Total Least Squares RBF surface approximation using thin-plate splines

Radial basis functions using thin-plate splines as basis functions offer the great advantage that an approximation of arbitrary surfaces is possible without specifying a user-defined shape parameter. If the coordinate components x , y and z of the input data are introduced as observations subject to random errors, a non-linear adjustment problem of type Total Least Squares results. In this paper, a solution is elaborated by means of an iteratively linearised Gauss-Helmert model. Two solutions are developed for thin-plate splines without and with consideration of additional polynomial terms. With regard to the processing of large amounts of data, it is shown how the system of normal equations can be converted into an equivalent system with a smaller dimension. Numerical examples are given to demonstrate the application of the developed approaches and a solution is shown for cases in which the iterative solution of the adjustment problem does not converge.