Weighted coordinate transformation formulated by standard least-squares theory

Mihajlović, Dušan; Cvijetinović, Željko

doi:10.1080/00396265.2016.1173329

Cited by 6 publications

(4 citation statements)

References 36 publications

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“…In that case, [9] has shown that the total least-squares solution can be obtained easily from a rigorous evaluation of the Gauss-Helmert model. A good discussion about total leastsquares algorithms can be seen in [9,10,18]. Different from the approaches mentioned so far, in the following sections, we present an alternative method based on the Monte Carlo method.…”

Section: The Gauss-markov Model In the Context Of Coordinate Transformationmentioning

confidence: 99%

Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

Rofatto¹,

Matsuoka

Klein

et al. 2019

Sci. Plena

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One of the main tasks of professionals in the Earth sciences is to convert coordinates from one reference frame into another. The coordinates transformation is widely needed and applied in all branches of modern geospatial activities. To convert from one reference frame to another, it is necessary initially to determine the parameters of a coordinate transformation model. A common way to estimate the transformation parameters is using the least-squares theory within a linearized Gauss-Markov Model (GMM). Another approach arises from a numerical method. Here, a Monte Carlo Method (MCM) is used to infer the uncertainty of estimators. This method is based on: (i) the assignment of probability distributions to the coordinates in the two reference frames, (ii) the determination of a discrete representation of the probability distribution for the transformation parameters, and (iii) the determination of the associated uncertainties from this discrete representation of the estimates of the transformation parameters. In this contribution, we compare the weighted least-squares within the GMM (WLSE-GM) and the proposed method regarding the transformation problem of computing 2D similarity transformation parameters. The results show that, the transformation parameters uncertainties are higher for the LS-MC than the WLSE-GM. This is due to the fact that the WLSE-GM solution does not take into account the uncertainties associated with the system matrix. In future studies the Monte Carlo method should be applied to the nonlinear least-squares solution.

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Section: The Gauss-markov Model In the Context Of Coordinate Transformationmentioning

confidence: 99%

Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

Rofatto¹,

Matsuoka

Klein

et al. 2019

Sci. Plena

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show abstract

“…So far, a large number of algorithms of three-dimensional coordinate transformation have been presented (cf., e.g., Aydin et al 2018;Grafarend and Awange 2003;Kanatani and Niitsuma 2012;Kurt 2018;Ligas and Prochniewicz 2019;Mahboub 2016;Marx 2017;Mercan et al 2018;Mihajlović and Cvijetinović 2017;Păun et al 2017;Shen et al 2006;Uygur et al 2020;Walker et al 1991;Wang et al 2014;Zeng 2015;Yi 2010, 2011;Zeng et al 2016Zeng et al , 2018Zeng et al , 2019. They can be classified into iterative algorithm or analytical algorithm.…”

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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Weighted coordinate transformation formulated by standard least-squares theory

Cited by 6 publications

References 36 publications

Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

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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