260 
2) Perspective trasformation, that gives the undistor 
ted position of P onto the image plane. 
f , ^ 
\ umf T 
K 
r'x 
(2) 
\h ur = k l *u(u 2 + v 2 ) + k 2 *u(u 2 + v 2 ) 2 
[S vr = k x *v(u 2 +v 2 ) + k 2 *v(u 2 + v 2 ) 2 ^ 
2) Decentering distortion: 
|8«/ = Pi *( 3 « 2 +v 2 ) + 2p 2 *uv 
[£>vd = 2p x uv + p 2 *v(w 2 +3v 2 ) 
3) Change of image reference system in order to rela 
te the metric image coordinates (u,v) of point P 
with the corresponding pixel coordinates (r,c) in 
the digitized image. 
r-r 0 =s u *u 
S„ * V 
(3) 
where (r 0 ,c 0 ) denotes the pixel position of the 
principal point O’, while s u and Sv are determined 
by the CCD cells dimension as follows: 
5 
Ac * 
Ncc 
~Nfi 
(4) 
where 
Ar, Ac center to center distance between adja 
cent sensor elements of the CCD array, 
in the Y and X direction respectively; 
Ncc number of sensor elements (columns of 
CCD array) in the horizontal direction 
(Y axis); 
Nfc number of pixels in a line as sampled by 
the frame-grabber; 
s image scale factor, this is an additional 
uncertainty parameter introduced to take 
into account various source of error in 
the CCD array sampling, performed by 
the frame-grabber [3]. 
As regards the lens distortion, in our camera model we 
considered three major kind of lens distortions namely: 
radial, decentering and thin-prism. However, the cali 
bration procedure was implemented in such a way to 
incorporate eventually further geometrical distortions, 
although this lead to a more complex camera model 
and requires an higher computational effort. 
The corresponding set of distortion parameters that we 
have adopted, is reported below: 
1) Radial distortion: 
3) Thin-prism distortion: 
Su P =s M 2 +v 2 ) 
I* , 2 2^ ( 7 ) 
5 V , =s 2 (u +v ) 
4) Total distortion: when all the above distortions are 
present, the effective distortion can be modeled by 
addition of the corresponding expressions [3]. 
Therefore combining (5), (6), (7) we obtain the 
total amount of lens distortions along the u and v 
axes, 
5 ut = k x *u(u 2 +v 2 ) + k 2 *u{u 2 +v 2 ) 2 + p x 
* (3u 2 + V 2 ) + 2p 2 *UV + 5, (w 2 + V 2 ) 
8 W =k x *v(u 2 +v 2 ) + k 2 *v(u 2 +v 2 ) 2 (8) 
+ 2p x u\ + p 2 * (w 2 + 3v 2 ) + s 2 (w 2 + v 2 ) 
Taking into account the distortion along the u and v 
axes, the relationship between distortion free image 
coordinates (u,v) and its corresponding pixel locations 
(r,c) becomes 
u + 6 u (u,v) = ^—^~ 
S _ U (9) 
v + 8 v (w,v) = —2^- 
l 5 v 
where 5 u (u,v), 5 v (u,v) can represent the total distortion 
or a combination of the above mentioned factors, ac 
cording to the purposes of the calibration. 
Note that the lens distortions are computed according 
to image coordinates (u,v) that are unknown. 
Now, following the procedure presented by [5], if we 
introduce the new variables (u’,v’), that represent the 
distorted location of image projections of target points 
onto a normalized image plane (Z c =l), 
^_u+8Ju,v) ^_r-r 0 
f fu 
v ,_ v + 5y(^v) = c-c 0 
f /v 
(10)