▶ Gathers original articles on topics in earth and planetary sciences ▶ Coverage includes geomagnetism, aeronomy, space science, seismology, volcanology, geodesy and planetary science ▶ Official journal of the Society of Geomagnetism and Earth, Planetary and Space Sciences, The Seismological Society of Japan, The Volcanological Society of Japan, The Geodetic Society of Japan, and The Japanese Society for Planetary Sciences

「Earth,Planets and Space」のオープンアクセス出版

In the Helmert transformation model, the rotation is more difficult to be treated in terms of representation, estimation, and error analysis. First, two classes of representations of the rotation, i.e. the redundant class including the direction cosine matrix and the unit quaternion, and the minimum class including the rotation vector, the Gibbs vector, the modified Rodrigues parameters, and the Euler angles, are reviewed. It is concluded that in general the redundant class should be preferred as they are transcendental-function-free, singularity-free, and discontinuity-free. Second, two classes of estimation errors, i.e. the additive and the multiplicative errors, are defined and compared in detail. While the multiplicative errors are more convenient, the relationship among different representations and the relationship with their additive counterparts are also explored from first principle. It can be seen as a review paper; however, the content concerning the relationship between the additive and the mu...

Representation of the rotation parameter estimation errors in the Helmert transformation model

On least-squares solution to 3D similarity transformation problem under Gauss-Helmert model

The standard forward transformation equation plays a major role in coordinate transformation between global and local datums. Thus, it is a prerequisite step in the forward conversion of geodetic coordinates into cartesian coordinates in coordinate transformation from global to local datum and vice versa. Numerous studies have been carried out on converting cartesian coordinates to geodetic coordinates (reverse procedure) through the application of iterative, approximate, closed form, vector-based and computational intelligence algorithms. However, based on literature covered pertaining to this study, it was realized that the existing researches do not fully address the issue of applying and testing alternative techniques in the case of the forward conversion. Hence, the purpose of this present study was to explore the coordinate conversion performance of two different artificial neural network approaches (backpropagation artificial neural network (BPANN) and radial basis function neural network (RBFNN)) and multiple linear regression (MLR). The statistical findings revealed that the BPANN, RBFNN and MLR offered satisfactory prediction of cartesian coordinates. However, the RBFNN compared to BPANN and MLR showed better stability and more accurate prediction results. Furthermore, in terms of maximum three-dimensional position error, the RBFNN attained 0.004 m while 0.011 and 0.627 m were achieved, respectively, by MLR and BPANN. By virtue of the success achieved in this study, the main conclusion drawn here is that RBFNN provides a promising alternative in the forward conversion of geodetic coordinates into cartesian coordinates. Therefore, the capability of artificial neural network as a powerful tool for solving majority of function approximation problems in geodesy has been demonstrated.

Capability of Artificial Neural Network for Forward Conversion of Geodetic Coordinates $$(\phi ,\lambda ,h)$$ ( ϕ , λ , h ) to Cartesian Coordinates ( X , Y , Z )

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
 ).

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A dual quaternion algorithm of the Helmert transformation problem

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

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).

Iterative solution of Helmert transformation based on a unit dual quaternion

The 3D similarity coordinate transformation is fundamental and frequently encountered in many areas of work such as geodesy, engineering surveying, LIDAR, terrestrial laser scanning, photogrammetry, machine vision, etc. The algorithms of 3D similarity transformation are divided into two categories. One is a closed-form algorithm that is straightforward and fast. However, it cannot provide the accuracy information for the transformation parameters. The other category of algorithm is iterative, and this can offer the accuracy information for the transformation parameters. However, the latter usually needs a good initial value of the unknown. Considering the accuracy information for transformation parameters is essential or indispensable from the viewpoint of uncertainty, this contribution proposes a weighted total least squares (WTLS) iterative algorithm of the 3D similarity coordinate transformation based on Gibbs vectors. It is fast in terms of fewer iterations, reliable and does not need good initial values of transformation parameters. Two cases including the registration of LIDAR points with big rotation angles and a geodetic datum transformation with small rotation angles are demonstrated to validate the new algorithm.

/pdf/wtls-iterative-algorithm-of-3d-similarity-coordinate-w37jqosejg.pdf

WTLS iterative algorithm of 3D similarity coordinate transformation based on Gibbs vectors

This paper presents a new form of quartic equation based on Lagrange’s extremum law and a Groebner basis under the constraint that the geodetic height is the shortest distance between a given point and the reference ellipsoid. A very explicit and concise formulae of the quartic equation by Ferrari’s line is found, which avoids the need of a good starting guess for iterative methods. A new explicit algorithm is then proposed to compute geodetic coordinates from Cartesian coordinates. The convergence region of the algorithm is investigated and the corresponding correct solution is given. Lastly, the algorithm is validated with numerical experiments.

/pdf/explicitly-computing-geodetic-coordinates-from-cartesian-58jic1ahdj.pdf

Huaien Zeng

Papers

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

A dual quaternion algorithm of the Helmert transformation problem

Iterative solution of Helmert transformation based on a unit dual quaternion

WTLS iterative algorithm of 3D similarity coordinate transformation based on Gibbs vectors

Explicitly computing geodetic coordinates from Cartesian coordinates