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

Abstract

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.

Extended WTLS iterative algorithm of 3D similarity transformation based on Gibbs vector

“…For n control points, the 3D similarity transformation in EIV model is established as follows in Zeng et al (2020),…”

“…e.g. Grafarend and Awange (2003), Felus and Burtch (2009), Zeng (2015), Zeng et al (2020), the point weight as follows is employed.…”

“…where w c i is the weight of control point i . The WTLS solution of is derived as follows in Zeng et al (2020).…”

“…( 4) into Eq. ( 1) and carrying out arrangements, Zeng et al (2020) eliminates from the transformation model, and the simplified model is where…”

Application of mathematical programming and modeling methods for monitoring the technical condition of underwater crossings of main gas pipelines

“…This rapid development of technology allowed surveyors to receive and process a huge amount of data about the objects under study. Thanks to the development of information technologies and the introduction of programming methods in geodetic production, it is possible to solve many problems using new methods, this idea is confirmed in the articles [1][2][3][4][5][6][7][8][9]. The application of information technologies in geodesy allows us to automate the calculation process in any software environment, which saves the final result from gross errors.…”