General Total Least Squares Theory for Geodetic Coordinate Transformations

Qin, Yuxin; Fang, Xing; Zeng, Wen; Wang, Bin

doi:10.3390/app10072598

Cited by 10 publications

(5 citation statements)

References 21 publications

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“…We consider that besides the classical least squares (Least Squares -LS), there are other methods such as the total least squares method (Total Least Squares -TLS) using the GM model (Gauss-Markov), EIV model (Errors) in variables),…. [1,2,3,6,7]. However, using the EIV model, the GM model must inverse the matrix has a large dimension when evaluating the accuracy of the unknowns and the measured values.…”

Section: Discussionmentioning

confidence: 99%

“…Therefore, in this condition, it may not be reasonable to use the classical least squares method to deal with this problem. A suitable approach is the total least squares method (Total Least Squares -TLS) [2,3,6,7]. The total least squares method has been developed in many fields of numerical analysis; but named of orthogonal regression or errors in variables (Errors In Variables -EIV), this method has a long history in the statistical literature.…”

Section: The Eio Modelmentioning

confidence: 99%

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Introduction of Eio Model in TLS Method to Calculate the Coordinates Transformation by Helmert’s Formula

Quoc¹,

Truong²,

Tram

2021

interconf

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The EIO (Errors In Observations) model is used in the total least squares method to calculate, process geodetic data. Next to the classical least squares method, it is applied to solve more solutions. When we use the EIO model in calculus and process, performing a matrix inverse has a large dimension will be avoided. Moreover, the calculation and accuracy evaluation steps are based on the iterative algorithm to get the results. In this paper, the authors use the procedure of calculating and evaluating the accuracy of the EIO model in the experimental calculation of the coordinate transformation according to the Helmert formula

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Section: Discussionmentioning

confidence: 99%

Section: The Eio Modelmentioning

confidence: 99%

Introduction of Eio Model in TLS Method to Calculate the Coordinates Transformation by Helmert’s Formula

Quoc¹,

Truong²,

Tram

2021

interconf

View full text Add to dashboard Cite

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“…(cf. Arun et al 1987;Besl and McKay 1992;Chang 2016;Chen et al 2004;Crosilla and Beinat 2002;Fan et al 2015;Fang 2015;Felus and Burtch 2009;Horn 1987;Horn et al 1988;Ioannidou and Pantazis 2020;Li et al 2022;Odziemczyk 2020;Ruffhead 2021;Qin et al 2020;Závoti and Kalmár 2016). The threedimensional coordinate transformation usually adopts seven-parameter similarity transformation model.…”

Section: Introductionmentioning

confidence: 99%

Analytical dual quaternion algorithm of the weighted three-dimensional coordinate transformation

Zeng

Wang

et al. 2022

Earth Planets Space

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Considering that a unit dual quaternion can describe elegantly the rigid transformation including rotation and translation, the point-wise weighted 3D coordinate transformation using a unit dual quaternion is formulated. The constructed transformation model by a unit dual quaternion does not need differential process to eliminate the three translation parameters, while traditional models do. Based on the Lagrangian extremum law, the analytical dual quaternion algorithm (ADQA) of the point-wise weighted 3D coordinate transformation is proved existed and derived in detail. Four numerical cases, including geodetic datum transformation, the registration of LIDAR point clouds, and two simulated cases, are studied. This study shows that ADQA is valid as well as the modified procrustes algorithm (MPA) and the orthonormal matrix algorithm (OMA). ADQA is suitable for the 3D coordinate transformation with point-wise weight and no matter rotation angles are small or big. In addition, the results also indicate that if the distribution of common points degrades from 3D or 2D space to 1D space, the solvable correct transformation parameters decrease. In other words, all common points should not be located on a line. From the perspective of improving the transformation accuracy, high accurate control points (with small errors in the coordinates) should be chosen, and it is preferred to decrease the rotation angles as much as possible. Graphical Abstract

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“…e.g. Chen et al (2004), El-Habiby et al (2009, Neitzel (2010), Yi (2011), Fang (2015), Mahboub (2016), Zeng et al (2016), Mihajlović and Cvijetinović (2017), Kurt (2018), Mercan et al (2018), Zeng et al (2019), Qin et al (2020), Ioannidou and Pantazis (2020). They require initial values of transformation parameters and linearization and iterative computation.…”

Section: Introductionmentioning

confidence: 99%

Extended WTLS iterative algorithm of 3D similarity transformation based on Gibbs vector

Zeng

He²,

Le-geng

et al. 2021

Acta Geod Geophys

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Considering coordinate errors of both control points and non-control points, and different weights between control points and non-control points, this contribution proposes an extended weighted total least squares (WTLS) iterative algorithm of 3D similarity transformation based on Gibbs vector. It treats the transformation parameters and the target coordinate of non-control points as unknowns. Thus it is able to recover the transformation parameters and compute the target coordinate of non-control points simultaneously. It is also able to assess the accuracy of the transformation parameters and the target coordinates of non-control points. Obviously it is different from the traditional algorithms that first recover the transformation parameters and then compute the target coordinate of non-control points by the estimated transformation parameters. Besides it utilizes a Gibbs vector to represent the rotation matrix. This representation does not introduce additional unknowns; neither introduces transcendental function like sine or cosine functions. As a result, the presented algorithm is not dependent to the initial value of transformation parameters. This excellent performance ensures the presented algorithm is suitable for the big rotation angles. Two numerical cases with big rotation angles including a real world case (LIDAR point cloud registration) and a simulative case are tested to validate the presented algorithm. Keywords3D similarity transformation • Weighted total least squares (WTLS) • Iterative algorithm • Initial value of parameter • Big rotation angels * Huaien Zeng

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General Total Least Squares Theory for Geodetic Coordinate Transformations

Cited by 10 publications

References 21 publications

Introduction of Eio Model in TLS Method to Calculate the Coordinates Transformation by Helmert’s Formula

Introduction of Eio Model in TLS Method to Calculate the Coordinates Transformation by Helmert’s Formula

Analytical dual quaternion algorithm of the weighted three-dimensional coordinate transformation

Extended WTLS iterative algorithm of 3D similarity transformation based on Gibbs vector

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