A new method for estimating Rational Function Coefficients



Rational functions are of great interest to engineers and geoscientists. The rational polynomial coefficient (RPC) model as a generalized sensor model has been introduced as an alternative for the rigorous sensor model of the satellite imaging.
Numerical instability of normal equations is the only single obstacle to the implementation of these functions. Practically, estimating rational function coefficients using available control points is mostly an ill-posed problem. Condition number of the normal matrix in the linear parametric model is relatively large. Therefore, a regularization method has to be employed in order to stabilize the equations. Implementation of the regularization technique improves the solution in the linear parametric model. The optimum value of the regularization parameter is estimated using the generalized cross validiation technique.
Moreover, simplification of the observation equations leads to a linear observation model which is the most frequently utilized approach for estimation of the unknown coefficients. However, rigorous modeling is recast in a combined adjustment model. Due to nonlinearity of the combined model, the initial values of unknown parameters are needed. The initialization process can be done using the estimated parameters from the linear parametric model.
Here, rational function coefficients are estimated using a combined model. Furthermore, the Tikhonov regularization method is employed for regularization of the problem in the combined model. Five different methods are implemented and their performances are compared.
Comparison of the root mean squared errors shows that the implementation of the combined model with an appropriate regularization parameter significantly improves the accuracy of the estimated coefficients. The regularized combined model gives the minimum root mean squared errors which is about half the value of the linear parametric model. The proposed method outperforms the already existing ones from an accuracy and computational point of view.