The Geometrical Comparisons of RSM and RFM for FORMOSAT-2 Satellite Images

In this paper, we compare the geometrical performance between the rigorous sensor model (RSM) and rational function model (RFM) in the sensor modeling of FORMOSAT-2 satellite images. For the RSM, we provide a least squares collocation procedure to determine the precise orbits. As for the RFM, we analyze the model errors when a large amount of quasi-control points, which are derived from the satellite ephemeris and attitude data, are employed. The model errors with respect to the length of the image strip are also demonstrated. Experimental results show that the RFM is well behaved, indicating that its positioning errors is similar to that of the RSM.  Sensor orientation modeling is a prerequisite for the georeferencing of satellite images or 3D object reconstruction from satellite stereopairs. Nowadays, most of the high-resolution satellites use linear array pushbroom scanners. Based on the pushbroom scanning geometry, a number of investigations have been reported regarding the geometric accuracy of linear array images (Westin, 1990; Chen and Lee, 1993; Li, 1998; Tao et al., 2000; Toutin, 2003; Grodecki and Dial, 2003). The geometric modeling of the sensor orientation may be divided into two categories, namely, the rigorous sensor model (RSM) and the rational function model (RFM) (Toutin, 2004). Capable of fully delineating the imaging geometry between the image space and object space, the RSM has been recognized in providing the most precise geometrical processing of satellite images. Based on the collinearity condition, an image point corresponds to a ground point using the employment of the orientation parameters, which are expressed as a function of the sampling time. Due to the dynamic sampling, the RSM contains many mathematical calculations, which can cause problems for researchers who are not familiar with the data preprocessing. Moreover, with the increasing number of Earth resource satellites, researchers need to familiarize themselves with the uniqueness and complexity of each sensor model. Therefore, a generic sensor model of the geometrical processing is needed for simplification. (Dowman and Michalis, 2003). The RFM is a generalized sensor model that is used as an alternative for the RSM. The model uses a pair of ratios of two polynomials to approximate the collinearity condition equations. The RFM has been successfully applied to several high-resolution satellite images such as Ikonos (Di et al., 2003; Grodecki and Dial, 2003; Fraser and Hanley, 2003) and QuickBird (Robertson, 2003). Due to its simple impleThe Geometrical Comparisons of RSM and RFM for FORMOSAT-2 Satellite Images Liang-Chien Chen,  Teo, and Chien-Liang Liu mentation and standardization (NIMA, 2000), the approach has been widely used in the remote sensing community. Launched on 20 May 2004, FORMOSAT-2 is operated by the National Space Organization of Taiwan. The satellite operates in a sun-synchronous orbit at an altitude of 891 km and with an inclination of 99.1 degrees. It has a swath width of 24 km and orbits the Earth exactly 14 times per day, which makes daily revisits possible (NSPO, 2005). Its panchromatic images have a resolution of 2 meters, while the multispectral sensor produces 8 meter resolution images covering the blue, green, red, and NIR bands. Its high performance provides an excellent data resource for the remote sensing researchers. The major objective of this investigation is to compare the geometrical performances between the RSM and RFM when FORMOSAT-2 images are employed. A least squares collocation-based RSM will also be proposed in the paper. In the reconstruction of the RFM, rational polynomial coefficients are generated by using the on-board ephemeris and attitude data. In addition to the comparison of the two models, the modeling error of the RFM is analyzed when long image strips are used. Rigorous Sensor Models The proposed method comprises essentially of two parts. The first involves the development of the mathematical model for time-dependent orientations. The second performs the least squares collocation to compensate the local systematic errors. Orbit Fitting There are two types of sensor models for pushbroom satellite images, i.e., orbital elements (Westin, 1990) and state vectors (Chen and Chang, 1998). The orbital elements use the Kepler elements as the orbital parameters, while the state vectors calculate the orbital parameters directly by using the position vector. Although both sensor models are robust, the state vector model provides simpler mathematical calculations. For this reason, we select the state vector approach in this investigation. Three steps are included in the orbit modeling: (a) Initialization of the orientation parameters using on-board ephemeris data; (b) Compensation of the systematic errors of the orbital parameters and attitude data via ground control points (GCPs); and (c) Modification of the orbital parameters by using the Least Squares Collocation (Mikhail and Ackermann, 1982) technique. PHOTOGRAMMETRIC ENGINEER ING & REMOTE SENS ING May 2006 573 Center for Space and Remote Sensing Research National Central University, Chung-Li, Taiwan (lcchen@csrsr.ncu.edu.tw). Photogrammetric Engineering & Remote Sensing Vol. 72, No. 5, May 2006, pp. 573–579. 0099-1112/06/7205–0573/$3.00/0