# General Total Least Squares Theory for Geodetic Coordinate Transformations
## 5 Numerical Examples  
In this part, two numerical examples are provided to verify the proposed constrained TLS algorithm. In the first example, data from [21] presents a typical photogrammetry problem which consists of the 2-D coordinates that were measured and respectively calibrated of four side fiducial marks in a photograph, as seen in Table 1. Here, the coordinates in the target system \(\mathbf{x_{r}}\), \(\mathbf{y_{t}}\) (\"Calibrated Coordinates\") and the coordinates in the source system \(\mathbf{x_{s}}\), \(\mathbf{y_{s}}\) (\"Measured Coordinates\") are presented in Figure 2 to clarify their positions, and we regard them as equally weighted uncorrelated observations. The similarity transformation for a dataset was adjusted for the EIV model before by [22] and by [6] in two different parameterizations, i.e., either with or without shift/translation parameters, all in 2D. They both need complicated preprocessing, i.e., either the reduction of parameters or the selection of independent random elements within the coefficient matrix. Furthermore, these authors only implemented the similarity transformation. By applying Algorithm 1 to the data set of Table 1, the estimated 2D transformation parameters for all kinds of transformations and the TLS objective function were determined and are presented in Table 2. The four parameters are completely different in the transformation matrix \(\mathbf{\Xi}_{2\times 2}\) in the affine and orthogonal transformations, whereas the four parameters are pairwise different in the similarity and rigid transformations. The results successfully correspond with the transformation models, as the affine and orthogonal transformations have different scale factors. The estimated values of the objective functions become larger and larger from the affine transformation to the rigid transformation (see the last row of Table 2), since more and more constraints are considered. The standard deviation \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Point No.** & \(\mathbf{x_{t}}\) & \(\mathbf{y_{t}}\) & \(\mathbf{x_{s}}\) & \(\mathbf{y_{s}}\) \\ \hline
1 & \(-\)117.478 & 0 & 17.856 & 144.794 \\
2 & 117.472 & 0 & 252.637 & 154.448 \\
3 & 0.015 & \(-\)117.41 & 140.089 & 32.326 \\
4 & \(-\)0.014 & 117.451 & 130.40 & 267.027 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The common points of the 2D transformation. Figure 2: The positions of the source and target coordinates in the 2D transformations. of the parameter estimates is given in Table 3. The table reveals that the precision of the parameter estimates becomes higher with more constraints. However, in the last row of Table 3, the transformation models expressed by larger numbers of constraints do not provide larger estimates of the unit standard deviation, which depends on both the redundant number and the objective function values. In the second example, a dataset for 3D transformations, shown in Table 4, was given by the Coordinate Systems Analysis Team (CSAT) at the National Geospatial-Intelligence Agency [9]. CSAT preserves and releases a set of datum transformation parameter estimates for different countries via the CSAT website. [PERSON] and [PERSON] provided the adjustment result for similarity transformation based on Procrustes analysis on the assumption that the covariance matrix must be expressed by the Kronecker product. Here, we implemented our proposed algorithm for all kinds of transformations by applying the dataset in Table 4. It is important to note that our algorithm can adjust the dataset with an arbitrary covariance matrix, which is beyond the inherent assumption in Procrustes analysis. With the dataset in Table 4, four types of 3D transformation models were adjusted for the six control points. The estimation results for the affine, orthogonal, similarity, and rigid transformations are presented in Table 5, Table 6, Table 7, and Table 8, respectively.