Converted Total Least Squares Method and Gauss-Helmert Model with Applications to Coordinate Transformations

Abstract

In this paper, the three kind of solutions of TLS problem, the common solution by singular value decomposition (SVD), the iteration solution and Partial-EIV model are firstly reviewed with respect to their advantages and disadvantages. Then a newly developed Converted Total Least Squares (CTLS) dealing with the errors-in-variables (EIV) model is introduced. The basic idea of CTLS has been proposed by the authors in 2010, which is to take the stochastic design matrix elements as virtual observations, and to transform the TLS problem into a traditional Least Squares problem. This new method has the advantages that it cannot only easily consider the weight of observations and the weight of stochastic design matrix, but also deal with TLS problem without complicated iteration processing, if the suitable approximates of parameters are available, which enriches the TLS algorithm and solves the bottleneck restricting the application of TLS solutions. CTLS method, together with all the three TLS models reviewed here has been successfully integrated in our coordinate transformation programs and verified with the real case study of 6-parameters Affine coordinate transformation. Furthermore, the comparison and connection of this notable CLTS method and estimation of Gauss-Helmert model are also discussed in detail with applications of coordinate transformations.