A Statistical Variable Selection Solution for RFM Ill-Posedness and Overparameterization Problems

Parameters of a rational function model (RFM), known as rational polynomial coefficients, are commonly redundant and highly correlated, leading to the problems of overparameterization and ill-posedness, respectively. In this paper, an innovative two-stage statistical method, called an uncorrelated and statistically significant RFM (USS-RFM), is presented to deal directly with these two problems. In the first stage, the proposed method employs a novel correlation analysis, which aims to exclude highly correlated coefficients. In the second stage, a new iterative significance test is applied to detect and remove unnecessary coefficients from the RFM. The proposed method is implemented on eight real data sets captured by Cartosat-1, GeoEye-1, Pleiades, Spot-3, and WorldView-3 platforms. The results are evaluated in terms of the positioning accuracy, model degrees of freedom, processing time, and figure condition analysis. Experimental results prove the efficiency of the proposed method, showing that it could achieve subpixel accuracy even for cases with five ground control points. The proposed USS-RFM is compared to an $\ell _{1}$ -norm regularization (L1R) technique and a particle swarm optimization (PSO) algorithm in the terrain-dependent case of the RFM. The results demonstrate the superiority of the USS-RFM, which performs better than the alternative methods in terms of positioning accuracy by more than 50% on average. Moreover, the RFMs resulted from the USS-RFM demonstrate to have higher degrees of freedom and, as a result, higher level of reliability. From the perspective of processing time, USS-RFM and L1R are similar while both are much faster than PSO.