The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
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last section deals with the evaluation of the fitness accuracy 
obtained on the stereo IRS-P5 images acquired over a test site. 
1.1 Rational Function Model 
The concept of RFM was developed by Gyer (Sadeghian, 
2001).This model project the 3D object space coordinate into a 
2D image coordinate and could be used for any type of images; 
airborne or space borne. The Rational Function Model is 
expressed as a ratio of two polynomials: 
.. K(xj,z) 
f;(x,y ,z) (0 
n(xx,z) 
f;(x,y,z) 
Where: x ,y - are the normalized image coordinates 
X ,Y ,Z = are the normalized object coordinates 
F j = is the r order polynomials 
r is the order of polynomials which is usually equal to 3 so the 
rational model would be the ratio of two cubic polynomials and 
each polynomial has 20 terms making a total of 80 parameters 
which is reduced to 78: i.e. 
1.1.2 Terrain independent scenario: When the rigorous sensor 
model is available, the RFCs can be determined via 3D object 
grid with corresponding image points which are determined 
through rigorous model. In this case, the solution is actually 
terrain independent since no terrain information is used (Tao C 
V, Hu Y, 2001. The method has three main steps: 
First, determination of a grid of sufficient image points, 
Second, set up a 3D grid of object points via rigorous model, in 
this step, rigorous parameters -which are measured using 
onboard GPS receivers and gyros-are used to compute the 
corresponding object coordinates of image points. 
Third, RFM fitting, the RPCs are computed using image 
coordinates and their corresponding object coordinates by 
applying space resection. 
1.1.3 The Bias Compensated RPC: Exterior orientation 
parameters which comprise position and attitude are used to 
calculate RPCs in terrain independent mode. On-board GPS 
receivers determine the satellite ephemeris. Star tracker and 
gyros determine attitude as a function of time. Ephemeris and 
attitude have finite accuracy, about one meter for ephemeris and 
one or two arc-second for attitude ^ ^. As a result, the calculated 
RPCs in terrain independent mode have bias. Many research 
works have already been conducted as regards the methods and 
the accuracy of the bias removal. It has been demonstrated that 
the bias can be compensated via one shift parameter in line 
direction and one shift parameter in sample direction (Dial & 
Grodecki 2002). 
r _(l ZXY..Y i X 1 ).(a 0 a r . 1 qJ 
(iZXY ..7 3 Y 3 ).(lè,è 2 ..i) 19 ) 
(IZXY .■7 3 X 3 ).(c oC ,..r 19 ) 
y (iZXY ..7 3 Y 3 ).(W,rf 2 ..// l9 ) 
(2) 
x +A o 
y +B 0 
f;(x,y ,z) 
f;(x,y,z) 
f; (x,y,z) 
f;(x,y,z) 
(4) 
In order to increase the stability of equation, it is essential to 
normalize the two image coordinates and three object 
coordinates using shift and scale parameters to fit the range 
-1 ~ +1. The normalizing equation can be described as: 
» x ~ x 0 > y -yo 
x = , y = - — 
X s 
, r ..LiLL , r. 
X. Y. 
Z-Z„ 
(3) 
Where: x 0 , y 0 = are shift values for image coordinate 
x ,y — are the scale value for image coordinate 
X a J 0 ,Z 0 = are shift value for object coordinate 
X s ,Y S ,Z = are the scale value for object coordinate 
The unknown parameters involved in RFM can be determined 
with or without using the rigorous sensor model. Therefore two 
computational scenarios are present, terrain dependent and 
terrain independent (Grodecki et al., 2004). 
The bias compensated RPCs can be computed as: 
F{X,Y ,z)={a -è/) + (a ~bA o )x 
+... + (a a -b x A ).Z 1 
f(x,Y ,z) =(c -dB) + (c 2 -d b)x 
+... + (c -d B ).Z’ 
X 20 20 0 ' 
The accuracy of this model has been reported less then 0.5 
pixels (Hanley & Fraser, 2004). Parameters for drift can also be 
added. 
1.2 3D Affine Transformation 
Affine model is a linear transformation which maps the 3D 
object space into 2D image space through 8 parameters, as 
shown in equation (5): 
1.1.1 Terrain dependent scenario: When there is no rigorous 
sensor model at hand, one has to measure control points and 
check points from both images and the actual DEM or maps. In 
this case, the solution is heavily dependent on the actual terrain 
relief, number of control points and distribution of control 
points (Tao C V, Hu Y, 2001/ 
* =A t X +A/ +A 3 Z +A 4 
y =A S X +A/ +A 7 Z +A s 
Where A A g = affine parameters 
(5)