Symmetric similarity 3D coordinate transformation based on dual quaternion algorithm

IF 1.8 4区 地球科学 Q3 GEOCHEMISTRY & GEOPHYSICS
Sebahattin Bektas
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引用次数: 0

Abstract

Nowadays, dual quaternion algorithms are used in 3D coordinate transformation problems due to their advantages. The 3D coordinate transformation problem is one of the important problems in geodesy. This transformation problem is encountered in many application areas other than geodesy. Although there are many coordinate transformation methods (similarity, affine, projective, etc.), similarity transformation is used because of its simplicity. Asymmetric transformation is preferred over symmetric coordinate transformation because of its ease of use. In terms of error theory, the symmetric transformation should be preferred. This study discusses the topic of symmetric similarity 3D coordinate transformation based on the dual quaternion algorithm, as well as the bottlenecks encountered in solving the problem and using the solution method. A new iterative algorithm based on the dual quaternion is presented. The solution is implemented in two models: with and without constraint equations. The advantages and disadvantages of the two models compared to each other are also evaluated. Not only the transformation parameters but also the errors of the transformation parameters are determined. The detailed derivation of the formulas for estimating the symmetric similarity of 3D transformation parameters is presented step by step. Since symmetric transformation is the general form of asymmetric transformation, we can also obtain asymmetric transformation results with a simple modification of the model we developed for symmetric transformation. The proposed algorithm can perform both 2D and 3D symmetric and asymmetric similarity transformations. For the 2D transformation, replacing the z and Z coordinates in both systems with zero is sufficient.

基于对偶四元数算法的对称相似度三维坐标变换
目前,对偶四元数算法以其自身的优点被广泛应用于三维坐标变换问题中。三维坐标变换问题是大地测量学中的重要问题之一。这种转换问题在测量学以外的许多应用领域都遇到过。虽然坐标变换的方法有很多(相似变换、仿射变换、投影变换等),但由于相似变换简单,所以采用相似变换。非对称变换比对称坐标变换更容易使用。在误差理论方面,应优先采用对称变换。本研究讨论了基于对偶四元数算法的对称相似度三维坐标变换问题,以及在求解问题和使用求解方法时遇到的瓶颈。提出了一种新的基于对偶四元数的迭代算法。求解过程分为有约束方程和无约束方程两种模型。并对两种模型的优缺点进行了比较。不仅确定了变换参数,而且确定了变换参数的误差。详细推导了三维变换参数对称相似度估计公式。由于对称变换是不对称变换的一般形式,我们也可以通过对对称变换模型的简单修改得到不对称变换的结果。该算法可以进行二维和三维对称和非对称相似变换。对于二维变换,将两个系统中的z和z坐标替换为零就足够了。
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来源期刊
Acta Geodaetica et Geophysica
Acta Geodaetica et Geophysica GEOCHEMISTRY & GEOPHYSICS-
CiteScore
3.10
自引率
7.10%
发文量
26
期刊介绍: The journal publishes original research papers in the field of geodesy and geophysics under headings: aeronomy and space physics, electromagnetic studies, geodesy and gravimetry, geodynamics, geomathematics, rock physics, seismology, solid earth physics, history. Papers dealing with problems of the Carpathian region and its surroundings are preferred. Similarly, papers on topics traditionally covered by Hungarian geodesists and geophysicists (e.g. robust estimations, geoid, EM properties of the Earth’s crust, geomagnetic pulsations and seismological risk) are especially welcome.
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