On symmetrical three-dimensional datum conversion

Abstract

A 3-D similarity transformation is frequently used to convert GPS-WGS84-based coordinates to those in a local datum using a set of control points with coordinate values in both systems. In this application, the GaussMarkov (GM) model is often employed to represent the problem, and a least-squares approach is used to compute the parameters within the mathematical model. However, the Gauss-Markov model considers the source coordinates in the data matrix (A) as fixed or error-free; this is an imprecise assumption since these coordinates are also measured quantities and include random errors. The errorsin-variables (EIV) model assumes that all the variables in the mathematical model are contaminated by random errors. This model may be solved using the relatively new total least-squares (TLS) estimation technique, introduced in 1980 by Golub and Van Loan. In this paper, the similarity transformation problem is analyzed with respect to the EIV model, and a novel algorithm is described to obtain the transformation parameters. It is proved that even with the EIV model, a closed form Procrustes approach can be employed to obtain the rotation matrix and translation parameters. The transformation scale may be calculated by solving the proper quadratic equation. A numerical example and a practical case study are presented to test this new algorithm and compare the EIV and the GM models.

Errors-in-Variables Anisotropic Extended Orthogonal Procrustes Analysis

“…As previously reported, Felus and Burtch [10] introduced the EIV model in the EOPA problem, obtaining a TLS solution for a similarity transformation. Their method can deal also with the weighting by rows of the coordinate matrices, i.e., to each point, whose coordinates are known in two reference systems, a different weight is given.…”

“…On the contrary, for similarity transformations the equivalence between the LS solution of model (2) and the TLS solution of the EIV model (4) is disturbed by the scale factor ζ, as demonstrated in [10]. Both methods lead to the following solution for R and c:…”

“…Arun [9] showed that for rigid transformations the classical LS orthogonal Procrustes solution coincides with All the authors are with the DPIA -University of Udine, Via Delle Scienze, 208 -33100 Udine, Italy (e-mail: maset.eleonora@spes.uniud.it; (fabio.crosilla; andrea.fusiello)@uniud.it) the TLS one. Felus and Burtch [10] instead started from a EIV extended orthogonal Procrustes model to compute the unknown parameters of a similarity transformation (rotation matrix, translation vector and scale factor), showing that, in this case, LS and TLS solution are not coincident.…”

A weighted adjustment of a similarity transformation between two point sets containing errors

“…The direct solution method derived in section 3 and summarized in section 3.6 was applied to an example of a 3D similarity transformation of four points, see Table 1, using data from [7]. For comparison, the transformation parameters were determined additionally through an iterative solution based on the GHM (for a general description of this method see e.g.…”

“…Goryn and Hein [9] provided a solution for the 3D rigid transformation with homogenous and uncorrelated errors based on a common approach to the Procrustes problem. Felus and Burtch [7] dealt with the 3D similarity transformation with pointwise weights and uncorrelated errors; the determination of the rotation matrix, however, was based on a scale factor xed to one. Chang [4] o ered a rigorous solution for the case of an additional scaling between the weights of the source and the target coordinates.…”