A Quaternion-based Geodetic Datum Transformation Algorithm

Shen, Yunzhong; Chen, Y.; Zheng, D. H.

doi:10.1007/s00190-006-0054-8

Cited by 78 publications

(32 citation statements)

References 12 publications

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“…Centering: cx, cy,X,Ȳ by Eqs. (24), (27) In the borderline cases ky = and kx = , the formulas of steps 1, 2 and 4 are also valid. C. Marx …”

Section: Calculation Methodsmentioning

confidence: 97%

A weighted adjustment of a similarity transformation between two point sets containing errors

Marx¹

2017

Journal of Geodetic Science

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Abstract:For an adjustment of a similarity transformation, it is often appropriate to consider that both the source and the target coordinates of the transformation are affected by errors. For the least squares adjustment of this problem, a direct solution is possible in the cases of speci c weighing schemas of the coordinates. Such a problem is considered in the present contribution and a direct solution is generally derived for the m-dimensional space. The applied weighing schema allows (fully populated) pointwise weight matrices for the source and target coordinates, both weight matrices have to be proportional to each other. Additionally, the solutions of two borderline cases of this weighting schema are derived, which only consider errors in the source or target coordinates. The investigated solution of the rotation matrix of the adjustment is independent of the scaling between the weight matrices of the source and the target coordinates. The mentioned borderline cases, therefore, have the same solution of the rotation matrix. The direct solution method is successfully tested on an example of a 3D similarity transformation using a comparison with an iterative solution based on the Gauß-Helmert model.

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“…Centering: cx, cy,X,Ȳ by Eqs. (24), (27) In the borderline cases ky = and kx = , the formulas of steps 1, 2 and 4 are also valid. C. Marx …”

Section: Calculation Methodsmentioning

confidence: 97%

A weighted adjustment of a similarity transformation between two point sets containing errors

Marx¹

2017

Journal of Geodetic Science

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“…, the bilinear form (35) can be rewritten so that the rotation matrix R is replaced with the unknown quaternion q ¼ q 0 ; q Shen et al (2006) give the following equation for the rotation matrix R and the q quaternion:…”

Section: Determination Of the Linear And Shift Parametersmentioning

confidence: 99%

“…The maximum of the quadratic form (38) is obtained when q is an eigenvector of N, and then its value is equal to the eigenvalue of N (Shen et al 2006), and thus we have to determine the maximal eigenvalue of N and the eigenvector q (the quaternion we are looking for) belonging to this eigenvalue.…”

Section: Determination Of the Linear And Shift Parametersmentioning

confidence: 99%

A comparison of different solutions of the Bursa–Wolf model and of the 3D, 7-parameter datum transformation

Závoti

Kalmár

2015

Acta Geod Geophys

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The present work deals with an important theoretical problem of geodesy: we are looking for a mathematical dependency between two spatial coordinate systems utilizing common pairs of points whose coordinates are given in both systems. In geodesy and photogrammetry the most often used procedure to move from one coordinate system to the other is the 3D, 7 parameter (Helmert) transformation. Up to recent times this task was solved either by iteration, or by applying the Bursa-Wolf model. Producers of GPS/GNSS receivers install these algorithms in their systems to achieve a quick processing of data. But nowadays algebraic methods of mathematics give closed form solutions of this problem, which require high level computer technology background. In everyday usage, the closed form solutions are much more simple and have a higher precision than earlier procedures and thus it can be predicted that these new solutions will find their place in the practice. The paper discusses various methods for calculating the scale factor and it also compares solutions based on quaternion with those that are based on rotation matrix making use of skew-symmetric matrix.

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“…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. Han (2010) presented a stepwise approach to individually calculate the transformation parameters by the physical interpretation of similarity transformation.…”

Section: Introductionmentioning

confidence: 99%

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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A Quaternion-based Geodetic Datum Transformation Algorithm

Cited by 78 publications

References 12 publications

A weighted adjustment of a similarity transformation between two point sets containing errors

A weighted adjustment of a similarity transformation between two point sets containing errors

A comparison of different solutions of the Bursa–Wolf model and of the 3D, 7-parameter datum transformation

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

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