]{\mathit{\boldsymbol{\hat A}}^{\rm{T}}}$ (26)

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat A}}}^{\rm{T}}} = {{\left[ {\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat X}}}&{{\mathit{\boldsymbol{I}}_n}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{I}}_n}} \end{array}} \right]} \right]}^{ - 1}} \cdot \left[ {\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat X}}}&{{\mathit{\boldsymbol{I}}_n}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{L}}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{A}}^{\rm{T}}}} \end{array}} \right]} \right]}\\ { = {{\left[ {\mathit{\boldsymbol{\hat X}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} + {\mathit{\boldsymbol{I}}_n}} \right]}^{ - 1}} \cdot \left[ {\mathit{\boldsymbol{\hat X}}{\mathit{\boldsymbol{L}}^{\rm{T}}} + {\mathit{\boldsymbol{A}}^{\rm{T}}}} \right]} \end{array}$ (27)

 $\mathit{\boldsymbol{\hat A}} = \left[ {\mathit{\boldsymbol{L}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} + \mathit{\boldsymbol{A}}} \right] \cdot {\left[ {\mathit{\boldsymbol{\hat X}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} + {\mathit{\boldsymbol{I}}_n}} \right]^{ - 1}}$ (28)

 $\mathit{\boldsymbol{\hat X}} = {\left( {{{\mathit{\boldsymbol{\hat A}}}^{\rm{T}}}\mathit{\boldsymbol{\hat A}} + \alpha \mathit{\boldsymbol{R}}} \right)^{ - 1}} \cdot {\mathit{\boldsymbol{\hat A}}^{\rm{T}}}\mathit{\boldsymbol{L}}$ (29)

 $\left\{ \begin{array}{l} {{\mathit{\boldsymbol{\hat A}}}^{\left( k \right)}} = \left[ {\mathit{\boldsymbol{L}}{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right){\rm{T}}}} + \mathit{\boldsymbol{A}}} \right] \cdot {\left[ {{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}{{\mathit{\boldsymbol{\hat X}}}^{\left( k \right){\rm{T}}}} + {\mathit{\boldsymbol{I}}_n}} \right]^{ - 1}}\\ {{\mathit{\boldsymbol{\hat X}}}^{\left( {k + 1} \right)}} = {\left( {{{\mathit{\boldsymbol{\hat A}}}^{\left( k \right){\rm{T}}}}{{\mathit{\boldsymbol{\hat A}}}^{\left( k \right)}} + \alpha \mathit{\boldsymbol{R}}} \right)^{ - 1}} \cdot {{\mathit{\boldsymbol{\hat A}}}^{\left( k \right){\rm{T}}}}\mathit{\boldsymbol{L}} \end{array} \right.$ (30)

$\left\| {{{\mathit{\boldsymbol{\hat X}}}^{\left( {k + 1} \right)}} - {{\mathit{\boldsymbol{\hat X}}}^{\left( k \right)}}} \right\| < \varepsilon$时，迭代终止。本文病态TLS靶向奇异值修正法是将式(30)中R替换成R1, 具体解算步骤如下：1)采用最小二乘估计求出迭代初值${{\mathit{\boldsymbol{\hat X}}}_0}$；2)根据式(30)求出改正的系数矩阵${\mathit{\boldsymbol{\hat A}}}$；3)用改正的系数矩阵${\mathit{\boldsymbol{\hat A}}}$求出新的法矩阵，并根据式(12)求出靶向矩阵R1；4)根据求得的R1用L-曲线法[1]求出相应的正则化参数α；5)再利用式(30)求出参数估值，迭代得出最优估值。

2.3 2种新方法比较

 ${\mathit{\boldsymbol{\hat A}}_1} = \mathit{\boldsymbol{A}} + \left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A\hat X}}} \right){\left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}} + 1} \right)^{ - 1}}{\mathit{\boldsymbol{\hat X}}^{\rm{T}}}$ (31)

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat A}}}_1} = \mathit{\boldsymbol{A}} + \mathit{\boldsymbol{L}}{{\left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}} + 1} \right)}^{ - 1}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} - }\\ {\mathit{\boldsymbol{A\hat X}}{{\left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}} + 1} \right)}^{ - 1}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}} \end{array}$ (32)

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat A}}}_2} = \mathit{\boldsymbol{L}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} \cdot }\\ {{{\left[ {\mathit{\boldsymbol{\hat X}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} + {\mathit{\boldsymbol{I}}_n}} \right]}^{ - 1}} + \mathit{\boldsymbol{A}}{{\left[ {\mathit{\boldsymbol{\hat X}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} + {\mathit{\boldsymbol{I}}_n}} \right]}^{ - 1}}} \end{array}$ (33)

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat A}}}_1} - {{\mathit{\boldsymbol{\hat A}}}_2} = \mathit{\boldsymbol{A}} + \left( {\mathit{\boldsymbol{L}} - \mathit{\boldsymbol{A\hat X}}} \right){{\left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}} + 1} \right)}^{ - 1}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} - }\\ {\mathit{\boldsymbol{L}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} \cdot {{\left[ {\mathit{\boldsymbol{\hat X}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} + {\mathit{\boldsymbol{I}}_n}} \right]}^{ - 1}} - \mathit{\boldsymbol{A}}{{\left[ {\mathit{\boldsymbol{\hat X}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} + {\mathit{\boldsymbol{I}}_n}} \right]}^{ - 1}}} \end{array}$ (34)

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat A}}}_1} - {{\mathit{\boldsymbol{\hat A}}}_2} = \mathit{\boldsymbol{A}}\left\{ {{\mathit{\boldsymbol{I}}_n} - {{\left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}} + 1} \right)}^{ - 1}}\mathit{\boldsymbol{\hat X}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} - \left[ {\mathit{\boldsymbol{\hat X}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} + } \right.} \right.}\\ {\left. {\left. {{{\left. {{\mathit{\boldsymbol{I}}_n}} \right]}^{ - 1}}} \right\} + \mathit{\boldsymbol{L}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}{{\left[ {{{\left( {{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}}\mathit{\boldsymbol{\hat X}} + 1} \right)}^{ - 1}}{\mathit{\boldsymbol{I}}_n} - \left\{ {\mathit{\boldsymbol{\hat X}}{{\mathit{\boldsymbol{\hat X}}}^{\rm{T}}} + {\mathit{\boldsymbol{I}}_n}} \right.} \right]}^{ - 1}}} \right\}} \end{array}$ (35)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{M}}^{\rm{T}}} - \sum\limits_{i = 1}^n {{\mathit{\boldsymbol{M}}_i}\frac{{{\rho _i}}}{{1 + \sum\limits_{i = 1}^n {{\rho _i}} }}} \mathit{\boldsymbol{M}}_i^{\rm{T}} - \sum\limits_{i = 1}^n {{\mathit{\boldsymbol{M}}_i}\frac{1}{{1 + {\rho _i}}}} \mathit{\boldsymbol{M}}_i^{\rm{T}} = }\\ {\sum\limits_{i = 1}^n {{\mathit{\boldsymbol{M}}_i}\left( {1 - \frac{{{\rho _i}}}{{1 + \sum\limits_{i = 1}^n {{\rho _i}} }} - \frac{1}{{1 + {\rho _i}}}} \right)} \mathit{\boldsymbol{M}}_i^{\rm{T}} = }\\ {\sum\limits_{i = 1}^n {{\mathit{\boldsymbol{M}}_i}\left( {\frac{{{\rho _i} \cdot \sum\limits_{i = 1}^n {{\rho _i}} - \rho _i^2}}{{\left( {1 + \sum\limits_{i = 1}^n {{\rho _i}} } \right) \cdot \left( {1 + {\rho _i}} \right)}}} \right)} \mathit{\boldsymbol{M}}_i^{\rm{T}}} \end{array}$ (36)

3 算例及分析 3.1 算例1

 $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {2.000\;0}&{ - 5.000\;0}&{1.000\;0}&{1.000\;0}&{ - 9.500\;0}\\ { - 2.000\;0}&{4.000\;0}&{1.000\;0}&{ - 1.050\;0}&{8.500\;0}\\ { - 2.000\;0}&{1.000\;0}&{1.000\;0}&{ - 1.000\;0}&{2.400\;0}\\ { - 1.000\;0}&{2.500\;0}&{4.000\;0}&{ - 0.500\;0}&{7.000\;0}\\ { - 1.000\;0}&{3.200\;0}&{4.000\;0}&{ - 0.500\;0}&{8.400\;0}\\ {1.000\;0}&{1.000\;0}&{ - 3.000\;0}&{0.400\;0}&{0.490\;0}\\ {3.000\;0}&{7.000\;0}&{ - 3.000\;0}&{1.500\;0}&{12.700\;0}\\ {5.000\;0}&{ - 1.000\;0}&{ - 2.000\;0}&{{\rm{2}}{\rm{.500}}\;{\rm{0}}}&{ - 3.000\;0}\\ {4.000\;0}&{2.000\;0}&{ - 2.000\;0}&{{\rm{2}}.0{\rm{1}}0\;0}&{3.000\;0}\\ {4.000\;0}&{3.000\;0}&{ - 2.000\;0}&{{\rm{2}}.000\;0}&{{\rm{5}}.000\;0} \end{array}} \right],\mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} { - 10.500\;0}\\ {10.450\;0}\\ {1.400\;0}\\ {12.000\;0}\\ {14.100\;0}\\ { - 0.110\;0}\\ {21.200\;0}\\ {1.500\;0}\\ {9.010\;0}\\ {12.000\;0} \end{array}} \right]$

 $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {1.881\;2}&{ - 5.118\;6}&{1.012\;9}&{1.080\;6}&{ - 9.533\;1}\\ { - 2.220\;2}&{3.894\;4}&{1.0656}&{ - 1.026\;8}&{8.415\;6}\\ { - 1.901\;4}&{1.147\;2}&{0.883\;2}&{ - 1.099\;0}&{2.449\;8}\\ { - 1.051\;9}&{2.505\;6}&{3.953\;9}&{ - 0.366\;0}&{7.148\;8}\\ { - 0.967\;3}&{3.078\;3}&{3.973\;8}&{ - 0.471\;0}&{8.345\;4}\\ {1.023\;4}&{0.995\;9}&{ - 3.121\;3}&{0.547\;9}&{0.405\;3}\\ {3.002\;1}&{6.887\;2}&{ - 3.131\;9}&{1.613\;8}&{12.6750}\\ {4.899\;6}&{ - 1.134\;9}&{ - 1.906\;9}&{2.431\;6}&{ - 2.933\;7}\\ {3.905\;3}&{1.973\;9}&{ - 1.998\;9}&{1.880\;8}&{2.914\;6}\\ {3.962\;6}&{3.095\;3}&{ - 2.064\;5}&{1.992\;7}&{4.879\;9} \end{array}} \right],\mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} { - 10.512\;0}\\ {10.443\;0}\\ {1.448\;5}\\ {11.940\;0}\\ {14.085\;0}\\ { - 0.153\;5}\\ {21.192\;0}\\ {1.653\;5}\\ {8.949\;4}\\ {11.865\;0} \end{array}} \right]$

 图 1 新方法迭代次数 Fig. 1 New method iterations

 图 2 新方法差值范数 Fig. 2 New method difference norm

 图 3 2种方法差值范数之差 Fig. 3 The difference between the difference norms of the two methods
3.2 算例2

 图 4 控制点点位 Fig. 4 Control points bitmap

3.3 算例分析

4 结语

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New Methods of Ill-Posed Total Least-Squares with Targeting Singular Value Corrections
WU Guangming1,2     LU Tieding1,2,3
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, MNR, 418 Guanglan Road, Nanchang 330013, China;
3. Jiangxi Province Key Lab for Digital Land, 418 Guanglan Road, Nanchang 330013, China
Abstract: The general ill-posed problem is that there are several singular eigenvalues in the coefficient matrix, and the singular value can be corrected with the target matrix in the calculation process. In the total least squares iteration process, the coefficient matrix is constantly changing, so the target matrix should also change accordingly. For target matrix changing, this paper deduces two new methods of ill-posed total least-squares targeting singular value corrections. By finding the new coefficient matrix and then finding the target matrix, the iteration is calculated with the parameter estimate and used in the example. The results show that these methods have some advantages.
Key words: ill-posed; total least squares; coefficient matrix; targeting correction