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Substituting (1.6) in (1.3), we obtain Equation 1.1.
Note that
S
[π]
O(3),det(
S
[π]
)=
1andthat(
D
[π]
)
−
1
=
D
[π]
.
∈
−
1.2.2
The Virtual Camera and the Projection Equivalence
Proposition 1.1 of Section 1.2.1 shows how the 3D point
X
is mirrored by
Π
at
u
onto the image plane of the camera
. Analogously to the concept of virtual point,
we can introduce the geometrically intuitive notion of
virtual camera
c
(dashed in
Figure 1.2(b)), whose reference frame is simply reflected with respect to
v
c
.The
proof of the next proposition follows directly from the observation that
X
v
=
X
[π]
(1.7)
where
X
v
is the point
X
in
.
Proposition 1.2 (Projection equivalence).
Let
u
be the perspective projection in
v
c
of a 3D point
X
after its reflection by
Π
. Then,
u
=
u
[π]
,
(1.8)
being
u
[π]
the perspective projection of
X
onto the image plane of the virtual
camera
.
Proposition 1.2 states that the perspective projection
u
of
X
[π]
coincides with the
perspective projection
u
[π]
of
X
v
. In other words, the camera projections of the re-
flected points correspond to the virtual camera projections of the real points.
v
1.2.3
Reflective Epipolar Geometry
In this section we study the imaging geometry relating cameras
.Note
that this is different from [5, section 3.1], where the epipolar geometry between the
virtual cameras has been investigated.
c
and
v
Proposition 1.3 (Reflective epipolar constraint).
Let us consider the setup in Fig-
ure 1.3 and let d
π
and
n
π
be the distance and the normal of the mirror
Π
measured
3
be the homogeneous representation of the
projection of a 3D point in the image plane of the views
from
c
, respectively. Let
u
r
,
u
π
∈
R
, respectively,
and let the camera calibration matrix
K
be the identity. Then, the
reflective epipolar
constraint
is given by
c
and
v
u
T
π
E
[π]
u
r
= 0
where
E
[π]
= 2
d
π
[
n
π
]
×
is the
reflective essential matrix
,being
[
n
π
]
×
the skew-symmetric matrix associated
with the vector
n
π
.
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