1.1 The Pinhole Model 5
nate system W into the camera coordinate system C
i
. The transformation
C
i
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 =
C
i
W
T
W
x
=R
i
W
x +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
1
r
2
+k
3
r
4
+k
5
r
6
I
x +d
t
, (1.3)
where
I
x =( ˆu, ˆv)
T
and r
2
=ˆu
2
+ˆv
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
d
t
=
2k
2
ˆu ˆv +k
4
(r
2
+2ˆu
2
)
k
2
(r
2
+2ˆv
2
) +2k
4
ˆu ˆv
(1.4)
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, ˆu and ˆv 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
S
x =
S
I
T
I
x
(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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