ACCURACY EVALUATION OF RATIONAL POLYNOMIAL COEFFICIENTS 
SOLUTION FOR QUICKBIRD IMAGERY BASED ON AUXILIARY GROUND 
CONTROL POINTS 
Yun Zhan a Chun Liu a,b Gang Qiao 3 
a Department of Surveying and Geo-Informatics, Tongji University, Shanghai, China 
b Key Laboratory of Advanced Engineering Surveying of State Bureau of Surveying and Mapping, China 
Commission VI, WG VII/6 
KEYWORDS: Rational polynomial coefficients, Batch iterative least-squares solution with regularization, 
Incremental discrete Kalman filtering, Geo-positioning accuracy 
ABSTRACT: 
Relative to the rigorous physical model, rational polynomial coefficient (RPC) has been adopted as an alternative common sensor 
model data for image Geometric correction exploitation. In this paper, based on collected QuickBird imagery in Shanghai region, the 
iterative least-squares solution with regularization(ILSR) is derived to determine the RPCs by using 50 fair distributed ground 
control points (GCPs) firstly. Two methods are then used to refine determined RPCs under different circumstance as: 1) when both 
the original and the additional GCPs are available, the RPCs will be recomputed using the batch iterative least-squares solution with 
regularization (BILSR) method; and 2) when only the new GCPs are available, incremental discrete Kalman filtering (IDKF) method 
has been described. Meanwhile, check points are used to evaluate their geo-positioning accuracy, and their comparison is conducted. 
Finally, some conclusion is then achieved when handing the high resolution imagery in metropolitan area. 
1. INTRODUCTION 
Satellite Imagery such as QuickBird, IKONOS has been widely 
used with the development of high resolution satellite 
technology. Collinearity based rigorous sensor model is the 
basis of geometric positioning for high resolution satellite 
imagery (HRSI). Dependence on physical parameters and 
satellite orbit parameters makes the rigorous sensor model much 
more complicated, thus hard to be applied with. The RPC, a 
mathematical model which is sensor independent and not 
rigorous, has been used widely by satellite companies for the 
survey process of HRSI and as the alternative of the rigorous 
sensor model. The image coordinates are denoted as the third 
polynomial expression in RPC. RPC, by providing a simple and 
exact relation for vendors and customers to describe the 
relationship of object and image, has been successfully 
employed in the terrain modeling, orthographic projection and 
feature extraction. A lot of research work has been done about 
the geometric correction and 3D reconstruction of IKONOS 
imagery using RPC (Tao and Hu, 2001, 
2002(1), 2002(2); 
Dowman, 2000; Fraser, 2002; Clive, 2002). 
Two methods are used for the calculation of RPC, terrain 
dependant approach and terrain independent approach (Yong 
Hu et al., 2004). The terrain dependent approach, without 
setting up grids, is to obtain GCP through topographical 
measurement or field survey to fit the imagery geometry using 
sufficient parameters. Its accuracy is determined by 
hypsography and the GCP number and distribution (Fraser,2006; 
Liu,2006). The relativity between the RPC parameters may 
result in the singularity of design matrix for normal equation. 
The regularization method can improve the condition number of 
the design matrix, thus avoiding the numerical instability of 
least square solution (Tao and Hu,2001). 
The RPC direct correction method was put forward to improve 
the positioning accuracy and meet the demand of high accuracy 
users. Different mathematical methods were applied for the 
RPC accuracy improvement when the physical sensor model 
was unknown. When the original and auxiliary GCP were both 
available, BILSR was used to recalculate the RPC (Hu and 
Tao,2002; Di et al.,2003). Here the original GCP denotes the 
GCP used for calculating the original RPC, while the auxiliary 
GCP means the auxiliary collected GCP that never used for 
RPC calculation. The correction process is to include all the 
GCP into the RPC solution with different power to the new and 
original GCP. When there are only auxiliary GCP, the IDKF 
can be employed to improve the RPC accuracy (Hu and 
Tao,2002; Bang et al.,2003), which means the accuracy of RPC 
is improved through the inclusion of new GCP with proper 
power. 
Based on the QuickBird imagery in Shanghai, China, this paper 
mainly discusses the solution of RPC and accuracy after 
correction applied in metropolitan area without obvious 
hypsography. Experiences of application for similar data could 
be learned from the models and data this paper employed. 
2. RPC MATHEMATICAL MODEL 
The RPC of QuickBird imagery denotes the image coordinates 
as the ratio of polynomials based on the variable of longitude, 
latitude and height, which is as equation (1): 
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