Image Processing Reference
In-Depth Information
E
(
−−−→
O
L
E
L
)
L
=
TR
(
−−−→
O
L
E
L
)
L
=
−−−−→
−−−→
O
L
E
L
=
0
O
R
O
L
×
(13.108)
because it is parallel to
−−−−→
O
R
O
L
, and multiplication by
T
is a cross-product that is
fully determined by
−−−−→
O
R
O
L
, Eq. (13.98). Yet, (
−−−→
O
L
E
L
)
L
is a homogenized version of
(
−−−−→
O
L
E
L
)
L
, so that the latter is also in the null space of
E
. Similarly, the homogenized
−−−−→
O
R
E
R
must be in the null space of
E
T
because
E
T
−−−−→
O
R
E
R
=
R
T
T
T
−−−−→
O
R
E
R
(13.109)
which, noting that
T
T
=
−
T
, translates to:
E
T
−−−−→
R
T
T
−−−−→
R
T
−−−−→
−−−−→
O
R
E
R
=
0
O
R
E
R
=
O
R
E
R
=
O
R
O
L
−
−
×
(13.110)
Evidently, the 3
×
3 matrices
E
,
E
T
are rank-deficient. We summarize these results
in the following lemma.
Lemma 13.10.
Let a stereo system be characterized by a rotation matrix
R
such
that
R
(
−−−→
O
L
P
)
L
=(
−−−→
O
L
P
)
R
, and by a displacement vector
t
=(
X, Y, Z
)
T
=
(
−−−−→
O
R
O
L
)
R
, such that the matrix
T
is determined by the elements of
t
as
⎛
⎞
0
−
ZY
⎝
⎠
T
=
Z
0
−
X
(13.111)
−
YX
0
Then, two points
P
L
and
P
R
in the left and the right (analog) image coordinates
are images of the same world point
P
if and only if
(
−−−−−→
O
R
P
R
)
R
E
(
−−−−→
O
L
P
L
)
L
=0
,
with
E
=
TR
.
(13.112)
Furthermore, the homogenized (analog) image coordinates of the left and the right
epipoles
,
E
L
,
E
R
, are in the null spaces of the
essential matrix
,
E
, and its transpose,
E
T
, respectively,
E
T
(
−−−−→
E
(
O
L
E
L
)
L
=
0
,
O
R
E
R
)
R
=
0
(13.113)
Now we show that the relative position and gaze of the cameras are not needed
to find the epipoles, and that the latter implicitly encode the extrinsic parameters of
the stereo system. We note first that (
−−−−−→
O
L
P
L
)
L
is represented in the analog image
coordinates of the left frame. It can also be expressed in the digital image coordinates,
i.e., in column and row counts via
