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Title
International cooperation and technology transfer
Author
Mussio, Luigi

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)