ANALYSIS OF RATIONAL FUNCTION DEPENDENCY TO THE HEIGHT 
DISTRIBUTION OF GROUND CONTROL POINTS IN GEOMETRIC CORRECTION OF 
AERIAL AND SATELLITE IMAGES 
M. Hosseini, 
Department of Geomatics Engineering, Faculty of Engineering, Tehran University, hoseinm@ut.ac.ir 
(Centre of Excellence for Geomatics Engineering & Natural Disasters Management) 
ABSTRACT: 
One of the existence mathematical models is Direct Linear Transformation (DLT). These equations are being regarded because of 
their simplicity as they are direct. When the height distribution of ground control points (GCPs) is inappropriate, height accuracy of 
DLT is low. This problem was not obvious about rational functions. To assess this case, the accuracy of rational functions has been 
tested in three different cases of GCPs distribution including over sampling, optimum sampling and under sampling. At last we have 
come to conclusions that the accuracy of rational functions in over sampling and optimum sampling are more than under sampling. 
But the accuracy of over sampling has not a significant difference with the accuracy of optimum sampling. In all cases, to compare 
that the accuracy of direct solution is more than the accuracy of indirect solution. All done tests are in terrain-dependent case of 
rational functions. 
1996). Selecting one of these models depends on the required 
accuracy and the available sensor ephemeris rigorous models 
are based on collinearity equations. One of the difficulties of 
rigorous models is their dependency to sensor. In other words Where r n and c n are normalized row and column pixel 
these models have changed for different sensors. Because the coordinates in image space and Xn, Yn and Zn are normalized 
number of different aerial and satellite sensors like frame, coordinates in ground space. For minimizing calculation errors, 
pushbroom and their applications are increasing, it is necessary two iag e coordinates and three ground coordinates are 
that existed software be changed for the analysis of their normalized such tha being in (-1,1) (NIMA, 2000). 
different data. Also for using rigorous models it is necessary 
that imaging parameters like orbital parameters, satellite Ay k „ by k , Cy k and dy k are polynomial coordinates and were 
ephemeris, earth curvature, atmospheric refraction and lens named rational faction coefficients. For normalizing 
distortion be known. It is essential that linearize these models coordinates we can use below relations (OGC, 1999): 
because of their non-linearity. But generic models are in 
jj j because of its independence from position and „ v 
orientation of sensor. Generally it isn’t essential to know _ r ~ r o _ c c o yy _ ^ ^ o 
sensor’s geometry for using generic models and it is possible to ” r ’ ” C ’ ” X 
use them for different types of sensors. In generic models, s s s 
relationship between image space and object space is making by 
In rational functions, image pixel coordinates (r,c) are ratio of image coordinate scale numbers. Similarly, Xo, Y 0 and Z 0 are 
the direct and indirect solution of rational function, we solved rational functions with both mentioned methods. At last it was clear 
1 INTRODUCTION 
ml m2 m3 
There are a lot of mathematical models for photogrammetric 
P3(X n ,Y n ,Z n ) _ /=0 j= o k=o 
processings. These models show the geometrical relationship C n 
between 2D image space and 3D ground space. Generally these 
models are divided to rigorous and generic models (McGlone, 
rational functions. 
2 RATIONAL FUNCTIONS 
Where r 0 and c 0 are image coordinate shifts and rs and cs are 
polynomials of ground coordinates (X,Y,Z) (OGC, 1999): 
Pl(X n ,Y n ,Z n ) _ m) y=o k=o 
La La La 
/=0 j=0 k=0 
ml m2 m3 
ground coordinate offsets and Xs, Ys and Zs are image 
coordinate scale numbers. Inverse rational functions are 
transformations from image space to ground space (Tao and Hu, 
2001b):