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)
