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nate system
W
into the camera coordinate system
C
i
. The transformation
C
W
T
is
composed of a rotational part
R
i
, corresponding to an orthonormal matrix of size
3
3 determined by three independent parameters, e.g. the Euler rotation angles
(Craig,
1989
), and a translation vector
t
i
denoting the offset between the coordinate
systems. This decomposition yields
C
i
x
×
W
T
W
x
=
R
i
W
x
C
i
=
+
t
i
.
(1.2)
Furthermore, the image formation process is determined by the intrinsic parameters
{
c
j
}
i
of each camera
i
, some of which are lens-specific while others are sensor-
specific. For a camera described by the pinhole model and equipped with a digital
sensor, these parameters comprise the principal distance
b
, the effective number of
pixels per unit length
k
u
and
k
v
along the horizontal and the vertical image axes,
respectively, the pixel skew angle
θ
, and the coordinates
u
0
and
v
0
of the prin-
cipal point in the image plane (Birchfield,
1998
). For most modern camera sen-
sors, the skew angle amounts to
θ
90
◦
=
and the pixels are of quadratic shape with
k
u
=
k
v
.
For a real lens system, however, the observed image coordinates of scene points
may deviate from those given by (
1.1
) due to the effect of lens distortion. In this
work we employ the lens distortion model by Brown (
1966
,
1971
) which has been
extended by Heikkilä and Silvén (
1997
) and by Bouguet (
1999
). According to
Heikkilä and Silvén (
1997
), the distorted coordinates
I
x
d
of a point in the image
plane are obtained from the undistorted coordinates
I
x
by
I
x
d
=
1
k
5
r
6
I
x
k
1
r
2
k
3
r
4
+
+
+
+
d
t
,
(1.3)
where
I
x
v
2
. If radial distortion is present, straight lines in
the object space crossing the optical axis still appear straight in the image, but the
observed distance of a point in the image from the principal point deviates from the
distance expected according to (
1.1
). The vector
v)
T
and
r
2
u
2
=
(
u,
ˆ
ˆ
=ˆ
+ˆ
2
k
2
ˆ
k
4
(r
2
u
2
)
u
v
ˆ
+
+
2
ˆ
d
t
=
(1.4)
k
2
(r
2
v
2
)
+
2
ˆ
+
2
k
4
ˆ
u
v
ˆ
is termed tangential distortion. The occurrence of tangential distortion implies that
straight lines in the object space crossing the optical axis appear bent in some direc-
tions in the image.
When a film is used as an imaging sensor,
v
directly denote metric dis-
tances on the film with respect to the principal point, which has to be determined by
an appropriate calibration procedure (cf. Sect.
1.4
). When a digital camera sensor is
used, the transformation
u
and
ˆ
ˆ
I
T
I
x
S
x
S
=
(1.5)
from the image coordinate system into the sensor coordinate system is defined in the
general case by an affine transformation
S
I
T
(as long as the sensor has no 'exotic'
architecture such as a hexagonal pixel raster, where the transformation would be
still more complex). The corresponding coordinates
S
x
(u, v)
T
=
are measured in
pixels.
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