Updating Solutions of the Rational Function Model Using Additional Control Information

The rational function model (RFM) is a generic form of many sensor models. The universal real-time image geometry model recommended by the OpenGIS consortium is essentially based on the RFM. Many data vendors are or will be supporting the RFM based solutions by providing rational function coefficients (RFCs). With known RFCs, the end users will be able to rectify the imagery using the RFM without a need of knowing the rigorous sensor model. In fact, the RFCs will be calculated by the vendors using existing or user-provided control information. The entire calculation process would not be transparent to the users in most cases. This paper addresses a practical issue that if one has additional control information, whether or not the RFM solutions (i.e., the values of RFCs) provided by the vendor can be improved or updated based on the additional control information. INTRODUCTION Rational function is a function, which can be represented as the quotient of polynomials. Mathematically speaking, all polynomials are rational functions (Newman, 1978). The rational function model (RFM) in this context is a sensor model that is developed to represent the imaging geometry between the object space and the image space. Compared to the polynomial models that have been used widely in remote sensing, the RFM is essentially a more generic expression (Tao and Hu, 2000a). Unlike the rigorous physical sensor models such as the collinearity-based models, the RFM needs no knowledge specific to each type of imaging sensor. It has been noticed that the rigorous sensor models are not always available, especially for imagery from commercial satellites (e.g., IKONOS), where the rigorous sensor model is hidden to the end user. Therefore, the RFM has gained considerable interests in remote sensing community, and has been recommended as an alternative to physical sensor models by OpenGIS Consortium (OGC, 1999). A comprehensive investigation on RFM including solutions, numerical stability, accuracy, and the required number and distribution of ground control points (GCPs), etc., can be found in Tao and Hu (2000a). It has been reported that the RFCs will be provided by vendors as a component of the universal real-time image geometry support data (OGC, 1999). Many satellite imagery vendors will also adopt RFM as part of the image transfer format. The RFCs are solved using a 3D object grid of which points are computed using the appropriate rigorous sensor model (Tao and Hu, 2000b). Tests have shown that the RFM could achieve a very high approximation accuracy and is capable of replacing the rigorous sensor models for photogrammetric restitution (Tao and Hu, 2000a). Since there is no way back to the rigorous model from RFM, that is, physical parameters of the rigorous sensor models cannot be recovered from RFCs, the information pertaining to the physical sensor models can be hidden. Thus, the end users will be able to rectify the imagery and achieve a reasonably high accuracy using the RFM without knowing the rigorous sensor models. However, what happens if users have own additional control information? One possible solution is to include the additional control information in an adjustment process by using the existing RFM solution as an input. Technically, this process can be used to update or verify the solution provided by vendors. In this paper, we present an approach to improve or update the solution of the RFM using additional control information. The discrete Kalman filter is a powerful technique for efficiently computing statistically optimal estimates of time varying process from series of measurements. We therefore employed this theory to develop this approach. Real data sets have been used to test this approach. This paper is organized as follows. In the next section, we briefly address the rational function model by introducing the weighted least squares solutions, which give the initial estimate to RFCs. In Section 3, we present the model to update the initial solution by discrete Kalman filtering assimilating new control information. In Section 4, we show the results computed by both the weighted least squares method and discrete Kalman filtering when adding new control points, and provide a qualitative comparison between both results. Finally, a summary based on our tests is provided. A LEAST SQUARES SOLUTION TO THE RATIONAL FUNCTION MODEL RFM uses a ratio of two polynomial functions of ground coordinates to compute the row image coordinate, and a similar ratio to compute the column image coordinate. The two image coordinates (row and column) and three ground coordinates (e.g., latitude, longitude and height) are each offset and scaled to fit the range from –1.0 to 1.0 over an image or image section. A detailed description on this normalization process can be found from OGC ( 1999). For an image, the defined ratios of polynomials have the form (Greve, 1992):