264 
where 
r. - r n 
f. 
C; ~Cr 
/» 
(25) 
are the measured distorted image coordinates of the N 
control points, as derived by egde detection algorithm. 
Obviously in case we consider a camera model with a 
reduced set of distortion parameters, the corresponding 
columns in matrix A have to be eliminated. 
4. THE CALIBRATION SOFTWARE GUI 
In order to make more easy and understandable the 
calibration procedure from teaching and user final 
point of view, we have provided the algorithm with a 
Graphic User Interface. It was implemented in Matlab, 
because this is a development environment best suited 
for mathematical applications and it don’t requires a 
specific knowledge about programming language like 
Fortran, C or C++. Of course Matlab has own syntax, 
anyway a complex software can be structured through 
scripts readable with a common text editor, making 
therefore easy to manage and modify the software 
itself. 
An example of the GUI is depicted in Fig. 5 showing 
the DATA window, where all input parameters can be 
set up, namely: CCD camera construction features 
(menu Camera), number of used target planes (up to 6, 
menu Target) and the calibration mode (menu Option), 
listing the distortion parameters taken into account. 
The user can view the results of calibration process 
both in numeric form, through the PARAMETERS 
window selecting the param submenu (Fig.6), and in 
graphical form selecting the submenu result. The aim 
of using such graphic windows should allow the user to 
assess in easier way the quality and accuracy of the 
calibration. 5 
5. TEST AND RESULTS 
In order to evaluate the overall accuracy of our method, 
we have performed a calibration test using a target 
plane with 48 black squares on white background, each 
having lateral dimension of 50mm and horizontal and 
vertical spacing of 150mm (Fig. 7). The vertices of 
these squares were employed as control points. 
To this aim the Canny edge detection algorithm [1] was 
applied to the target squares, then the corresponding 
lines were recovered from resulted edge points by 
cubic splines interpolation. Finally the image coordi 
nates of the vertices were determined as the points of 
edge lines intersections. 
For the test we used a Kodak DCS-410 professional 
digital camera, employing a full-frame CCD image 
measuring 1524x1012 pixels, with a lateral dimension 
of CCD cells of 9pm. The target plane, located at a 
distance of = 2m from the camera, was taken from dif 
ferent points of view (up to 6 positions) in order to get 
a larger spatial information about the perspective tran 
sformations experienced by control points. Employing 
an objective of 24mm the focal length was set up to 
infinity and the diagfram to 11. 
Considering each time a different combination of dis 
tortion parameters, we have therefore performed seve 
ral calibrations, which results are listed in tables 1, 2 
and 3. The values of internal and distortion parameters 
are reported in table 1, according to 4 calibration test 
(rad2 means the estimate of both radial distortion coef 
ficients). Instead, in table 2 (stdr,stdc) represent the 
errors along rows (r) and columns (c) in L rc , between 
image points positions, as derived by features extrac 
tion procedure, and the image coordinates of same 
points, as computed by the model. As both position are 
affected by geometrical distortions of the camera, the 
discrepancy can be regarded as an estimate of the noise 
superimposed on the image. The following four para 
meters (mean X w , mean Y w , std X w , std Y w ) represent 
the mean and the stdev of the position errors along X w 
and Y w axes in reference system X w . These values are 
calculated by differences between measured 3D coor 
dinates of control points and backprojected positions 
on the target of corresponding image points, which 
locations were corrected through the estimated camera 
model. Finally, in the same way, the means and stdev 
of position errors between coordinates points in the 
camera reference system X c were computed, which 
results are listed in table 3. 
Table 1: calibration results about internal and 
distortion parameters 
Internal 
parameters 
rad + dec 
rad + thin 
rad2 + dec 
+ thin 
I'O 
521.13 
504.79 
517.34 
cp 
768.25 
757.54 
758.51 
f 
23.95 
23.95 
23.97 
s 
0.99923 
0.99925 
0.99991 
Distortion 
parameters 
ki 
0.12650 
0.12597 
0.16914 
h. 
0 
0 
-0.69756 
Pi 
0.002332 
0 
0.001 S99 
P2 
-0.001526 
0 
-0.001 651 
Si 
0 
U.ÜÜ3U53 
U.000648 
S2 
0 
-0.001948 
-0.001 753 
Table 2: position errors of backprojected control points 
rad + dec 
rad + thin 
rad2 + dec + thin 
std-r 
0.209 
0.209 
0.206 
std-c 
0.24 7 
0.248 
0.244 
mean X v 
0.185 
0.185 
0.242 
mean Y, r 
0.232 
0.232 
0.322 
std X*, 
0.130 
0.120 
0.216 
std Y v 
0.161 
0.162 
0.257