CIPA 2003 XIX th International Symposium, 30 September - 04 October, 2003, Antalya, Turkey 
336 
MULTI-COLLIMATOR 
picture, and see how distortion distribution changes according 
to the f variations set by user. 
Figure 6. LDS’s user interface. 
The effects of related variations are graphically shown and will 
help the user to understand the distortions appearing on images. 
4. RADIAL DISTORTION MODELS. 
The mathematic model is as follows: 
MX -X s ) + r a (y-Y s )+r n {Z -Z,) 
(x-x„)(\ + a,r + “i r + <V ) = -/ 
(y-yOO + aS 2 +a 2 r* + a 3 r b ) = -f 
r = J(x-x 0 ) 2 +(y-y B ) 2 
where the following is assumed 
r,AX -X s ) + r n (Y -Y s ) + r„(Z-Z s ) 
r 2i (X-X s ) + r 22 (Y -K,) + r 21 (Z -Z,) 
r M (X-X s ) + r n (Y -Y s ) + r 3J (Z-Z s ) 
[6] 
We have already pointed out to some difficulties concernig this 
problem. In fact, there is not a unique model to explain the 
relationship between the radius and the radial displacement. 
Distortion correction: 
A r = F{r) 
Radial distortion value: 
dr = F(r) HO] 
1. Brown’s model: 
x s " 
CO 
V 
= 0 ; 
Y s 
= 0 ; 
(p 
_To_ 
.Zs. 
_X_ 
where the coordinates (X,Y)j of the grid nodes being Z x . So that 
the model can be written as below: 
- dr = Ar - a x r 3 + a 2 r 5 + a 3 r 7 
2. USGS’s model: 
- dr = Ar = a 0 r + a x r J + a 2 r 5 + a^r 1 
3. ISPRS’s model: 
[H] 
[12] 
Z(1 + a x r 2 + a 2 r 4 + a 3 r 6 ) ' 
v= zL V 
Z(1 + a x r 2 + a 2 r 4 + a 3 r ò ) ' 
r = ^(x i -x 0 ) 2 +(y. -y 0 ) 2 
- dr = Ar = a,r(r 2 - r 0 2 ) + a 2 r(r 4 - r () 4 ) a 3 r(r ( ’ - r 0 6 ) 
[13] 
[8] 
Notice that the third one can easily be identified with the 
USGS’s if the following is assumed: 
-a 0 =r 0 2 *(a x +a 2 *r;) [14] 
As it is said before, while the distorted model remains the same, 
any change in focal length brings about a new distortion scheme 
(distribution) and also new coefficients (ai) as a consequence: 
where (x„ y,) are image coordinates affected by radial 
distortion. 
Once the expressions of the image coordinates have been 
isolated, it is seen that in both right side terms there are 
unknowns: the distorted coordinates. As long as we have set up 
perfect coordinates as a starting for simulating their distortions, 
it is needed to use an iterative strategy to evaluate that system. 
The user is allowed to modify the four basic parameters (a l5 a 2 , 
a 3 , f) and evaluate the effect of their changes. It’s also possible 
to arrange a grid of image points just simulating any taken 
% = 2il = 2l ns] 
/, A f 
In LDS special attention has been paid to the comparison 
between Gaussian model of distortion correction versus that so 
called balanced one (ISPRS) due to the meaning of the last one. 
Lets reformulate the first expression above to avoid the 
confusion caused by the use of the same letters for the 
coefficients: