VIGOROUS 3D ANGULAR RESECTION MODEL USING LEVENBERG – MARQUARDT METHOD

Yaseen T. Mustafa
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Abstract

The resection in 3D space is a common problem in surveying engineering and photogrammetry based on observed distances, angles, and coordinates. This resection problem is nonlinear and comprises redundant observations which is normally solved using the least-squares method in an iterative approach. In this paper, we introduce a vigorous angular based resection method that converges to the global minimum even with very challenging starting values of the unknowns. The method is based on deriving oblique angles from the measured horizontal and vertical angles by solving spherical triangles. The derived oblique angles tightly connected the rays enclosed between the resection point and the reference points. Both techniques of the nonlinear least square adjustment either using the Gauss-Newton or Levenberg – Marquardt are applied in two 3D resection experiments. In both numerical methods, the results converged steadily to the global minimum using the proposed angular resection even with improper starting values. However, applying the Levenberg – Marquardt method proved to reach the global minimum solution in all the challenging situations and outperformed the Gauss-Newton method.
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利用levenberg - marquardt方法建立三维有力角切除模型
基于观测距离、角度和坐标的三维空间交会是测量工程和摄影测量中常见的问题。这种切除问题是非线性的,包含冗余观测值,通常使用迭代方法中的最小二乘法来解决。在本文中,我们引入了一种强有力的基于角度的切除方法,即使在非常具有挑战性的未知起始值下,该方法也收敛到全局最小值。该方法是通过求解球面三角形,由测得的水平线和垂直角求出斜角。导出的斜角紧密地连接了封闭在切点和参考点之间的射线。采用高斯-牛顿法和Levenberg - Marquardt法两种非线性最小二乘平差技术分别进行了两个三维切分实验。在这两种数值方法中,即使初始值不合适,使用所提出的角切除,结果也稳定收敛到全局最小值。然而,应用Levenberg - Marquardt方法证明在所有具有挑战性的情况下都能达到全局最小解,并且优于高斯-牛顿方法。
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