A structured total least squares algorithm for spatial straight line fitting

2015 ◽  
Author(s):  
Xiao-yuan Deng ◽  
Tie-ding Lu ◽  
Xiao-tao Chang ◽  
Xiao-yong Zhu
Keyword(s):  
Least Squares ◽  
Total Least Squares ◽  
Straight Line ◽  
Structured Total Least Squares ◽  
Least Squares Algorithm ◽  
Line Fitting

scholarly journals Weighted Total Least Squares (WTLS) Solutions for Straight Line Fitting to 3D Point Data

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1450
Author(s):  
Georgios Malissiovas ◽  
Frank Neitzel ◽  
Sven Weisbrich ◽  
Svetozar Petrovic
Keyword(s):  
Least Squares ◽  
Iterative Solution ◽  
Total Least Squares ◽  
Dispersion Matrix ◽  
Random Errors ◽  
Direct Solution ◽  
Weighted Total Least Squares ◽  
Straight Line ◽  
Point Data ◽  
Line Fitting

In this contribution the fitting of a straight line to 3D point data is considered, with Cartesian coordinates xi, yi, zi as observations subject to random errors. A direct solution for the case of equally weighted and uncorrelated coordinate components was already presented almost forty years ago. For more general weighting cases, iterative algorithms, e.g., by means of an iteratively linearized Gauss–Helmert (GH) model, have been proposed in the literature. In this investigation, a new direct solution for the case of pointwise weights is derived. In the terminology of total least squares (TLS), this solution is a direct weighted total least squares (WTLS) approach. For the most general weighting case, considering a full dispersion matrix of the observations that can even be singular to some extent, a new iterative solution based on the ordinary iteration method is developed. The latter is a new iterative WTLS algorithm, since no linearization of the problem by Taylor series is performed at any step. Using a numerical example it is demonstrated how the newly developed WTLS approaches can be applied for 3D straight line fitting considering different weighting cases. The solutions are compared with results from the literature and with those obtained from an iteratively linearized GH model.

Robust structured total least squares algorithm for passive location

2015 ◽  
Vol 26 (5) ◽  
pp. 946-953 ◽  
Author(s):  
Hao Wu ◽  
Shuxin Chen ◽  
Yihang Zhang ◽  
Hengyang Zhang ◽  
Juan Ni
Keyword(s):  
Least Squares ◽  
Total Least Squares ◽  
Passive Location ◽  
Structured Total Least Squares ◽  
Least Squares Algorithm

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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