Updating Solutions of the Rational Function Model Using Additional Control Information Yong Hu and C. Vincent Tao Abstract shown that the RFM can achieve a very high fitting accuracy to The rational function model (RFM) is a sensor model that the physical and is capable of replacing the rigor- allows users to peqorm ~ r t h o - ~ ~ c t i f i c a t i ~ ~ and 30 feature sensor models for photogrammetric restitution ( ~ a d a n i , extraction from imagery without knowledge of the physical lg99; DOwman and Dolloff,2000; Yang, 2000; Tao and Hu, sensor model. It is a fact that the RFM is determined by the 2001~). It was reported in Grodecki (2001)that the ~konos ra- vendor using a proprietary physical sensor model. The ac- tional differs no more than 0.04 pixel fromthe ~ h~si- curacy of the RFM solutions is dependent on the availability cal withtheRMS error 0.01 pixel. and the usage of ground control points (GCPS). In order to The RFM solutions are determined by the data vendor us- obtain a more accurate RFM solution, the user may be asked ing a proprietary physical enso or model. The accuracy of the to supply GCPS to the data vendor. However, control infor- RFM solutions is dependent on the availability and the usage of mation may not be available at the time of data processing the GCPS. If accurate RFM solutions are required, GCPs are need- or cannot be supplied due to some reasons (e.g., politics or ed and are incorporated into the RFM solution process. In this confidentiality). This paper addresses a means to update or case, the user may be a ~ k e d to supply the GCPs to the data ven- improve the existing RFM solutions when additional GCP~ are dor. However, the GCPS may not be available at the time of proc- available, without knowing the physical sensor model. From essing or cannot be supplied due to some reasons (e-g., politics a linear estimation perspective, the above issue can be tackled or confidentiality). using a phased estimation theory. In this paper, two methods If additional GCPS are available, one may ask if it is possible are proposed: a batch iterative least-squares (BILS) method to update or improve the existing RFM solutions (provided,for and an incremental discrete Kalman filtering (IDKF) method. example, by the vendor). In this paper, we present an approach Detailed descriptions of both methods are given. The feasi- to update and/or improve the existing RFM solutions when ad- bility of these two methods is validated and their perform- ditional GCPs are available, given that the physical sensor ances are evaluated. Some results concerning the updating model is unknown. In the next section, we briefly describe the of Ikonos imagery are also discussed. RFM by introducing a least-squares solution as well as two com- putation scenarios for RFM determination. In the following Introduction section, we present two methods for updating the initial RFM A rational function is a function that can be represented as the solution, namely, a batch iterative least-squares (sns) method quotient of two polynomials. Mathematically speaking, all and an incremental discrete Kalman filtering (IDKF) method us- polynomials are rational functions (Newman, 1978).The ratio- ing additional GCPS. Finally, we show the results computed by nal function model (RFM) in this context is a sensor model rep- both the sns and IDKF methods to demonstrate the feasibility resenting the imaging geometry between the object space and and the performance of each method. An aerial photograph the image space. and an Ikonos stereo pair were used in the experiments. The The RFM has gained considerable interest recently mainly left and right images of the Ikonos stereo pair were updated due to the fact that Space Imaging Inc. (Thornton, Colorado) and three-dimensional reconstruction was done to check the has adopted the RFM as a replacement sensor model for image possibility to update the rational function coefficients (RFCS) exploitation. The RFM is provided to end users for photogram- without further information about their covariance. metric processing instead of the Ikonos physical sensor model. Such a strategy can serve two purposes. On the one hand, the use of an RFMmay help keep information about the sensor con- SO~U~~O~S to the Rational Function Model fidential because it is difficult to derive the physical sensor parameters from the RFM. On the other hand, rational function Direct and Iterative LeastSquares Solutions models facilitate the exploitation of high-resolution satellite The WM uses a ratio of two polynomial functions of ground co- imageryby end users. With the RFM provided, users and devel- ordinates to compute the row image coordinate, and a similar opers are able to perform photogrammetric processing such as ratio to compute the column image coordinate. The two image ortho-rectification, 3D feature extraction, and DEM generation coordinates (row and column) and three ground coordinates hom imagery without knowing the complex physical sensor (e.g., latitude, longitude, and height) are each offset and scaled model (Tao and Hu, 2001a; Tao and Hu, 2001b). Tests have to fit the range from -1.0 to 1.0 over an image or an image sec- Y. Hu is with the Department of Geomatics Engineering, The University of Calgary, 2500 University Dr. NW, Calgary, Alberta T2N 1N4 Canada (yhu@ucalgary.ca]. C.V. Tao is with the Geospatial Information and Communica- tion Technology (Geo-ICT) Lab, Department of Earth and Atmospheric Science, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada (tao@yorku.ca) PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING Photogrammetric Engineering & Remote Sensing Vol. 68, No. 7, July 2002, pp. 715-723. 0099-1112/02/6807-715$3.00/0 O 2002 American Society for Photogrammetry and Remote Sensing