Empirical Affine Reference Frame Transformations by Weighted Multivariate TLS Adjustment

In order to determine the transformation parameters between two reference frames empirically, a sufficient number of point coordinates (or possibly higher dimensional features such as, e. g., straight lines or conics) need to be observed in both systems. A proper adjustment of the observed data must take the different variances and covariances into account. The resulting adjustment model is of type Errors-in-Variables rather than of type Gauss-Markov because both sets of coordinates are associated with errors. In the homoscedastic case, the least-squares principle thus leads to the Total Least-Squares Solution (TLSS) rather than the standard Least-Squares Solution (LESS). Here, we generalize the TLSS by allowing the individual variances to be different and the covariances to be non-zero, while still maintaining a certain reasonable variance-covariance structure. This leads to the Weighted TLSS. We show experimentally that the Weighted TLSS yields slightly better estimates (in terms of precision) than LESS or TLSS, but significantly more accurate dispersion measures.