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a rotation matrix. To overcome this problem, the correct rotation matrix should be
computed as
UV
T
, where matrices
U
and
V
are obtained from the singular value
decomposition of the initial estimate of
R
c
w
.
Remark 1.4.
In Section 1.3.2 we have seen that Equation 1.13 relates the pose of
two cameras
with the pose of their virtual counterparts. An analogous
equation, that will be instrumental in proving the second statement in Proposition
1.8, relates the pose of
c
1
and
c
2
c
and
w
with that of the virtual cameras
v
1
and
v
2
.
In fact, as from Figure 1.6, assuming that the camera frame
c
1
is coincident with
w
and
c
2
is coincident with
c
,weget
H
v
v
1
=
H
c
D
[1
w
D
[2
w
H
c
w
.
(1.18)
By inverting and then pre-multiplying (1.18) by
H
c
w
, we obtain the
Sylvester equa-
tion
(with unknown
H
c
w
)
H
c
w
H
v
v
2
=
D
[2
w
D
[1
w
H
c
w
(1.19)
where
D
[1
w
and
D
[2
w
are the reflection transformations about the two mirrors written
in
w
:
⎡
⎣
⎤
⎦
,
1000
0
D
[2
w
=
S
[2]
0
100
0010
0001
−
D
[1
w
=
w
0
T
1
being
n
1
=[010]
T
0]
T
and
n
2
=[
−
sin
θ
cos
θ
in
w
.
2
be the vector containing the
first two components of the projection of
t
c
w
on the plane
(
t
c
w
)
Consider the setup in Figure 1.7(a) and let
℘
∈
R
Γ
. In the next proposition
(
t
c
w
),
i.e.
,
℘(
t
c
w
)
we present two methods for estimating the direction of
℘
.Thefirst
(
t
c
w
)
℘
method uses the fundamental matrix
F
[12]
between the virtual cameras
v
1
,
v
2
and
the second one the Equation 1.19. Let
Σ
be the plane defined by the mirrors screw
axis and the center of
(see Figure 1.7(b)). In the
next proposition
R
z
(γ) denotes a rotation about the
z
-axis by an angle
c
,andlet
n
Σ
be the normal to
Σ
γ
.
Proposition 1.8 (Estimation of
℘(
t
c
w
)
℘
(
t
c
w
)
is given by
).
The direction of
℘
(
t
c
w
)
(
t
c
w
)
℘
t
c
w
℘
=
(1.20)
(
t
c
w
)
t
c
w
where
t
c
w
=[
a
(1)
a
(2)]
T
(1.21)
with
a
=
R
z
(
−
0
, or alternatively,
t
c
w
=
a
(1)(1
−
n
2
(2)) +
n
2
(1)
n
2
(2)
a
(2)
90
deg
)
n
Σ
,
n
Σ
(1)
>
T
a
(2)
n
2
(1)
a
(1)
n
2
(2)
2
n
2
(1)
−
(1.22)
2
n
2
(1)
R
c
w
t
v
v
2
,where
n
2
is the normal vector to the second mirror, written in
with
a
=
−
w
.
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