The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part Bl. Beijing 2008 
1304 
where 
GM 
~2 .(X 2 0 +Y 0 2 +Z 2 o f' 2 
and ti and t 2 is the acquisition time of the corresponding 
framelets in image 1 and 2 respectively. 
Combining the equations (4) based on (2), B x is defined as 
follows: 
B x =x Lr x Li 
— u x • (/| — t 2 — dt) + A • X 0 • {t 2 -F dt + 2 • ?2 ■ dt — ) 
The above equation clearly shows the important role of the 
UCL sensor model approach because it relates directly the B x 
only with the state vector of the center point of the first 
pushbroom image, which are the unknown parameters in this 
equation. 
Going one step further the acquisition time t 1 and t 2 can be 
described as: 
t x = x, • interval 
t 2 = x 2 ■ interval 
where interval is the acquisition time of a framelet which is 
assumed to be constant (Michalis and Dowman, 2004). 
Finally B x is defined as follows based on the previous 
equations (with exactly the same procedure the B y and B z are 
also calculated): 
B x -u x ■ [interval • (x, - x 2 ) - dt\ + 
+ A ■ X o ■ [interval 2 ■ (x 2 - x 2 ) + 2 • x 2 • interval ■ dt + dt 2 ] 
B y = u y • [interval ■ (x, - x 2 ) - dt\ + ^ 
+ A ■ Y o ■ [interval 2 ■ (x 2 - x, 2 ) + 2 • x 2 • interval ■ dt + dt 2 \ 
B z = u z ■ [interval • (x, - x 2 ) - dt] + 
+ A • Z 0 ■ [interval 2 ■ (x 2 - x 2 ) + 2 • x 2 • interval • dt + dt 2 ] 
The coplanarity equation f c is developed as a combination of 
equations (3) and (5) where the unknown parameters are the 
elements of the exterior orientation of both stereo images 
which are: 
• State vector of the center point of the first image ( 6 
unknowns) 
• The coefficients of the first order rotation 
polynomials of both images( 12 unknowns) 
Thus, the coplanarity equation f c is defined as follows: 
fc=UX 0 Jo,Z 0 ,u x ,u y ,u 2 , 
i i ■> IP\2 » $\ l ? &\2 •> ^*111 ^"l2 ’ ®21 ’ ^22 ’ fill ’ ^22 ’ ^*21 ’ ^*22 ) — ® 
5.3.3. Coplanarity role in exterior orientation 
determination and in DEM extraction process. The 
coplanarity equation can be used in combination with the 
collinearity equations during the computation of the exterior 
orientation providing additional equations and stability. In 
more detail in the resection process three equations (two 
collinearity equations plus one coplanarity equation) for each 
GCP and one coplanarity equation for each tie point can be 
applied, improving the accuracy and the stability of the 
solution. 
Traditionally in perspective center geometry the epipolarity is 
used for the production of normalized images. In this paper a 
different approach is followed as a first step. The coplanarity 
condition is a robust and rigorous equation which can be used 
easily and straightforwardly as a geometrical constraint in the 
matching procedure of the DEM generation process. In more 
detail: just after the correlation process the coplanarity 
equation can be applied to all the conjugate points that are 
extracted by the correlation in order to see if this equation is 
fulfilled, using a threshold related to RMSE of the resection 
solution. The points that do not pass this test are blunders or 
correlation errors in general. 
6. EVALUATION 
6.1. Introduction. 
6.1.1. RPCs model. The Cartosat data are distributed with the 
rational polynomials coefficients. In EPS there is a module 
where Cartosat RPCs could be imported and used for the 
orientation of the images. 
6.1.2. UCL model. The UCL sensor model could be solved 
directly using navigation data, without GCPs (Michalis and 
Dowman, 2004, 2005). Unfortunately, because in the case of 
Cartosat no navigation data is provided the exterior orientation 
parameters should be calculated using GCPs. The total number 
of exterior orientation parameters of the two Cartosat along 
track stereo images is eighteen. The state vector of the center 
framelet of the first image represents six of these unknown 
parameters, while the corresponding state vector of the second 
image is calculated from the previous one by the Kepler 
equation. The other twelve unknown parameters are the 
rotation angles of the two images; six rotations for each image 
which are the coefficients of the polynomials. Thus, at least 
five GCPs are needed for the solution, when the collinearity 
equations are used. It will be shown that with the simultaneous 
use of the coplanarity equation it is possible to reach accurate 
solution with four GCPs. Moreover even in the case where 
enough GCPs are available for a solution only with the 
collinearity equation the coplanarity ensures a more precise 
solution. 
6.2. Evaluation strategy 
As it shown in figure 1 the GCPs are well distributed on the 
images. Nine of the total 36 available points are used as GCPs 
in the evaluation process of both models, while the rest 25 (two 
are not well identified on the images) are used as ICPs 
(Independent Check Points). The GCPs location is not 
important for the exterior orientation solution (Michalis and