Application of PCA Analysis and QR Decomposition to Address RFM's Ill-Posedness

Abstract

Rational function model (<small>RFM</small>) is the most widely used sensor model in the remote sensing community. However, it suffers from ill-posedness, challenging its feasibility. This problem, i.e., ill-posedness, is mainly caused due to highly correlated coefficients of the <small>RFM</small>, which magnifies any small perturbations of observations, such as noise and instrumental error. This paper outlines a novel two-step method, called principal component analysis (<small>PCA</small>)-<small>RFM</small>, based on the integration of <small>PCA</small> and QR decomposition. In the first step, the <small>PCA</small>-<small>RFM</small> reduces the observational perturbations from the design matrix using the <small>PCA</small>. In the next step, the <small>RFM</small>'s coefficients are estimated using a <small>QR</small> decomposition with column pivoting and least square method. According to the results, the <small>PCA</small>-<small>RFM</small> is less sensitive than its rivals to the changes of the ground control point (<small>GCPs</small>) distribution. Geometrically speaking, in addition, <small>PCA</small>-<small>RFM</small> is more accurate than recently established methods even in the presence of the small number of <small>GCPs</small>.