1150 
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
In these equations planimetric ground space coordinates (X,Y) 
are the ratio of polynomials of image pixel coordinates (r,c) and 
vertical ground coordinate (Z). 
Rational function coefficients can be solved by sensor physical 
model or without it. If physical sensor model be known then we 
make a grid in image space. Then we used this grid and physical 
sensor models to produce another grid in 3D object space. Grid 
dimensions depend on ground dimensions and ground object’s 
height differences. In other words grid dimensions fill all 3D 
ground space. This grid has many layers. Each layer points in 
each layer have the same elevation. Number of layers should be 
more than three to avoid rank deficiency of design matrix (Tao 
and Hu, 2000). After making the grid we had used ground 
coordinates with their similar image coordinates to calculate 
rational function coefficients by least square method. In this 
method there isn’t any need to true ground information and it is 
named ground independent (Tao and Hu, 2000). This method 
were used for geometric correction of high resolution satellite 
images (Paderes et al., 1989; Madani, 1999; Yang et al., 2000; 
Baltsavias et al., 2001; Tao and Hu, 2000). We should know 
physical sensor model to produce 3D ground grid. For solving 
rational functions coefficients, we should used ground control 
points (GCPs) that were collected by general methods like map 
and DEM and calculating rational function coefficients. This 
method of solving rational functions was named terrain 
dependent (Tao and Hu, 2000). We used this method in remote 
sensing when physical sensor model is unknown (Toutin and 
Cheng, 2000; Tao and Hu, 2001a, b). There is limited research 
on solving rational functions by terrain dependent method that 
had done by Tao and Hu. 
3 EXPERIMENT AND RESULTS 
3.1 Simulated data set 
For making simulated data set, first we suppose of a grid in 
ground space. Number of points should be sufficient. For 
calculating left and right image coordinates of ground points, 
we 
used collinearity equations. Simulated is related to 1:10000 
image scale. There totally 96 points that make a 12*8 grid. 
Figure 1 shows a 3D view of ground surface and these ground 
points. Heights of points had been choose such that the ground 
be approximately £ . Heights of points are 
between 10-100m. 
Simulated data is related to 1:10000 image scale. There are 
totally 96 points that make a 12*8 grid. Figure 1 shows a 3D 
view of ground surface and these ground points. Heights of 
points have been chose such that the ground be nearly 
. They are between 10-100m. systematic error that 
have is 10pm. 
Figure 1. Simulated ground points 
3.2 Aerial data 
In the next step, we used true aerial data for testing the models. 
These images are stereo that show a part of Germany. We used 
Softcopy for measuring ground control points’ coordinates. 
These points are nearly a grid and fill the 
entire image surface. Then we used collinearity equations with 
interior and exterior parameters to calculate image coordinates 
of ground points on stereo images. Calibrated focal length of the 
camera for taken aerial images is 152.844 and the approximate 
scale of these images is 1:15000. Figure 2 shows a 3D view of 
ground surface and extracted points. Maximum height 
difference of existed points is 120 meters. 
Figure2. Extracted ground points from aerial images 
3.3 Space data 
Stereo images had taken by IRS-1C satellite. These images 
show Mashhad city of Iran. Size of each image is 4096*4096 
pixels and the overlap area of two images is 90 percents. There 
are 53 ground control points and their similar points in the 
images. Height of these points are between 930-1075m. 
3.4 Results of simulated data 
All the experiments have done in three different cases of control 
point’s height distribution. We used terrain dependent rational 
functions in these experiments.