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v
1
e
[12]
n
Γ
v
1
Γ
e
[21]
v
2
v
2
Σ
n
1
n
Σ
℘(
t
c
w
)
℘(
t
c
w
)
w
t
c
w
n
2
c
(
R
c
w
,
t
c
w
)
n
Σ
c
(a)
(b)
℘(
t
c
w
) is the projection of
t
c
w
on the plane
℘(
t
c
w
) using
Fig. 1.7
(a)
Γ
; and (b) computation of
the epipolar lines and
Σ
Proposition 1.7 ( R
c
w
estimation).
For every rigid-motion
(
R
c
w
,
t
c
w
)
, the following
equation holds true,
⎡
⎤
⎦
n
Γ
1
010
−
00
−
(
c
)
e
1
a
(
c
)
−
1
⎣
R
c
w
=
.
(1.17)
10 0
Proof.
Due to the assumption that the
x
-axis of
w
lies on
n
1
, for every pose of
c
the world
y
-axis expressed in the camera frame
c
corresponds to
e
1
. Consequently,
for every pose of
c
,
e
1
and
e
2
lie on the same plane
Γ
defined by the three camera
centers having normal vector
n
Γ
(
w
)
=[001]
T
in
w
(see Figure 1.7(a)), then
z
w
(
c
)
×
e
1
e
2
where
z
w
(
c
)
is the
z
-axis of the world reference frame expressed in the camera
frame
. The world frame
x
-axis can be easily obtained as the cross product of
e
1
and
z
w
(
c
)
:
c
x
w
(
c
)
e
1
×
z
w
(
c
)
.
Finally, we have that
⎡
⎤
−
10 0
010
00
⎣
⎦
=
R
c
w
[
x
w
(
c
)
e
1
z
w
(
c
)
]
−
1
from which (1.17) follows.
Note that, in absence of noise on the image points, Equation 1.17 provides us with
the
exact
R
c
w
. In the case of noisy data, the estimated
R
c
w
will not be, in general,
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