In: Paparoditis N., Pierrot-Deseilligny M.. Mallet C, Tournaire O. (Eds), 1APRS. Vol. XXXVIII. Part ЗА - Saint-Mandé, France. September 1-3, 2010 
76 
noise (i 
n pixel) 
0.3 
0.5 
1.0 
2.0 
— 
A 
f 
0.004 
0.006 
0.005 
0.04 
£ 
A Cpp A 
0.010 
0.020 
0.001 
0.20 
Ü 
A 
1 PPA 
0.022 
0.036 
0.014 
0.128 
<N 
A 
f 
0.013 
0.022 
0.052 
0.021 
£ 
A cpp a 
0.046 
0.077 
0.215 
0.308 
c 
A 
IpPA 
0.166 
0.276 
0.174 
0.348 
Table 1: Influence of noise on the intrinsic parameters on simu 
lated data. 
Tab. 1 shows the results that we obtained in our experiments. 
They show that our method can efficiently estimate the intrin 
sic parameters of the camera and that the noise only has a very 
slight influence on the calibration process. The results are quite 
similar for both short and long focal lengths. Even if there is an 
important difference between the results for the two cameras, the 
calibration can be considered very good since the parameters are 
estimated with an error bounded by 0.2 pixel. 
2.3 Distortion function 
Most of the cameras have a radial distortion which can be quite 
well modelled by a polynomial (see eq. 2). The unknowns to be 
estimated are thus: 
[ Ri, P , • • •, Rn, P , f, cppa,Ippa,cpps, Ipps, a, b, c ] (8) 
Tab. 2 summarizes the results of the tests that were conducted for 
Camera 1. The intrinsic parameters (/ and PPA) are very close 
to real values (about 0.1 pixel with 1 pixel noise). The error on 
the distortion parameters between the distortion function and the 
estimated model is around 1 pixel in the image comers for a noise 
of 1 pixel. 
0.3 
noise (i 
0.5 
l pixel) 
1.0 
2.0 
Д f 
0.01 
0.02 
0.03 
0.63 
Д Cpp A 
0.04 
0.09 
0.12 
0.21 
A IppA 
0.06 
0.11 
0.09 
0.14 
A cpps 
1.27 
2.36 
2.64 
9.18 
A Ipps 
1.59 
2.67 
3.75 
0.70 
Apixeis (image border) 
0.15 
0.23 
1.08 
4.7 
Table 2: Influence of noise on the intrinsic parameters and distor 
tion parameters on simulated data (camera with /=1000). 
Tab. 3 summarizes the results for Camera 2. We can note that 
the intrinsic parameters (/ and PPA) are very close to real val 
ues. The error on the distortion parameters between the distortion 
function and the estimated model is around 0.5 pixel in the image 
corners for a noise of 1 pixel. 
2.4 Influence of parallax 
To study the influence of the the parallax, we have simulated a 
failure in the position of the camera nodal point compared of the 
center of the pan-tilt system. The magnitude of this defect is 
in the range [—5; 5] centimetres on each of the 3 X, Y and Z 
components. Fig. 5, 6 and 7 represent the variations of /, cppa 
and Ippa when A', Y or Z evolve. These results are just for 
Camera 2. 
0.3 
noise (i 
0.5 
n pixel) 
1.0 
2.0 
Д f 
0.07 
0.11 
0.02 
0.91 
Д Cpp A 
0.02 
0.04 
0.47 
1.80 
A Ip PA 
0.68 
1.14 
1.59 
2.96 
A cpps 
0.04 
0.08 
0.52 
3.24 
A Ipps 
1.39 
2.33 
3.81 
6.98 
Apixeis (image border) 
0.14 
0.24 
0.48 
0.72 
Table 3: Influence of noise on the intrinsic parameters and distor 
tion parameters on simulated data (camera w'ith /=3000). 
Variation of intrinsic parameters (X) 
)5 -0 
)4 -0 
)3 -0 
)2 -d 
И 
0. 
>1 о. 
)2 0. 
>3 0. 
W 0. 
X (m) 
♦ f ■ C PPA L PPA 
Figure 5: f(X), cppa(X), l PPA (X) 
Variation of intrinsic parameters (Y) 
Variation of intrinsic parameters (Z) 
Figure7: f(Z),e.ppA(Z),lppA(Z) 
Д f 
Д Cpp A 
Д IppA 
Camera 1 
min 
-9.19 
-8.97 
-6.57 
max 
7.39 
8.61 
8.10 
Camera 2 
min 
-1.18 
-1.23 
-1.03 
max 
1.18 
1.09 
1.26 
Table 4: Influence of parallax on the intrinsic parameters on sim 
ulated data. 
When we vary the parallax on the 3 axes simultaneously, we get