A Graph-Based Solution for RFM Problem via Accelerated ADMM

Abstract

The rational function model (RFM) is crucial for the geometric processing of remote sensing satellite imagery. However, estimating the unknown rational polynomial coefficients (RPCs) remains an ill-posed and over-parameterized problem due to severe multicollinearity. Existing regularization methods predominantly rely on the assumption of independent and identically distributed (i.i.d.) observational errors, treating ground control points (GCPs) as independent measurements while neglecting their intrinsic spatial topological relationships. In practice, imaging distortions exhibit strong spatial correlation, indicating that neighboring GCPs share similar geometric deformation patterns. Neglecting this topological prior may limit the accuracy of RFM estimation, especially when the GCPs are insufficient. To address this limitation, this article proposes a graph-based RFM (G-RFM) estimation method that explicitly encodes spatial dependencies among GCPs within a Bayesian maximum a posteriori (MAP) framework. Specifically, the observation residuals are modeled as a Gaussian–Markov random field (GMRF) governed by a graph Laplacian matrix, which constrains estimation errors between spatially adjacent GCPs. To ensure computational efficiency for large-scale applications, an accelerated alternating direction method of multipliers (ADMM) algorithm is developed. Comprehensive experiments demonstrate that the proposed method achieves competitive and often superior performance compared with existing methods in both terrain-dependent (TD) and terrain-independent (TID) cases.