Optimization of RFM Problem Using Linearly Programed ℓ₁-Regularization

Due to the existence of a large number of highly correlated parameters in the rational function model (RFM), known as rational polynomial coefficients (RPCs), RFM suffers from both ill-posedness and overparameterization problems. To tackle these problems, recently, ℓ₁-regularized least-squares, in which ℓ₂-norm of residuals and ℓ₁-norm of RPCs are minimized, has been introduced. Nonetheless, RFM still suffers from these two phenomena, especially in the presence of a limited number of ground control points (GCPs). This study proposes a new parameter-free linear framework for RFM optimization. In this framework, called linearly programed ℓ₁-regularized RFM framework (LPRFM), the ℓ₁-norms of RPCs and residuals are simultaneously minimized through a linear objective function. To solve LPRFM as a linear optimization problem, the commonly used dual-simplex method is applied. LPRFM can be implemented for both RPC estimation and correction. Our experimental results indicated the superiority of the LPRFM against well-known competing methods in both estimation and correction of RPCs. The results showed that, in the estimation of RPCs, the proposed framework has led to accurate and robust results with only five GCPs. Moreover, accurate coordinates were obtained after correcting vendor-provided RPCs with only one GCP.