The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
1320 
dr F 
dw T 
x F ~r F (<p 0 ,A 0 h 0 ) 
dp f 
y F -p F (<p 0 ,A 0 h 0 ) 
dw 
dr A 
dw T 
,dw =\dX,d(p,dh\ ,L = 
X A -r A (<p 0 ,A 0 h 0 ) 
& £ 
1 
y A -P A (<P 0 ,*0 h o). 
The unknown object space coordinate is solved for iteratively. 
At the first iteration, the initial value for coordinates is needed 
which could be determined through linear equations such as 3D 
affine or DLT. Alternatively, truncated RPCs can also be used. 
At each iteration step, application of least squares, results in 
correction of approximate values of object coordinates, i.e.: 
dw = (a t A ) A L 
(12) 
(13) 
2.2 Affine Intersection 
For evaluating the object space coordinate using affine 
transformation, one should calculate its 8 parameters of each 
image at first by applying at least four GCPs: 
X ,Y ,Z,0,0,0 
0,0,0,X ,Y ,Z 
U 
(14) 
After evaluation of the parameters, by measuring image 
coordinate of any point in a stereo image, its object coordinates 
could be determined via the following relations: 
a f 
1 
a f 
5 
a a 
1 
A* 
5 
V -A'' 
( x ) 
y F -< 
Y 
= 
A .A 
X “ 4 
A A 
(15) 
Iterative least squares solution of Equation 10 yields the object 
coordinates for the virtual GCPs. This step is then followed by 
the solution of Equation 14 through which 3D affine parameters 
are derived using any combination of the well distributed virtual 
GCPs. Subsequently, for any virtual GCP whose scan and line 
coordinates is already measured, new object coordinates can be 
determined via Equation 15. The final fitting accuracy 
evaluation is then performed by comparing the object 
coordinates of the GCPs calculated by Equation 10 and the 
object coordinates computed by Equation 15. Next section 
presents the result of the tests conducted on IRS-P5 stereo 
image. 
3. EXPERIMENTAL RESULT 
3.1 Data Set 
A stereo orthokit image of the IRS-P5 satellite imagery over the 
city of Arak, Iran which were acquired in 2007, are used in the 
test. The RPC parameters were available for both forward and 
afterward images. Figure 1 shows parts of the stereo dataset 
used in the project. 
Figure 1. Afterward (left) and forward (right) images of stereo 
dataset. 
As mentioned before, for the solution of Equation 10 
approximate abject coordinates are required. The image 
coordinates of 18 virtual GCPs are measured. Three methods 
are implemented to generate the approximate coordinates, 
namely: 3D affine, DLT and truncated RPCs. The achieved 
results for RPC intersection by applying different method for 
evaluating the initial values have been presented in Table 1. 
Method 
AE(m) 
AN(m) 
Ah(m) 
Iteration 
3D affine 
2.42 
3.11 
6.40 
11 
DLT 
2.42 
3.11 
6.40 
14 
Truncated RPC 
2.42 
3.11 
6.40 
15 
Table2. Results of RPC intersection, applying different methods 
for evaluating initial value. 
As the Table 1 indicates, all methods of deriving approximate 
values have produced reasonably accurate initial values and the 
equations have been successfully converged. It is interesting to 
note that the affine model for calculating approximate values 
has lead to smaller number of iterations in the solution of 
Equation 10. This implies that the affine model generates more 
accurate results as compared to the other two approaches. 
Distribution of the generated virtual GCPs is presented in 
Figure 2. 
Having determined the object coordinates of the virtual GCPs, 
in the next step the 3D affined parameters (Equation 14) are 
solved to derive the 8 affine parameters by applying different 
number of GCPs and CPs. This is followed by the solution of 
Equation 15 to derive the object coordinates of all virtual GCPs. 
The residual vectors for the height and planimetric coordinates 
when 5 GCPs were used are presented in Figures 3 and 4 
respectively.