scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Iterative solution of Helmert transformation based on a unit dual quaternion

01 Mar 2019-Acta Geodaetica Et Geophysica Hungarica (Springer International Publishing)-Vol. 54, Iss: 1, pp 123-141
TL;DR: In this paper, the authors presented a rigid iterative algorithm of Helmert transformation using a unit dual quaternion and showed that the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange.
Abstract: The rigid motion involving both rotation and translation in the 3D space can be simultaneously described by a unit dual quaternion. Considering this excellent property, the paper constructs the Helmert transformation (seven-parameter similarity transformation) model based on a unit dual quaternion and then presents a rigid iterative algorithm of Helmert transformation using a unit dual quaternion. Because of the singularity of the coefficient matrix of the normal equation, the nine parameter (including one scale factor and eight parameters of a dual quaternion) Helmert transformation model is reduced into five parameter (including one scale factor and four parameters of a unit quaternion which can represent the rotation matrix) Helmert transformation one. Besides, a good start estimate of parameter is required for the iterative algorithm, hence another algorithm employed to compute the initial value of parameter is put forward. The numerical experiments involving a case of small rotation angles i.e. geodetic coordinate transformation and a case of big rotation angles i.e. the registration of LIDAR points are studied. The results show the presented algorithms in this paper are correct and valid for the two cases, disregarding the rotation angles are big or small. And the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange (J Geod 77:66–76, 2003), the orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015), and the algorithm from Wang et al. (J Photogramm Remote Sens 94:63–69, 2014).
Citations
More filters
Journal ArticleDOI
TL;DR: Recently, it has been shown how quaternion-based representation of a rotation matrix has advantages over conventional Eulerian representation in 3D similarity transformations as discussed by the authors, and the iterative estimat...
Abstract: Recently, it has been shown how quaternion-based representation of a rotation matrix has advantages over conventional Eulerian representation in 3D similarity transformations. The iterative estimat...

10 citations

Journal ArticleDOI
TL;DR: In this paper, the authors discuss the quality of Global Positioning System data, as well as effects of phase center variation (PCV) model and attitude for GRACE and GRACE-FO.
Abstract: As the gravity formation flying spacecraft jointly developed by National Aeronautics and Space Administration and German Research Centre for Geosciences, the Gravity Recovery and Climate Experiment (GRACE) and GRACE follow-on (GRACE-FO) satellites have adopted the same K-band ranging system and orbit design in order to detect the Earth’s gravity field information. Different from the BlackJack receiver onboard GRACE, a TriG GNSS receiver is loaded on the GRACE-FO satellites. The orbit determination accuracy of GRACE is better than 2.5 cm, but for GRACE-FO there is no comprehensive assessment of orbit accuracy. We discuss the quality of Global Positioning System data, as well as effects of phase center variation (PCV) model and attitude for GRACE and GRACE-FO. The results show that there is no significant difference in terms of rate of change in ionospheric delay (IOD) and multipath effect, which suggests that the performance of TriG receiver is as excellent as that of BlackJack receiver. After using PCV corrections, the root mean square (RMS) errors of kinematic and reduced-dynamic (RD) orbit residuals decrease by 0.4–0.5 and 0.6–0.9 mm, respectively. Satellite laser ranging RMS errors for RD orbit solutions are lower than 2.59 cm whether PCV corrections are used or not. The effect of attitude data on kinematic and RD orbits indicates that nominal attitude data can reliably replace measured attitude data in GRACE-FO orbit determination.

9 citations

Journal ArticleDOI
TL;DR: A weighted total least squares (WTLS) iterative algorithm of the 3D similarity coordinate transformation based on Gibbs vectors is proposed that is fast in terms of fewer iterations, reliable and does not need good initial values of transformation parameters.
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.

8 citations

Journal ArticleDOI
TL;DR: In this article, the authors proposed an extended weighted total least squares (WTLS) iterative algorithm of 3D similarity transformation based on Gibbs vector, which treats the transformation parameters and the target coordinate of non-control points as unknowns.
Abstract: Considering coordinate errors of both control points and non-control points, and different weights between control points and non-control points, this contribution proposes an extended weighted total least squares (WTLS) iterative algorithm of 3D similarity transformation based on Gibbs vector. It treats the transformation parameters and the target coordinate of non-control points as unknowns. Thus it is able to recover the transformation parameters and compute the target coordinate of non-control points simultaneously. It is also able to assess the accuracy of the transformation parameters and the target coordinates of non-control points. Obviously it is different from the traditional algorithms that first recover the transformation parameters and then compute the target coordinate of non-control points by the estimated transformation parameters. Besides it utilizes a Gibbs vector to represent the rotation matrix. This representation does not introduce additional unknowns; neither introduces transcendental function like sine or cosine functions. As a result, the presented algorithm is not dependent to the initial value of transformation parameters. This excellent performance ensures the presented algorithm is suitable for the big rotation angles. Two numerical cases with big rotation angles including a real world case (LIDAR point cloud registration) and a simulative case are tested to validate the presented algorithm.

2 citations

Journal ArticleDOI
TL;DR: A new 3D Cartesian coordinate transformation with the dual quaternion method is explained in detail, and its advantages over the classical transformation problem algorithm are emphasized.

1 citations

References
More filters
Journal ArticleDOI
Paul J. Besl1, H.D. McKay1
TL;DR: In this paper, the authors describe a general-purpose representation-independent method for the accurate and computationally efficient registration of 3D shapes including free-form curves and surfaces, based on the iterative closest point (ICP) algorithm, which requires only a procedure to find the closest point on a geometric entity to a given point.
Abstract: The authors describe a general-purpose, representation-independent method for the accurate and computationally efficient registration of 3-D shapes including free-form curves and surfaces. The method handles the full six degrees of freedom and is based on the iterative closest point (ICP) algorithm, which requires only a procedure to find the closest point on a geometric entity to a given point. The ICP algorithm always converges monotonically to the nearest local minimum of a mean-square distance metric, and the rate of convergence is rapid during the first few iterations. Therefore, given an adequate set of initial rotations and translations for a particular class of objects with a certain level of 'shape complexity', one can globally minimize the mean-square distance metric over all six degrees of freedom by testing each initial registration. One important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model, prior to shape inspection. Experimental results show the capabilities of the registration algorithm on point sets, curves, and surfaces. >

17,598 citations

Journal ArticleDOI
TL;DR: A closed-form solution to the least-squares problem for three or more paints is presented, simplified by use of unit quaternions to represent rotation.
Abstract: Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task . It finds applications i n stereoph and in robotics . I present here a closed-form solution to the least-squares problem for three or more paints . Currently various empirical, graphical, and numerical iterative methods are in use . Derivation of the solution i s simplified by use of unit quaternions to represent rotation . I emphasize a symmetry property that a solution to thi s problem ought to possess . The best translational offset is the difference between the centroid of the coordinates i n one system and the rotated and scaled centroid of the coordinates in the other system . The best scale is equal to th e ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids . These exact results are to be preferred to approximate methods based on measurements of a few selected points . The unit quaternion representing the best rotation is the eigenvector associated with the most positive eigenvalue o f a symmetric 4 X 4 matrix . The elements of this matrix are combinations of sums of products of correspondin g coordinates of the points .

4,522 citations

Journal ArticleDOI
TL;DR: An algorithm for finding the least-squares solution of R and T, which is based on the singular value decomposition (SVD) of a 3 × 3 matrix, is presented.
Abstract: Two point sets {pi} and {p'i}; i = 1, 2,..., N are related by p'i = Rpi + T + Ni, where R is a rotation matrix, T a translation vector, and Ni a noise vector. Given {pi} and {p'i}, we present an algorithm for finding the least-squares solution of R and T, which is based on the singular value decomposition (SVD) of a 3 × 3 matrix. This new algorithm is compared to two earlier algorithms with respect to computer time requirements.

3,862 citations

Journal ArticleDOI
TL;DR: The proposed theorem is a strict solution of the problem, and it always gives the correct transformation parameters even when the data is corrupted.
Abstract: In many applications of computer vision, the following problem is encountered. Two point patterns (sets of points) (x/sub i/) and (x/sub i/); i=1, 2, . . ., n are given in m-dimensional space, and the similarity transformation parameters (rotation, translation, and scaling) that give the least mean squared error between these point patterns are needed. Recently, K.S. Arun et al. (1987) and B.K.P. Horn et al. (1987) presented a solution of this problem. Their solution, however, sometimes fails to give a correct rotation matrix and gives a reflection instead when the data is severely corrupted. The proposed theorem is a strict solution of the problem, and it always gives the correct transformation parameters even when the data is corrupted. >

2,123 citations


Additional excerpts

  • ...2012; Teunissen 1986; Teunissen 1988; Umeyama 1991; Yang 1999; Zeng 2014; Zeng et al....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a closed-form solution to the least square problem for three or more points is presented, which requires the computation of the square root of a symmetric matrix, and the best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids.
Abstract: Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical, and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper [ J. Opt. Soc. Am. A4, 629 ( 1987)]. Since orthonormal matrices are used more widely to represent rotation, we now present a solution in which 3 × 3 matrices are used. Our method requires the computation of the square root of a symmetric matrix. We compare the new result with that obtained by an alternative method in which orthonormality is not directly enforced. In this other method a best-fit linear transformation is found, and then the nearest orthonormal matrix is chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.

1,101 citations