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exploiting the fact that straight lines in the real world remain straight in the im-
age. Radial and tangential distortions can be directly inferred from deviations from
straightness in the image. These first calibration methods based on bundle adjust-
ment, which may additionally determine deviations of the photographic plate from
flatness or distortions caused by expansion or shrinkage of the film material, are
usually termed 'on-the-job calibration' (Clarke and Fryer,
1998
).
1.4.2 The Direct Linear Transform (DLT) Method
In its simplest form, the direct linear transform (DLT) calibration method introduced
by Abdel-Aziz and Karara (
1971
) aims for a determination of the intrinsic and ex-
trinsic camera parameters according to (
1.1
). This goal is achieved by establishing
an appropriate transformation which translates the world coordinates of known con-
trol points in the scene into image coordinates. This section follows the illustrative
presentation of the DLT method by Kwon (
1998
). Accordingly, the DLT method
assumes a camera described by the pinhole model, for which, as outlined in the
introduction given in Sect.
1.1
, it is straightforward to derive the relation
⎛
⎞
⎛
⎞
u
ˆ
ˆ
x
−
x
0
⎝
⎠
=
⎝
⎠
.
v
−
cR
y
−
y
0
(1.26)
b
z
−
z
0
In (
1.26
),
R
denotes the rotation matrix as described in Sect.
1.1
,
v
the metric
pixel coordinates in the image plane relative to the principal point, and
x
,
y
,
z
are
the components of a scene point
W
x
in the world coordinate system. The values
x
0
,
y
0
, and
z
0
can be inferred from the translation vector
t
introduced in Sect.
1.1
, while
c
is a scalar scale factor. This scale factor amounts to
u
and
ˆ
ˆ
b
c
=−
z
0
)
,
(1.27)
r
31
(x
−
x
0
)
+
r
32
(y
−
y
0
)
+
r
33
(z
−
where the coefficients
r
ij
denote the elements of the rotation matrix
R
. Assuming
rectangular sensor pixels without skew, the coordinates of the image point in the
sensor coordinate system, i.e. the pixel coordinates, are given by
u
−
u
0
=
k
u
ˆ
u
and
v
v
, where
u
0
and
v
0
denote the position of the principal point in the sensor
coordinate system. Inserting (
1.27
)into(
1.26
) then yields the relations
−
v
0
=
k
v
ˆ
b
k
u
r
11
(x
−
x
0
)
+
r
12
(y
−
y
0
)
+
r
13
(z
−
z
0
)
u
−
u
0
=−
r
31
(x
−
x
0
)
+
r
32
(y
−
y
0
)
+
r
33
(z
−
z
0
)
(1.28)
b
k
v
r
21
(x
−
x
0
)
+
r
22
(y
−
y
0
)
+
r
23
(z
−
z
0
)
v
−
v
0
=−
r
31
(x
−
x
0
)
+
r
32
(y
−
y
0
)
+
r
33
(z
−
z
0
)
Rearranging (
1.28
) results in expressions for the pixel coordinates
u
and
v
which
only depend on the coordinates
x
,
y
, and
z
of the scene point and 11 constant pa-
rameters that comprise intrinsic and extrinsic camera parameters:
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