Total least L1- and L2-norm estimations of a coordinate transformation model with a structured parameter matrix

Abstract

Total least L1- and L2-norm estimations of a symmetrical coordinate transformation model with a structured parameter matrix are proposed, with the aim to account for the relationships between the transformation parameters. In the model, the errors in the coordinates of the measured points in both the source and target coordinate systems in the transformation model are taken into account. The solution of the proposed symmetrical coordinate transformation model is derived by the use of the total least L1- and L2-norm estimations. In addition, the variance-covariance matrices of the estimated parameters and the adjusted coordinates of the points are further derived in the two proposed methods. A numerical experiment in coordinate transformation is conducted to test the proposed methods. The results show that the proposed total least L2-norm estimation method is suitable for resolving the transformation model when the coordinates of the points in both the source and target systems are contaminated only by random errors. However, in the case of gross errors in the coordinates of the points, the proposed total least L1-norm estimation method performs better than the total least L2-norm estimation, resulting in higher precision of the estimated parameters.