Multivariate total least - squares adjustment for empirical affine transformations

In Geodetic Science it occurs frequently that, for a given set of points, their coordinates have been measured in two (or more) different systems, and empirical transformation parameters need to be determined by some sort of adjustment for a defined class of transformations. In the linear case, these parameters appear in a matrix that relates one set of coordinates with the other, after correcting them for random errors and centering them around their mid-points (to avoid shift parameters). In the standard approach, a structured version of the Errors-in-Variables (EIV) model would be obtained which would require elaborate modifications of the regular Total Least-Squares Solution (TLSS). In this contribution, a multivariate (but unstructured) EIV model is proposed for which an algorithm has been developed using the nonlinear Euler-Lagrange conditions. The new algorithm is used to estimate the TLSS of the affine transformation parameters. Other types of linear transformations (such as the similarity transformation, e.g.) may require additional constraints.