261 
and replace in (10) the arguments of modeled distor 
tions by u\v\ we obtain the relationship between undi 
storted unknown image coordinates (u,v) and the dis 
torted, but known, image coordinates (u’,v’) : 
nates (r i? Cj), computed by features extraction algo 
rithm, and the corresponding coordinates 
C\{m,d)] as derived by 3D coordinates of target points 
(X w i , Y wi , Z wi ) and the parameters (m,d), defining the 
camera model: 
4 = " ,+ 5 „■(“>') 
/ 
y = v’+S V) 
(11) 
The complete camera model is reported in (12) at the 
bottom of the page, where one can note that the full 
expression is linear respect to the considered distortion 
coefficients: this will simplify their estimation. 
3. THE CALIBRATION ALGORITHM 
Denoting with m the set of internal and external undi 
stortion parameters 
(where a,P and y denote the three indipendent parame 
ters of rotation matrix R) and with d the set of distor 
tion parameters 
d = (k li k 2 ,p l ,p 2 ,S l9 S 2 ) 
in order to perform a camera calibration we have to de 
termine the optimal estimate of parameters vectors m 
and d, given a set of visible target points (X w-i , Y Wji , 
Z w>i ) and the set of corresponding pixel locations (r’ i? 
c’O. Due to the noise and the lens distortions that affect 
the image-points positions, the optimal estimate means 
computing the set of calibration parameters (m*,d*) 
which minimize the following merit function: 
X {h ( w > d ) ~ r , f + k d ) ~ c , f } (14) 
/=1 
As the calibration parameters are related by nonlinear 
relationship (12), the minimization of (14) has to be 
performed through a nonlinear optimization algorithm, 
which will converge provided that a good initial guess 
of the parameters themselves is available. In order to 
meet these requirements, we approached the calibration 
process by combining the procedures proposed in [3] 
and [5], as previously mentioned in section 1. 
The main structure is based on two-steps method like 
in [5], which can be summarized as follows: 
1) First consider a distortion free camera model(c/=0). 
To this purpose only image central points are used 
because they are less affected by lens distortions; 
2) Compute vector m, which minimizes F(Q, co,m,d) 
with d fixed: 
minF(Q,co,w,£/) (15) 
m 
3) Then compute vector d, which minimizes F(Q, co, 
m, d) with m fixed as current estimate: 
minF(Q,co,77?, d) (16) 
d 
4) Go back to step 2), using as fixed value of d the 
solution of minimization in previous step 3). The 
loop is performed up to certain number of itera 
tions, and the procedure terminates. 
F(Q,co, m*,d*) = min F(Q,co, m, d) (13) 
m,d 
where Q and co represent the two sets of target and 
corresponding image points respectively. 
In our case, we considered as objective function F the 
sum of squared discrepancy between the image coordi 
Referring to this method, we have introduced following 
modifications: 
r '= r o + fu 
C '~ C 0 + fy 
r U X * +r n Y yy+ r U Z » +T X 
V r 3i X w +r 32 Y w +r 33 Z w +T :yl 
( r 2x X w + r 22 Y w +r 22 Z w +T y '\ 
\r 2] X w + r 32 Y w +r 33 Z„+T : 
~ f v \k\U'(u' 2 +V 2 ) + k 2 u'(iï 2 +v a ) 2 +2p 2 u'v' 
+/?,(3w' 2 +v ,2 ) + s 1 (y 2 +v' 2 )] 
-/v[v( w ' 2+v ' 2 ) + k 2 v'(u' 2 +v' 2 ) 2 +2p x u'v' 
+ p 2 (u' 2 +3v' 2 ) + s 2 (u' 2 +v' 2 )] 
(12)