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Proceedings ArticleDOI

A new analytical solution of nonlinear geodetic datum transformation

18 Jun 2010-pp 1-5
TL;DR: A new analytical solution is proposed that expresses the scale parameter and translation parameters as the function of rotation matrix, through the derivation process, and obtains the analytical solution to the seven parameters.
Abstract: Nonlinear geodetic datum transformation is particular suitable for the case of large rotation angles, and has been an active area of research in recent years. In this paper, we propose a new analytical solution. The method expresses the scale parameter and translation parameters as the function of rotation matrix, through the derivation process. Then, according to the orthogonality of rotation matrix, the method adopts Rodrigues matrix to represent it. Using the relative formula of Rodrigues matrix with anti-symmetric matrix which is formed by a rotation vector, we derive the solution formula of the rotation vector. Finally, we obtain the analytical solution to the seven parameters. The case studies indicate that the proposed method is valid, regardless of whether the coordinates are polluted by noise or not, and whether rotation angles are big or small.
Citations
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Journal ArticleDOI
TL;DR: The paper adopts an unit quaternion to represent three-dimensional rotation matrix, then puts forward a quaternION-based iterative solution of the problem, showing that the quaternions-based solution has no dependence on the initial value of parameter and desirable result with fast speed.
Abstract: Three-dimensional coordinate transformation problem is the most frequent problem in photogrammetry, geodesy, mapping, geographical information science (GIS), and computer vision. To overcome the drawback that traditional solution of the problem based on rotation angles depends strongly on initial value of parameter, which makes the method ineffective in the case of super-large rotation angle, the paper adopts an unit quaternion to represent three-dimensional rotation matrix, then puts forward a quaternion-based iterative solution of the problem. The cases study shows that the quaternion-based solution has no dependence on the initial value of parameter and desirable result with fast speed. Thus it is valid for three-dimensional coordinate transformation of any rotation angle.

21 citations


Cites methods from "A new analytical solution of nonlin..."

  • ...The authors presented a new analytical algorithm based on optimization process and the good properties of Rodrigues matrix ([14])....

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Journal ArticleDOI
TL;DR: The paper investigates the stability of the presented algorithm in the cases that the common points are distributed in 3D, 2D, and 1D spaces including the approximate 2D and1D spaces, and gives the corresponding modified formula of rotation matrix.
Abstract: Based on the Lagrangian extremum law with the constraint that rotation matrix is an orthonormal matrix, the paper presents a new analytical algorithm of weighted 3D datum transformation It is a stepwise algorithm Firstly, the rotation matrix is computed using eigenvalue-eigenvector decomposition Then, the scale parameter is computed with computed rotation matrix Lastly, the translation parameters are computed with computed rotation matrix and scale parameter The paper investigates the stability of the presented algorithm in the cases that the common points are distributed in 3D, 2D, and 1D spaces including the approximate 2D and 1D spaces, and gives the corresponding modified formula of rotation matrix The comparison of the presented algorithm and classic Procrustes algorithm is investigated, and an improved Procrustes algorithm is presented since that the classic Procrustes algorithm may yield a reflection rather than a rotation in the cases that the common points are distributed in 2D space A simulative numerical case and a practical case are illustrated

19 citations


Cites methods from "A new analytical solution of nonlin..."

  • ...Zeng and Yi (2010) presented a new analytical algorithm based on the good properties of Rodrigues matrix and Gibbs vector....

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Journal ArticleDOI
TL;DR: In this article, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion and a rigid iterative algorithm is presented.
Abstract: Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Zavoti and Kalmar (Acta Geod Geophys 51:245–256, 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77:66–76, 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.org/10.1186/s40623-015-0263-6 ).

14 citations


Cites methods from "A new analytical solution of nonlin..."

  • ...2006a, b; Zeng and Yi 2011), algorithms based on Rodrigues matrix and Gibbs vector see (e.g., Zeng and Huang 2008; Zeng and Yi 2010; Zeng et al. 2016), algorithms based on dual quaternion (see, e....

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  • ...(2006a, b), Zeng and Huang (2008), Han (2010), Zeng and Yi (2010, 2011), Zeng (2015), Závoti and Kalmár (2016), (58)s = 1 2c (B− C)r (59) 1 c ( CT − BT ) (B− C)r − 2 Ar + β1r = 0....

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  • ...2006a, b; Zeng and Yi 2011), algorithms based on Rodrigues matrix and Gibbs vector see (e.g., Zeng and Huang 2008; Zeng and Yi 2010; Zeng et  al....

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Journal ArticleDOI
TL;DR: In this paper, the authors presented a rigid iterative algorithm of Helmert transformation using a unit dual quaternion and showed that the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange.
Abstract: The rigid motion involving both rotation and translation in the 3D space can be simultaneously described by a unit dual quaternion. Considering this excellent property, the paper constructs the Helmert transformation (seven-parameter similarity transformation) model based on a unit dual quaternion and then presents a rigid iterative algorithm of Helmert transformation using a unit dual quaternion. Because of the singularity of the coefficient matrix of the normal equation, the nine parameter (including one scale factor and eight parameters of a dual quaternion) Helmert transformation model is reduced into five parameter (including one scale factor and four parameters of a unit quaternion which can represent the rotation matrix) Helmert transformation one. Besides, a good start estimate of parameter is required for the iterative algorithm, hence another algorithm employed to compute the initial value of parameter is put forward. The numerical experiments involving a case of small rotation angles i.e. geodetic coordinate transformation and a case of big rotation angles i.e. the registration of LIDAR points are studied. The results show the presented algorithms in this paper are correct and valid for the two cases, disregarding the rotation angles are big or small. And the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange (J Geod 77:66–76, 2003), the orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015), and the algorithm from Wang et al. (J Photogramm Remote Sens 94:63–69, 2014).

10 citations


Cites background from "A new analytical solution of nonlin..."

  • ...2006; Zeng and Yi 2011; Závoti and Kalmár 2016), Rodrigues matrix and Gibbs vector see (e.g. Zeng and Huang 2008; Zeng and Yi 2010; Závoti and Kalmár 2016; Zeng et al....

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Journal ArticleDOI
TL;DR: Two kinds of iterative approach of 3D datum transformation with a non-isotropic weight are presented, one of which uses the Lagrangian minimum function in the variable of rotation matrix as the objective function and the other based on derivatives, and the result shows from the view of computing speed and reliability, the iteratives approach based on derivative is preferred.
Abstract: The analytical solution of 3D datum transformation with an isotropic weight has been elegantly presented based on Procrustes algorithm (singular value decomposition). But the existence of analytical solution of 3D datum transformation with a non-isotropic weight needs further investigation. Based on the Lagrangian extremum law, the paper derives the analytical formula for translation parameter and scale factor, but because the rotation matrix is unsolved, the analytical solution does not exist. For this reason, the paper presents two kinds of iterative approach of 3D datum transformation with a non-isotropic weight. One is the iterative approach dependent on the objective function value, which uses the Lagrangian minimum function in the variable of rotation matrix as the objective function, and the other is the iterative approach dependent on the derivative of function, which uses the 3D datum transformation model that eliminates the translation parameter. In order to improve the speed and reliability of iterative computation, the form of rotation matrix represented by Rodrigues matrix instead of rotation angles or unit quaternion is adopted for the two iterative approaches. A numerical experiment is demonstrated, and comparison analysis of the two iterative approaches is carried out. The result shows from the view of computing speed and reliability, the iterative approach based on derivatives is preferred.

9 citations


Cites background or methods from "A new analytical solution of nonlin..."

  • ...Zeng and Yi (2010a) presented an analytical algorithm based on Rodrigues matrix....

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  • ...(18) namely R is a matrix with 3 rows and 3 columns, usually represented by rotation angles (see e.g. El-Habiby et al. 2009; Zeng and Yi, 2011), unit quaternion (see e.g. Shen et al. 2006; Zeng and Yi 2011), and Rodrigues matrix (see e.g. Zeng and Yi 2010a)....

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  • ...Pattern search method is a kind of very popular direct search method, and for its more detail, the readers are referred to e.g. Torczon (1997), Lewis, et al. (2000), Dolan, et al. (2003), Al-Sumaita, et al. (2007), Zeng and Yi (2010b)....

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References
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Journal ArticleDOI
TL;DR: The problem of incorporating the stochasticity measures of both systems of coordinates involved in the seven parameter datum transformation problem [conformal group ℂ7(3)] which is free of linearization and any iteration procedure can be considered to be solved.
Abstract: The weighted Procrustes algorithm is presented as a very effective tool for solving the three-dimensional datum transformation problem In particular, the weighted Procrustes algorithm does not require any initial datum parameters for linearization or any iteration procedure As a closed-form algorithm it only requires the values of Cartesian coordinates in both systems of reference Where there is some prior information about the variance–covariance matrix of the two sets of Cartesian coordinates, also called pseudo-observations, the weighted Procrustes algorithm is able to incorporate such a quality property of the input data by means of a proper choice of weight matrix Such a choice is based on a properly designed criterion matrix which is discussed in detail Thanks to the weighted Procrustes algorithm, the problem of incorporating the stochasticity measures of both systems of coordinates involved in the seven parameter datum transformation problem [conformal group ℂ7(3)] which is free of linearization and any iterative procedure can be considered to be solved Illustrative examples are given

80 citations


"A new analytical solution of nonlin..." refers methods in this paper

  • ...By taking the partial derivative of (8) with respect to λ and setting it to be zero, we can derive the solution of λ as...

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Journal ArticleDOI
TL;DR: In this paper, the authors introduce quaternions to represent rotation parameters and derive the formulae to compute quaternion, translation and scale parameters in the Bursa-Wolf geodetic datum transformation model from two sets of co-located 3D coordinates.
Abstract: This paper briefly introduces quaternions to represent rotation parameters and then derives the formulae to compute quaternion, translation and scale parameters in the Bursa–Wolf geodetic datum transformation model from two sets of co-located 3D coordinates. The main advantage of this representation is that linearization and iteration are not needed for the computation of the datum transformation parameters. We further extend the formulae to compute quaternion-based datum transformation parameters under constraints such as the distance between two fixed stations, and develop the corresponding iteration algorithm. Finally, two numerical case studies are presented to demonstrate the applications of the derived formulae.

72 citations


"A new analytical solution of nonlin..." refers methods in this paper

  • ...Once v is solved, r is got by using (13), and R is obtained by using (12), then λ can be computed by using (9), and lastly t is computed by using (6)....

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  • ...(9) Obviously, if rotation matrix R is solved, we can get the solution of scale parameter λ and translation parameters t by using (9) and (6)....

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01 Jan 2003
TL;DR: In this article, a non-linear adjustment model of three-dimensional coordinate transformation that resolves the limited conditions with linear models is given, and the conditions with this linear model are discussed.
Abstract: In this paper, the limited conditions with this linear model are discussed. At the same time, non-linear adjustment model of three-dimensional coordinate transformation that resolves the limited conditions with linear models is given.

17 citations


"A new analytical solution of nonlin..." refers background or methods in this paper

  • ...MATHEMATICS MODEL OF DATUM TRANSFORMATION Usually, 3D datum transformation model uses Bursa-Wolf model, which can be written as [10] , i i Rp t s λ + = (1)...

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  • ...Secondly, compute coordinates in system A (true values) by using (1)....

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  • ..., (1) can be rewritten as follows by using the centrobaric coordinates (namely space vector) form....

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  • ...The transformation residuals is the differences between the calculated values and true values of coordinates in system A, of which the former is obtained by substituting coordinates in system B and solved parameters into (1)....

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Yao, Jili, Xu, Yufei, Xiao, Wei 
01 Jan 2007
TL;DR: In this paper, Molodensky et al. measured angles 0,0,1/s for the Lodrigues and WTUSMUSM, and showed that the angles were 0, 0,1,1 /s.
Abstract: 三个转变模型(囊狼, Molodensky,和 WTUSM ) 通常在二数据系统转变之间被使用。当旋转角度是小的时,线性模型被使用;然而,当旋转角度变得更大时,模型错误将被生产。在这篇论文,我们在场有三个主要术语的一个方法:传统的旋转 angles0,0,1/s 被代替与一, b,是三各自的价值在的 c 反对称或 Lodrigues 矩阵;(2 ) 直接并且精确地在旋转角度的任何价值计算七个参数的公式;并且一个相应调整模型被建立。这个方法不使用三角功能。相反它使用增加,减法,增加和分割,和方程的复杂性被减少,使计算容易、快。

9 citations

01 Jan 2006
TL;DR: In this article, a solution model only to regularize translation parameters is derived, and the results show that the precision of the transformed coordinates in peripheral area can be significantly improved by regularization; and will linearly decrease as the extension of extrapolating distance.
Abstract: The application of GPS is often needed to transform coordinates.If the transformation parameters are solved with the GPS data in a small area,the precision of transformation parameters may be very poor,especially the translation parameters.The reason is that the translation parameters and rotation parameters are high correlated in this case,which causes the solution model to be illposed,and regularization solution is an efficient method in dealing with ill-posed model.This paper discusses the regularization solution in solving 3-dimensional coordinate transformation parameters with small area's data,in order to improve the precision and extend the application range of the solved transformation parameters;a solution model only to regularize translation parameters is also derived in this paper.The model and algorithm are verified with 500 numerical simulated examples;and the results show that the precision of the transformed coordinates in peripheral area can be significantly improved by regularization;and will linearly decrease as the extension of extrapolating distance.

8 citations


"A new analytical solution of nonlin..." refers background in this paper

  • ...(10) According to the physical meaning of λ , i....

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  • ...i i s p λ = Δ Δ , where the symbol i represents the norm of space vector, (10) can be transformed as...

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