The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
Kim (2000) formulated exact epipolar curve for the linear array 
scanner scenes based on Orun and Natarajan’s orientation 
model and derived some useful properties of the epipolar curve. 
The shape of such epipolar curve is not a straight line but 
hyperbola-like non-linear curves. It can be approximated by a 
straight line for a small range but not for the entire image. 
Conjugate epipolar pairs do not exist for this kind of imagery, 
but exist “locally”. 
The epipolarity equation derived by Kim is useful in analysing 
the property, but it is difficulty to apply this equation to the 
process of linear push-broom scenes due to the need of 
orientation information. Morgan (2004) derived the epipolarity 
of this kind of scenes based on two-dimensional affine model, 
and obtained a linear model which can be used to resample the 
entire scene. This paper proposes a method to generate the 
approximate epipolar lines based on RFM without deriving 
epipolarity equation. 
2. DESCRIPTIONS OF RFM 
RFM relates object point coordinates (X, Y, Z) to image pixel 
coordinates (/, s') or vice visa, as physical sensor models, but in 
the form of rational functions that are ratios of polynomials 
(Yong, 2005). The RFM is essentially a generic form of the 
rigorous collinearity equations. The RFM is divided into 
forward RFM and inverse RFM according to the relationship 
between object space and image space. 
The forward RFM is a transformation from the coordinates in 
the object space to the row and column indices of pixel in the 
image space. The defined ratios have the form as follows: 
l n = P\( X n’Y n ,Z n )/p 2 (X„,Y n ,Z n ) 
s n =Pi(X n ,Y n ,Z n )/p<(X n ,Y n ,Z n ) 
The inverse RFM is to transform the image coordinates to the 
object coordinates, which can be presented as follows: 
X n=Ps(Ls n ,Z n )/p 6 (l„,s n ,Z n ) (2) 
Y n = pAL^n’ZJ/p s (i n ,s n ,z n ) 
where /„, s n are normalized image coordinates; 
X n , Y n , Z„ are normalized object space coordinates; 
Pi is a polynomial with the following form: 
p(X. Y, Z) = £ tt • X "‘ ■ Y "' Z * <3) 
;'=0 j=0 k=0 
Where N is the polynomial order, c ijk called rational polynomial 
coefficients. In order to improve the numerical stability and 
minimize the introduction errors during computation, both 
image coordinates and object space coordinates are normalized 
in the range of [-1 +1] by applying offsetting and scaling 
factors(NIMA, 2000). When N is three, Eq. (3) becomes a three 
dimensional polynomial with 20 coefficients, which is the most 
commonly used form by data and software vendors. The model 
used in this paper is in the form of three order polynomial. 
3. EPIPOLAR LINE GENERATION 
The epipolar geometry of linear array scanner scenes can be 
understood more easily with Figure 1. The p is a point on the 
left scene; S is the perspective centre of point p. The light ray 
pass through S and p hit the ground at point P. Every point on 
the light ray can be projected to the right scene, and the 
combination of these projected points can form a curve on the 
right scene, which is called the epipolar curve of point p. 
Figure 1. Epipolar geometry of linear array scanner scene 
According to the definition of the epipolarity for linear array 
scanner scene described above and the property summarized by 
Kim (2000), a method to generate the approximate epipolar line 
of point p based on RFM can be developed. 
The procedure of epipolar line generation is as follows. For a 
point p on the left scene give an elevation value Z and calculate 
the corresponding object space coordinate use the inverse RPCs 
of the left scene. Then project the object space point into the 
right scene use the forward RPCs of the right scene. Repeat this 
process for a quantity of times, each time the elevation of the 
object point should change with equal interval along the light 
ray connecting the perspective centre and image point on the 
left scene, a series of image point on the right scene can be 
obtained, fit them to be a line 1’. The line 1’ should be the 
approximate epipolar line of point p and the conjugate point of 
p should be located on the line 1’ or near from it. During this 
process, the actual elevation of the object space point is not 
required. In the experiment section some details such as how to 
change the elevation and if the elevation range and the times 
elevation changed have influence on the accuracy of the 
epipolar line will be discussed. 
4. EXPERIMENTS 
4.1 Data Description 
The dataset used in the experiment involves a stereo-pair 
captured by IKONOS-2 over the south of Australia with 1 
meter resolution. The specifications of these scenes are listed in 
Table 1. Residential areas, water areas and mountains are 
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