In: Paparoditis N., Pierrol-Deseilligny M„ Mallet C., Tournaire O. (Eds), IAPRS, Vol. XXXVIII, Part 3A - Saint-Mandé, France, September 1-3. 2010 
be achieved by minimising eq. 3: 
2 EXPERIMENTS WITH SYNTHETIC DATA 
argmin \^2{9i(c,l)-gj(c,l)) 2 \ (3) 
Ri,p-K NlP ,f,PPA Vc=0 ' 
where: 
• N is the number of couples of homologous points 
• fi (c, l) (resp. fj (c, /)) is the function defined in eq 1 ap 
plied to image i {resp. j). 
Of course, if there is some distortion, we must also find the pa 
rameters of the distortion function 
2.1 Simulation protocol 
To validate our calibration process regardless of possible mount 
ing errors on the acquisition system, we worked with synthetic 
data. This provides a ground truth to evaluate the calibration pa 
rameters that we have computed with our minimisation scheme. 
We have thus simulated a 3000 x 2000 pixels camera 1 with 2 dif 
ferent focal lengths: 
• a 1000 pixels short focal length (90° per 112°). 
• a 3000 pixels long focal length (36° per 53°). 
A points dataset is then created by regularly sampling points on 
a 10 meters radius sphere (see Fig. 3). Those sampled points are 
then back-projected in all the images to simulate a real acquisi 
tion (d,li). 
The camera is positioned at the center of the sphere to simulate 
d(cpps,lpps),a,b,c(.Cbilb) (Cci ¿c) 
(4) 
The cost function is minimized with a least squares optimization. 
Indeed, we are in a favourable case, since we can eliminate out 
liers using our initial solution provided by the pan-tilt system. 
As outliers are the main drawback of this kind of optimization, 
this choice is suitable for our application. Furthermore, as the 
calibration process does not need to be performed often, compu 
tation time is a minor concern. 
For example, with 3 images and some homologous points be 
tween images {1,2}, {2,3} and {1,3}, the system is defined by 
the matrix A with the unknown X in eq 5. 
h = gi (c, l) - gj(c,l) 
dh 
dh 
dh 
df 
df 
df 
dh 
dh 
dh 
dPP A 
dPPA 
dPPA 
dh 
dh 
dh 
dPPS 
dPPS 
dPPS 
dh 
dh 
dh 
da 
da 
da 
dh 
dh 
dh 
db 
db 
db 
dh 
dh 
dh 
dc 
dc 
dc 
dh 
0 
dh 
dR\ ? p 
9R\ iP 
dh 
dh 
0 
dR2.p 
dR2,p 
0 
dh 
dh 
dR3,p 
dR3.p 
\ 
X = 
df 
dPPA 
dPPS 
da 
db 
dc 
dJR\ ( p 
dR.2, P 
V dR.3, P / 
/ 
(5) 
The cost function defined by eq. 3 leads to solve the system of 
eq. 6. 
A n X n = 0 (6) 
In eq. 6, n denotes the iteration index. Solving this system is thus 
done iteratively using a least squares approach. For the first it 
eration (n = 0), we use a rough estimate of the parameters: / 
can take any arbitrary value, PPS and PPA are initialized at the 
center of the image, distortion coefficients are null and rotations 
are given by the pan-tilt system. The convergence of the min 
imisation process is detected when there are no changes of the 
parameters between two consecutive iterations. 
Figure 3: Sample points on sphere 
images. We used a 50% overlap in each dimension. It leads to an 
angle of 61° between horizontal images and 45° between vertical 
images (see Fig. 4). 
Figure 4: Illustration of the simulation protocol with nine images 
(in green) and the camera (in white). 
In all experiments presented in this section, we have added some 
noise following a normal distribution. The noise is centred on 
the true measure and its standard deviation is chosen in the set 
{0.3; 0.5; 1.0; 2.0} pixels. 
2.2 Intrinsic parameters 
We have first examined the influence of noise on the estimation 
of the focal length and of the PPA. The unknowns to be estimated 
are given by eq.7: 
[ R\, p ,... ,R n , P , f,cppA,lppA ] (7) 
1 {cppaJppa) = (1470,980). (cpp S ,lpPS) = (1530,1020) 
and a = 10~ 8 , b = 10~ 15 , c = 10~ 21