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A
−
1
2
x
T
t
x
=
S
2
RA
−
1
1
S
1
˜
×
˜
0
x
T
A
−
2
t
x
=
S
2
RA
−
1
1
S
1
˜
×
˜
0
(1.19)
S
2
x
T
F
S
1
˜
˜
x
=
0
,
where
A
−
T
2
EA
−
1
1
F
=
(1.20)
is termed the 'fundamental matrix' and provides a representation of both the intrin-
sic and the extrinsic parameters of the two cameras. The 3
3matrix
F
is always
of rank 2 (Hartley and Zisserman,
2003
); i.e. one of its eigenvalues is always zero.
Equation (
1.19
) is valid for all corresponding image points
×
S
1
S
2
x
and
˜
x
in the im-
˜
ages.
According to Hartley and Zisserman (
2003
), the fundamental matrix
F
relates
a point in one stereo image to the line of all points in the other stereo image that
may correspond to that point according to the epipolar constraint. In a projective
plane, a line
l
is defined such that for all points
l
x
T
0is
fulfilled. At the same time, this relation indicates that in a projective plane, points
and lines have the same representation and are thus dual with respect to each other.
Specifically, the epipolar line
x
on the line the relation
˜
˜
=
l
in image 2 which corresponds to a point
S
2
S
1
˜
x
in
l
image 1 is given by
S
2
F
S
1
˜
x
. Equation (
1.19
) immediately shows that this relation
must be fulfilled since all points
S
2
=
˜
x
in image 2 which may correspond to the point
l
. Accordingly, the line
l
S
1
S
2
S
1
F
TS
2
x
in image 1 are located on the line
˜
=
x
in
˜
S
1
image 1 is the epipolar line corresponding to the point
x
in image 2 (Birchfield,
˜
1998
; Hartley and Zisserman,
2003
).
Hartley and Zisserman (
2003
) point out that for an arbitrary point
S
1
˜
x
in image 1
l
e
2
in image 2 is a point on the epipolar line
S
2
˜
˜
=
except the epipole
e
1
, the epipole
F
S
1
e
2
are defined in the sensor coordinate system of camera 1
and camera 2, respectively, such that
x
. The epipoles
˜
e
1
and
˜
˜
e
2
(F
S
1
e
2
F)
S
1
S
1
˜
˜
x
)
=
(
˜
˜
x
=
0 for all points
x
˜
e
2
F
on the epipolar line, which implies
˜
=
0. Accordingly,
e
2
is the eigenvector
˜
belonging to the zero eigenvalue of
F
T
e
1
in
image 1 is given by the eigenvector belonging to the zero eigenvalue of
F
according
to
F
(i.e. its 'left null-vector'). The epipole
˜
e
1
=
˜
0 (i.e. the 'right null-vector' of
F
).
1.2.2.4 Projective Reconstruction of the Scene
This section follows the presentation by Hartley and Zisserman (
2003
). In the frame-
work of projective geometry, image formation by the pinhole model is defined by
the projection matrix
P
of size 3
4 as defined in (
1.13
). A projective scene re-
construction by two cameras is defined by
(P
1
,P
2
,
×
W
{
x
i
}
˜
)
, where
P
1
and
P
2
de-
note the projection matrix of camera 1 and 2, respectively, and
W
are the scene
points reconstructed from a set of point correspondences. Hartley and Zisserman
(
2003
) show that a projective scene reconstruction is always ambiguous up to a
{
x
i
}
˜
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