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×
projective transformation
H
, where
H
is an arbitrary 4
4 matrix. Hence, the pro-
W
{
x
i
}
˜
jective reconstruction given by
(P
1
,P
2
,
)
is equivalent to the one defined by
H
−
1
W
{
x
i
}
˜
(P
1
H,P
2
H,
)
.
It is possible to obtain the camera projection matrices
P
1
and
P
2
from the funda-
mental matrix
F
in a rather straightforward manner. Without loss of generality, the
projection matrix
P
1
may be chosen such that
P
1
=[
|
]
, i.e. the rotation matrix
R
is the identity matrix and the translation vector
t
is zero, such that the world coordi-
nate system
W
corresponds to the coordinate system
C
1
of camera 1. The projection
matrix of the second camera then corresponds to
P
2
=
[˜
I
0
e
2
.
e
2
]
×
F
| ˜
(1.21)
A more general form of
P
2
is
P
2
=
[˜
e
2
,
e
2
v
T
e
2
]
×
F
+ ˜
|
˜
λ
(1.22)
where
v
is an arbitrary 3
×
1 vector and
λ
=
0. Equations (
1.21
) and (
1.22
)show
that the fundamental matrix
F
and the epipole
e
2
, which is uniquely determined by
F
since it corresponds to the eigenvector belonging to the zero eigenvalue of
F
T
,
determine a projective reconstruction of the scene (Hartley and Zisserman,
2003
).
If two corresponding image points are situated exactly on their respective epipo-
lar lines, (
1.19
) is exactly fulfilled, such that the rays described by the image points
S
1
˜
S
2
W
x
which can be determined by triangulation in
a straightforward manner. We will return to this scenario in Sect.
1.5
in the context
of stereo image analysis in standard geometry, where the fundamental matrix
F
is
assumed to be known. The search for point correspondences only takes place along
corresponding epipolar lines, such that the world coordinates of the resulting scene
points are obtained by direct triangulation. If, however, an unrestricted search for
correspondences is performed, (
1.19
) is generally not exactly fulfilled due to noise
in the measured coordinates of the corresponding points, and the rays defined by
them do not intersect. Hartley and Zisserman (
2003
) point out that the projective
scene point
W
x
and
˜
x
intersect in the point
˜
˜
x
in the world coordinate system is obtained from
S
1
x
and
S
2
˜
˜
x
based
˜
on the relations
S
1
P
1
W
x
and
S
2
P
2
W
x
˜
=
˜
x
˜
=
x
. These expressions yield the relation
˜
G
W
x
˜
=
0
.
(1.23)
The cross product
S
1
(P
1
W
˜
˜
0
determines the homogeneous scale factor and
allows us to express the matrix
G
as
x
×
x
)
=
⎡
⎣
⎤
⎦
p
(
3
)T
1
p
(
1
)T
1
u
1
˜
− ˜
p
(
3
)T
1
p
(
2
)T
1
v
1
˜
− ˜
G
=
,
(1.24)
p
(
3
)T
2
p
(
1
)T
2
u
2
˜
− ˜
p
(
3
)T
2
p
(
2
)T
2
v
2
˜
−
˜
p
(j)T
where
S
1
(u
1
,v
1
,
1
)
T
,
S
2
(u
2
,v
2
,
1
)
T
, and
i
corresponds to the
j
th row
of the camera projection matrix
P
i
. Equation (
1.23
) is overdetermined since
W
˜
x
=
x
˜
=
˜
x
only
has three independent components due to its arbitrary projective scale, and generally
˜
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