A dual quaternion algorithm of the Helmert transformation problem

Huaien Zeng,Ronghua Yang,Guobin Chang,Xing Fang

doi:10.1186/s40623-018-0792-x

Abstract

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 Závoti and Kalmár (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).

Highlights

Helmert transformation problem is to determine the seven transformation parameters including three rotation angles, three translation parameters and one scale factor using a set of control points
The algorithms can be classified to a numerical iterative algorithm, e.g., El-Habiby et al (2009), Zeng and Yi (2011), Paláncz et al 2013, Zeng et al (2016), etc., and an analytical algorithm, e.g., Grafarend and Awange (2003), Shen et al (2006a, b), Zeng (2015), etc
This paper aims to construct the Helmert transformation based on quaternion and present a new dual quaternion algorithm for Helmert transformation which has explicit computation steps, no initial value problem of transformation parameters, fast computation and reliable result no matter how big the rotation angles are

Summary

Introduction

Helmert transformation (3D seven-parameter similarity transformation) is a frequently encountered task in geodesy, photogrammetry, geographical information science, mapping, engineering surveying, machine vision, etc. (see, e.g., Aktuğ (2009); Aktuğ 2012; Akyilmaz 2007; Arun et al 1987; Burša 1967; Chang 2015; Chang et al 2017; El-Habiby et al 2009; El-Mowafy et al 2009; Grafarend and Awange 2003; Han 2010; Horn 1986, 1987; Han and Van Gelder 2006; Horn et al 1988; Jaw and Chuang 2008; Jitka 2011; Kashani 2006; Krarup 1985; Leick 2004; Leick and van Gelder 1975; Mikhail et al 2001; Neitzel 2010; Soler 1998; Soler and Snay 2004; Soycan and Soycan2008; Teunissen 1986; Teunissen 1988; Wang et al 2014; Závoti and Kalmár 2016; Zeng 2014; Zeng 2015; Zeng et al 2016; Zeng and Yi 2011). Helmert transformation (3D seven-parameter similarity transformation) is a frequently encountered task in geodesy, photogrammetry, geographical information science, mapping, engineering surveying, machine vision, etc. Numerous algorithms of Helmert transformation have been presented. The algorithms can be classified to a numerical iterative algorithm, e.g., El-Habiby et al (2009), Zeng and Yi (2011), Paláncz et al 2013, Zeng et al (2016), etc., and an analytical algorithm, e.g., Grafarend and Awange (2003), Shen et al (2006a, b), Zeng (2015), etc. For the numerical iterative algorithm, initial values of transformation parameters are usually required. If a global optimization algorithm is employed to recover

Objectives

Conclusion