New Total Least Squares Algorithm and Model with Applications to the Transformation among ITRF Realizations

2020 ◽  
Author(s):  
Jianqing Cai ◽  
Dalu Dong ◽  
Nico Sneeuw
Keyword(s):  
Least Squares ◽  
Coordinate Transformation ◽  
Total Least Squares ◽  
Design Matrix ◽  
Matrix Elements ◽  
Least Squares Problem ◽  
Transformation Parameters ◽  
Least Squares Algorithm

<p>A newly developed Converted Total Least Squares (CTLS) algorithm is introduced, which is to take the stochastic design matrix elements as virtual observations, and to transform the TLS problem into a traditional Least Squares problem. This new algorithm has the advantages that it can not only easily consider the weight of observations and the weight of stochastic design matrix, but also deal with TLS problem without complicated iteration processing, which enriches the TLS algorithm and solves the bottleneck restricting the application of TLS solutions. The notable development of the CTLS reveals also that CTLS estimator is identical to Gauss-Helmert model estimator in dealing with EIV model, especially in the case of similarity coordinate transformation. CTLS has been successfully applied to the estimation of the transformation parameters, their rates and related transformed residuals between actual ITRF realizations of ITRF2014 and ITRF2008 with obvious improvement of their accuracies.</p>

Comparison of Total Least Squares and Least Squares for Four- and Seven-parameter Model Coordinate Transformation

2016 ◽  
Vol 10 (4) ◽  
Author(s):  
You Wu ◽  
Jun Liu ◽  
Hui Yong Ge
Keyword(s):  
Least Squares ◽  
Coordinate Transformation ◽  
Relative Effectiveness ◽  
Satellite System ◽  
Total Least Squares ◽  
Design Matrix ◽  
Errors In Variables ◽  
Random Errors ◽  
Global Navigation Satellite ◽  
Observation Vector

AbstractTotal least squares (TLS) is a technique that solves the traditional least squares (LS) problem for an errors-in-variables (EIV) model, in which both the observation vector and the design matrix are contaminated by random errors. Four- and seven-parameter models of coordinate transformation are typical EIV model. To determine which one of TLS and LS is more effective, taking the four- and seven-parameter models of Global Navigation Satellite System (GNSS) coordinate transformation with different coincidence pointsas examples, the relative effectiveness of the two methods was compared through simulation experiments. The results showed that in the EIV model, the errors-in-variables-only (EIVO) model and the errors-in-observations-only (EIOO) model, TLS is slightly inferior to LS in the four-parameter model coordinate transformation, and TLS is equivalent to LS in the seven-parameter model coordinate transformation. Consequently, in the four- and seven-parameter model coordinate transformation, TLS has no obvious advantage over LS.

scholarly journals PERFORMING 3D SIMILARITY TRANSFORMATION WITH LARGE ROTATION ANGLES USING CONSTRAINED MULTIVARIATE TOTAL LEAST SQUARES

2020 ◽  
Vol 26 (4) ◽  
Author(s):  
Wuyong Tao ◽  
Xianghong Hua ◽  
Shaoquan Feng
Keyword(s):  
Least Squares ◽  
Similarity Transformation ◽  
Coefficient Matrix ◽  
Total Least Squares ◽  
Translation Vector ◽  
Large Rotation ◽  
Parameter Matrix ◽  
Transformation Parameters ◽  
Least Squares Algorithm ◽  
3D Similarity

Abstract: 3D similarity transformation is frequently encountered operation in the field of geodetic data processing, and there are many applications that involve large rotation angles. In previous studies, the errors of the coefficient matrix were usually neglected and a least squares algorithm was applied to calculate the transformation parameters. However, the coefficient matrix is composed of the point coordinates in source coordinate system, i.e., the coefficient matrix is also contaminated by errors. Therefore, a total least squares algorithm should be applied. In this paper, a new method is proposed to address the 3D similarity transformation problem with large rotation angles. Firstly, the scale factor and rotation matrix are put together as the parameter matrix to avoid the rank-defect problem. Then, the translation vector is removed and the multivariate model is constructed. Finally, the constraints are introduced according to the properties of the parameter matrix and the constrained multivariate total least squares algorithm is derived to obtain the transformation parameters. The experimental results show that the proposed method has a high computational efficiency.

Position Estimation of Single Camera Visual System Based on Total Least Squares Algorithm

2012 ◽  
Vol 239-240 ◽  
pp. 1352-1355
Author(s):  
Jing Zhou ◽  
Yin Han Gao ◽  
Chang Yin Liu ◽  
Ji Zhi Li
Keyword(s):  
Visual System ◽  
Least Squares ◽  
Total Least Squares ◽  
Position Estimation ◽  
Feature Points ◽  
Perspective Theory ◽  
Optical Feature ◽  
The Matrix ◽  
Least Squares Algorithm ◽  
Set Up

The position estimation of optical feature points of visual system is the focus factor of the precision of system. For this problem , to present the Total Least Squares Algorithm . Firstly , set up the measurement coordinate system and 3D model between optical feature points, image points and the position of camera according to the position relation ; Second , build the matrix equations between optical feature points and image points ; Then apply in the total least squares to have an optimization calculation ; Finally apply in the coordinate measuring machining to have a simulation comparison experiment , the results indicate that the standard tolerance of attitude coordinate calculated by total least squares is 0.043mm, it validates the effectiveness; Compare with the traditional method based on three points perspective theory, measure the standard gauge of 500mm; the standard tolerance of traditional measurement system is 0.0641mm, the standard tolerance of Total Least Squares Algorithm is 0.0593mm; The experiment proves the Total Least Squares Algorithm is effective and has high precision.

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