How to solve errors-in-variables model for coordinate transformations in a classical adjustment way?

Abstract

In a coordinate transformation problem, the coordinates of both systems may have random errors. In such a case, the corresponding problem is considered within the Errors-In-Variables (EIV) model and solved by the method of Weighted Total Least-Squares (WTLS). However there are two main difficulties while applying the WTLS method: (1) A proper cofactor matrix for all elements in the design matrix of the EIV model is also to be formed. This is sometimes confusing work because some elements may be repeated twice or more in the same or different signs and some elements may be error-free coefficients in the design matrix depending on the type of the transformation problem. Hence setting the stochastic part of the model needs a special effort. (2) The derivation of the equations of the WTLS solution is complicated in contrast to the least-squares adjustment. So it may not be easy work to adapt some numerical and statistical methods and ideas to the solution. In order to remove these difficulties this study discusses the solution of the corresponding problem in a classical adjustment way. It is shown mathematically that the adjustment procedure derived for this aim is equivalent to the WTLS solution. Although the procedure is examined here for 2D Affine transformation, it may be easily adapted to other coordinate transformation problems.