Analytical algorithm of weighted 3D datum transformation using the constraint of orthonormal matrix

Huaien Zeng

doi:10.1186/s40623-015-0263-6

Abstract

Based on the Lagrangian extremum law with the constraint that rotation matrix is an orthonormal matrix, the paper presents a new analytical algorithm of weighted 3D datum transformation. It is a stepwise algorithm. Firstly, the rotation matrix is computed using eigenvalue-eigenvector decomposition. Then, the scale parameter is computed with computed rotation matrix. Lastly, the translation parameters are computed with computed rotation matrix and scale parameter. The paper investigates the stability of the presented algorithm in the cases that the common points are distributed in 3D, 2D, and 1D spaces including the approximate 2D and 1D spaces, and gives the corresponding modified formula of rotation matrix. The comparison of the presented algorithm and classic Procrustes algorithm is investigated, and an improved Procrustes algorithm is presented since that the classic Procrustes algorithm may yield a reflection rather than a rotation in the cases that the common points are distributed in 2D space. A simulative numerical case and a practical case are illustrated.

Highlights

Three-dimensional datum transformation is a frequently used work in geodesy, engineering surveying, photogrammetry, mapping, geographical information science (GIS), machine vision, etc., e.g., Aktuğ (2009), Akyilmaz (2007), El-Mowafy et al (2009), Ge et al (2013), Han and Van Gelder (2006), Horn (1986), Kashani (2006), Neitzel (2010), Paláncz et al (2013), Soler (1998), Soler and Snay (2004), Soycan and Soycan (2008), Zeng (2014)
The presented algorithm and classic Procrustes algorithm are compared, and an improved Procrustes algorithm is presented since that the classic Procrustes algorithm may yield a reflection rather than a rotation in the cases that the common points are distributed in 2D space
In order to investigate the stability performance of the presented algorithm (PA) in this paper, the classic Procrustes algorithm (CPA) and improved Procrustes algorithm (IPA) in the cases that the control points are distributed in 3D, 2D, and 1D spaces, six sets of control point in system B is first given in Table 1, of which set 1 is distributed in 3D

Summary

Introduction

Three-dimensional datum transformation is a frequently used work in geodesy, engineering surveying, photogrammetry, mapping, geographical information science (GIS), machine vision, etc., e.g., Aktuğ (2009), Akyilmaz (2007), El-Mowafy et al (2009), Ge et al (2013), Han and Van Gelder (2006), Horn (1986), Kashani (2006), Neitzel (2010), Paláncz et al (2013), Soler (1998), Soler and Snay (2004), Soycan and Soycan (2008), Zeng (2014). Zeng and Yi (2010) presented a new analytical algorithm based on the good properties of Rodrigues matrix and Gibbs vector. In the Methods section, a new analytical algorithm to weighted 3D datum transformation is derived in detail, based on the Lagrangian extremum law with the constraint that the rotation matrix is an orthonormal matrix. The presented algorithm and classic Procrustes algorithm are compared, and an improved Procrustes algorithm is presented since that the classic Procrustes algorithm may yield a reflection rather than a rotation in the cases that the common points are distributed in 2D space. In the Results and discussion section, a simulative numerical case and a practical case are given to demonstrate the presented algorithm, classic Procrustes algorithm, and improved Procrustes algorithm.

Methods

Results

Conclusion