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Proof.
Let us start by proving the first part of the statement.
t
c
w
and
(
t
c
w
) lie on the
℘
with normal vector
n
Σ
=[
n
Σ
(1)
n
Σ
(2) 0]
T
plane
. Since the corresponding
epipolar lines (computed from the fundamental matrix
F
[12]
) all intersect at a single
image line
m
, projection of the screw axis, we have that
Σ
in
w
n
Σ
=
R
c
w
(
K
T
m
)
where
K
T
m
is the normal vector to
.Since
n
Σ
⊥
℘(
t
c
w
), it is then sufficient
Σ
in
c
90 deg around the
z
-axis in order to obtain
t
c
w
in (1.21).
To prove the second part of the statement, consider the Equation 1.19. Let
R
c
w
be given (computed, for example, using Proposition 1.7) and
D
[2
w
,
n
2
be estimated
using one of the algorithms in Section 1.4.1. Let
H
v
v
2
also be given (see Section
1.3.1). From (1.19) we have that
R
c
w
t
v
v
2
+
t
c
w
=
S
[2
w
S
[1
w
t
c
w
. Collecting
t
c
w
on the left-
hand side of the equation, we get (
I
to rotate
n
Σ
of
−
S
[2
w
S
[1
w
)
t
c
w
=
R
c
w
t
v
v
2
,thatis
−
−
⎡
⎤
2
n
2
(1)
−
2
n
2
(1)
n
2
(2) 0
⎣
⎦
t
c
w
=
n
2
(2))
R
c
w
t
v
v
2
2
n
2
(1)
n
2
(2)
2(1
−
0
−
0
0
0
from which, after few manipulations, we obtain (1.22). Note that the normalization
of
t
c
w
in (1.20), removes the ambiguity due to the up to scale estimation of
t
v
v
2
.
1.5
Simulation and Experimental Results
In this section we present the results of the numerical simulations and real-data
experiments we performed to elucidate and validate the theory.
1.5.1
Simulations
Simulation experiments have been conducted with the
epipolar geometry toolbox
[14], in order to test the effectiveness of the algorithms presented in the previous
sections. The setup is composed of a pinhole camera with calibration matrix
⎡
⎣
⎤
⎦
951
.
8
0
640
.
66
K
=
0
951
.
8 605
.
11
0
0
1
57]
T
and two planar mirrors with normal vectors
n
1
=[0
.
60
.
55
−
0
.
and
n
2
=
37]
T
, corresponding to an angular displacement
−
.
.
−
.
[
= 60 deg. The
camera observes directly a total of 20 random points. For the sake of simplicity,
we will henceforth suppose that the correspondence matching problem is exact.
In order to solve the correspondence problem in practice, one might first use the
0
32 0
87
0
θ
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