Nonlinear analysis of the three-dimensional datum transformation [conformal group ? 7 (3)]

Grafarend, Erik W.; Awange, Joseph

doi:10.1007/s00190-002-0299-9

Cited by 81 publications

(72 citation statements)

References 20 publications

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“…The case data is from Grafarend and Awange (2003). The coordinates of control points in system B (local system) and A (WGS-84 system) is listed in Table 6.…”

Section: Actual Casementioning

confidence: 99%

“…When the weight matrix is an identity matrix, the computed results of seven parameters with PA and CPA are listed in Table 7. For the situation that the weight matrix is a point-wise matrix, i.e., every point has isotropic weight and is independent of each other, the point-wise matrix is generated by the way introduced in Grafarend and Awange (2003) and is listed in Table 8. The computed results of seven parameters with PA and CPA are listed in Table 9.…”

Section: Actual Casementioning

confidence: 99%

“…However, because of the complexity of mathematical derivation, only several analytical algorithms have been put forward. Grafarend and Awange (2003) presented the Procrustes algorithm which utilized the singular value decomposition technique. Shen et al (2006) presented a quaternion-based algorithm which utilized the quaternion property and eigenvalue-eigenvector decomposition.…”

Section: Introductionmentioning

confidence: 99%

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Analytical algorithm of weighted 3D datum transformation using the constraint of orthonormal matrix

Zeng

2015

Earth Planet Sp

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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.

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“…The case data is from Grafarend and Awange (2003). The coordinates of control points in system B (local system) and A (WGS-84 system) is listed in Table 6.…”

Section: Actual Casementioning

confidence: 99%

Section: Actual Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analytical algorithm of weighted 3D datum transformation using the constraint of orthonormal matrix

Zeng

2015

Earth Planet Sp

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“…(see, e.g., Aktuğ (2009) ;Aktuğ 2012;Akyilmaz 2007;Arun et al 1987;Burša 1967;Chang 2015;Chang et al 2017;El-Habiby et al 2009;El-Mowafy et al 2009;Grafarend and Awange 2003;Han 2010;Horn 1986Horn , 1987Han and Van Gelder 2006;Horn et al 1988;Jaw and Chuang 2008;Jitka 2011;Kashani 2006;Krarup 1985;Leick 2004;Leick and van Gelder 1975;Mikhail et al 2001;Neitzel 2010;Soler 1998;Soler and Snay 2004;Soycan and Soycan 2008;Teunissen 1986;Teunissen 1988;Wang et al 2014;Závoti and Kalmár 2016;Zeng 2014;Zeng 2015;Zeng et al 2016;Zeng and Yi 2011). Helmert transformation problem is to determine the seven transformation parameters including three rotation angles, three translation parameters and one scale factor using a set of control points (the number of control points should be equal to or more than three because three equations can be constructed for one point).…”

Section: Introductionmentioning

confidence: 99%

“…Numerous algorithms of Helmert transformation have been presented. On the one hand, the algorithms can be classified to a numerical iterative algorithm, e.g., El-Habiby et al (2009), Zeng and Yi (2011), Paláncz et al 2013, Zeng et al (2016, etc., and an analytical algorithm, e.g., Grafarend and Awange (2003), Shen et al (2006a, b), Zeng (2015), etc. For the numerical iterative algorithm, initial values of transformation parameters are usually required.…”

Section: Introductionmentioning

confidence: 99%

A dual quaternion algorithm of the Helmert transformation problem

Zeng

Fang

Chang

et al. 2018

Earth Planets Space

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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 Závoti and Kalmár (Acta Geod Geophys 51: [245][246][247][248][249][250][251][252][253][254][255][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][67][68][69][70][71][72][73][74][75][76] 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.

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