Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra

Stefania Ioannidou,George Pantazis

doi:10.3390/ijgi9090494

Stefania Ioannidou, George Pantazis

Open Access

https://doi.org/10.3390/ijgi9090494

Copy DOI

Abstract

The three-dimensional coordinate’s transformation from one system to another, and more specifically, the Helmert transformation problem, is one of the most well-known transformations in the field of engineering. In this paper, its solution, in reverse problem, was investigated for specific data using three different methods. It is presented by solving it with the method of Euler angles as well as with the use of quaternion and dual-quaternion algebra, after first giving some basic mathematical theory. After research, not only were three artificial sets of data used, which were structured in a specific way and forced into specific transformations to be solved, but also a real geodesy problem was tested, in order to identify the sensitivity and problems of each method. Statistical analysis of the results was performed by each method, while it was found that there were significant deviations in rotations and translations in the method of Euler angles and dual quaternions, respectively.

Highlights

Coordinate transformation is a well-known mathematical process that is often used in geodesy, photogrammetry, geographical information science (GIS), computer vision, and in many other branches of engineering
The Helmert transformation problem or 7-parametric transformation in the reverse problem was performed in this paper
It is a three-dimensional transformation consisting of three turns between the axes that defines the rotation matrix (R), a vector of translation of their origins as well as the uniform scale (λ) of one in relation to another

Summary

Introduction

Coordinate (system) transformation is a well-known mathematical process that is often used in geodesy, photogrammetry, geographical information science (GIS), computer vision, and in many other branches of engineering. Coordinates from the World Geodetic System 1984 (WGS84) to local systems from local systems to specific object systems or transformations of Lidar point clouds are only a few applications of that process [1] This can be used either as a direct problem or as a reverse. Rotations through all three axes, 3D vector displacement as well as single scale change, are calculated in order to transfer points from one system to another. These seven transformation parameters are known as the 7-parameter transformation or Helmert transformation problem, which is a well-known transformation, in engineering, and in most sciences [1,2]

Methods

Results

Conclusion