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Fig. 1.4
The camera
c
and the virtual cameras
v
1
and
v
2
v
1
(
R
,
t
)
v
2
u
[1]
i
u
[2]
i
n
1
θ
n
2
X
i
e
2
e
1
c
u
[2]
i
u
[1]
i
u
i
Rt
0
T
=
D
[1]
D
[2]
H
v
v
1
(1.12)
1
where (
R
,
t
) is the rigid body motion between
v
1
and
v
2
. Owing to Proposition
1.2, the points
u
[1]
i
and
u
[2]
i
in
c
are corresponding in both the virtual cameras
v
1
and
v
2
, (see Figure 1.4). This implies the existence of the epipolar geometry relat-
u
[1
i
= 0. The fundamental matrix
F
can be estimated
from a set of (at least) 8 image points and the epipoles
e
1
and
e
2
are obtained as the
right and left null-spaces of
F
[9]. Moreover, given the camera calibration matrix
K
,
from
F
we can compute the essential matrix
E
u
[2]
T
i
ing
v
1
and
v
2
,
i.e.
,
F
[
t
]
×
R
.Once
E
is known, a decom-
position [9] can be carried out to finally compute the matrix
R
and the vector
t
(up
to a scale factor).
Figure 1.5 shows the epipolar lines (white) relative to pairs of corresponding
points in the real and virtual views, on a sample image. Figure 1.5(a) reports the
epipolar lines between the virtual cameras
: the baseline lies far above
the edge of the image. As shown in [6], all corresponding epipolar lines intersect
at the image projection
m
of the mirrors screw axis (
i.e.
, the 3D line of intersection
between the mirrors). Figures 1.5(b-c) show the epipolar lines between
v
1
and
v
2
v
1
and
c
,
and between
v
2
and
c
, respectively.
1.3.2
Multiple-view Geometry
In this section we address the case of a moving camera that observes a set of
3D points
X
i
reflected by two mirrors, from two views
c
1
and
c
2
(see Figure
H
c
c
1
1.6). Let
H
R
be the homogeneous transformation matrix relating
c
1
and
,and
H
v
[2]
,
H
v
[2]
,
H
v
[1]
,
H
v
[2]
c
2
2
v
[2]
1
2
v
[1]
2
2
v
[1]
1
1
v
[1]
1
(with a slight abuse of notation since differently
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