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l
(t))
, which needs to be minimised
with respect to the parameter
t
. Hartley and Zisserman (
2003
) state that this min-
imisation corresponds to the determination of the real-valued zero points of a sixth-
order polynomial function. As the estimated points
l
(t))
can be formulated as
d
2
(
S
1
x
,
S
1
d
2
(
S
2
x
,
S
2
˜
+
˜
x
(e)
exactly fulfil
the epipolar constraint, an exact, triangulation-based solution for the corresponding
projective scene point
W
S
1
x
(e)
S
2
˜
and
˜
x
in the world coordinate system is obtained by inserting the
normalised coordinates
(u
(e)
1
˜
2
,v
(e
2
)
of
S
1
x
(e)
and
S
2
x
(e)
into (
1.24
).
The matrix
G
now has a zero singular value, to which belongs the singular vector
representing the solution for
W
,v
(e)
1
)
and
(u
(e)
x
.
Estimating the fundamental matrix
F
and, accordingly, the projective camera
matrices
P
1
and
P
2
and the projective scene points
˜
W
x
i
from a set of point cor-
respondences between the images can be regarded as the first (projective) stage of
camera calibration. Subsequent calibration stages consist of determining a metric
(Euclidean) scene reconstruction and camera calibration. These issues will be re-
garded further in Sect.
1.4.6
in the context of self-calibration of camera systems.
˜
1.3 The Bundle Adjustment Approach
In the following, the general configuration is assumed:
K
three-dimensional points
W
x
k
in the world appear in
L
images acquired from different viewpoints, and the
corresponding measured image points are denoted by their sensor coordinates
S
i
x
k
,
where
i
=
1
,...,K
(Triggs et al.,
2000
; Hartley and Zisserman,
2003
; Lourakis and Argyros,
2004
).
A nonlinear function
1
,...,L
and
k
=
(
C
W
T,
c
j
}
i
,
W
x
)
is defined such that it yields the mod-
elled image coordinates by transforming the point
W
x
in world coordinates into the
sensor coordinate system of camera
i
using (
1.1
)-(
1.5
) based on the camera pa-
rameters denoted by
C
W
T
and
Q
{
c
j
}
i
and the coordinates of the
K
three-dimensional
points
W
x
k
(Lourakis and Argyros,
2004
; Kuhl et al.,
2006
) (cf. also Sect.
5.1
). For
estimating all or some of these parameters, a framework termed 'bundle adjustment'
has been introduced, corresponding to a minimisation of the reprojection error
{
L
K
I
i
T
−
1
Q
C
W
T,
c
j
}
i
,
W
x
k
−
I
i
T
−
1
S
i
x
k
S
i
S
i
2
,
E
BA
=
{
(1.25)
i
=
1
k
=
1
which denotes the sum of squared Euclidean distances between the modelled and
the measured image point coordinates (Lourakis and Argyros,
2004
, cf. also Triggs
et al.,
2000
). The transformation by
S
i
I
i
T
−
1
in (
1.25
) ensures that the reprojection er-
ror is measured in Cartesian image coordinates. It can be omitted if a film is used for
image acquisition, on which Euclidean distances are measured in a Cartesian coor-
dinate system, or as long as the pixel raster of the digital camera sensor is orthogonal
(
θ
=
90
◦
) and the pixels are quadratic (
α
u
=
α
v
). This special case corresponds to
S
i
I
i
T
in (
1.5
) describing a similarity transform.
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