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):
