CIPA 2003 XIX"' International Symposium, 30 September - 04 October, 2003, Antalya, Turkey 
A r = v 1 + k 2 r 5 + k 3 r 7 
[16] 
On the other hand. 
Ar = a^(r 2 -r 0 2 ) + a 2 r(r 4 -r 4 ) + a 3 r(r 6 -r 0 6 ) t 17 ] 
that is equivalent to 
Ar = a 0 r + a x r : ’ + a 2 r 5 + a 3 r 1 
See in the figure 7, how the line Ar= aor serves to have the 
of 
A r = + k 2 r 5 + k 3 r 1 
in the form of this one: 
Ar = a 0 r + a x r' + a 2 r 5 + a 3 r 7 
The slope could be freely chosen in order to fit some needs, as 
it is for instance, to have the values of distortion arranged in a 
weight-balanced distribution, (equal areas of positive and 
negative distortion values or equal maximum and minimum). 
At the second zero of the curve (see the expression above) we 
can write ao = - air 0 2 - a 2 r 0 4 
So the same expression could be rewritten as follows: 
[19] 
[20] 
Ar= (- a,r 0 2 - a 2 r 0 4 ) r + air 3 + a 2 r 5 = a^ (r 2 - r 0 2 ) + a 2 r (r 4 - r 0 4 ) 
[21] 
r + — 1 —r +——r 
fb 1 ~b CIq 1 T Qq 
4(1 +a o) r + ky+k 2 r 5 
to conclude that: 
[25] 
that brings us 
f b =f,(l + a 0 ) 
a, = k, (l + a 0 ) [26] 
a 2 =k 2 (l + a 0 ) 
Our simulator allows the user to see how the Gaussian and 
balanced forms are related one to each other, showing 
equivalences between all the implied parameters in real time.. 
Then, lets call the principal distance corresponding to the 
Gaussian graph f g 
Ar g =k x -r +k 2 -r 
and ^ the same referred to the balanced form: 
Ar h = a 0 r + a x -r : + a 2 r' 
[22] 
[23] 
Lets notice that both families must comply the same relations. 
f + A/^ _ r + Ar h r + k t r 3 + k 2 r 3 _ (1 + a 0 )r + a 1 r 5 +a 2 r 5 [24] 
A 
f. 
A 
Figure 8. LDS’s Distortion graphing intergace. 
5. SIMULATION RESULTS. 
Once arranged all the mathematic models, we have tried various 
settings for testing several inner orientations for virtual 
cameras. After that, we have rendered the results to three- 
dimensional graphic outputs with the purpose of being of help 
for users to understand the models. (These plottings have been 
created with Mathlab®) 
The figures below are 2d and 3d plots representing a typical 
distortion distribution. Colour gradation enables a better 
visualisation of the shape and can also be intentionally set up to 
highlight zones where distortion surpasses usable levels. 
Therefore: