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the mirrors
n
1
and
n
2
at
u
[1]
i
and
u
[2]
i
(white dots), then two reflective fundamental
matrices
F
[1]
and
F
[2]
do exist and can be computed from the corresponding pairs
(
u
[1]
i
u
i
) and (
u
[2]
i
u
i
), respectively (cf. Proposition 1.2). From
F
[1]
and
F
[2]
, we can
then determine the epipoles
e
1
and
e
2
as their right null-spaces (black dots). Since
the direction of each epipole
e
j
is always parallel to
n
j
,
j
,
,
∈{
,
}
1
2
, we then obtain
the following:
Proposition 1.5 (Mirror calibration with the epipoles).
The angle
θ
between the
mirrors
n
1
and
n
2
is given by
= arccos
(
K
−
1
e
1
)
T
(
K
−
1
e
2
)
θ
.
K
−
1
e
1
K
−
1
e
2
In the next proposition the epipoles between the virtual cameras are used to solve
the mirror calibration problem (see Figure 1.7(a)). Let
F
[21]
be the fundamental ma-
trix computed from the corresponding points (
u
[1]
i
u
[2
i
) and let
,
γ
be the angle be-
tween the virtual epipoles
e
[12]
and
e
[21]
. It is easy to verify that
= arccos
(
K
−
1
e
[12]
)
T
(
K
−
1
e
[21]
)
.
γ
e
[12]
e
[21]
K
−
1
K
−
1
Proposition 1.6 (Mirror calibration with the virtual epipoles).
The angle
θ
be-
tween the mirrors
n
1
and
n
2
is given by
=
π
−
γ
2
θ
.
Remark 1.3.
It is worth emphasizing here that our calibration notion is different
from that considered in previous works (and notably in [6]). In fact, with “mirror
calibration” we mean the estimation of the
angle
between the mirrors, while in [6]
the authors mean the estimation the
focal length
of the camera and the orientation
of the mirrors
screw axis
.
1.4.2
Image-based Camera Localization
This section deals with the estimation of the rigid motion (
R
c
w
,
t
c
w
) of the camera
c
with respect to a world frame
w
. In the interest of simplicity, we will assume
that the
z
-axis of
w
coincides with the mirrors screw axis and the
x
-axis lies on
mirror
n
1
.
Proposition 1.7 provides a method to estimate the matrix
R
c
w
, while Proposition
1.8 presents two methods for computing the projection of
t
c
w
on the plane defined
by the camera centers. Consider the setup reported in Figures 1.7(a-b). If at least
two corresponding points exist between
,then
the fundamental matrices
F
[1]
and
F
[2]
exist. Let
e
1
and
e
2
be the epipoles having
unitary norm considered in Proposition 1.5,
z
w
(
c
)
c
and
v
1
and between
c
and
v
2
e
1
×
e
2
and
x
w
(
c
)
e
1
×
z
w
(
c
)
.
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