Image Processing Reference
In-Depth Information
(
−−−−−→
O
L
P
L
)
LD
=
M
I
(
−−−−−→
O
L
P
L
)
L
(13.114)
where
M
I
is the matrix encoding the intrinsic parameters of the left camera. Simi-
larly, we obtain
(
−−−−−→
O
R
P
R
)
RD
=
M
I
(
−−−−−→
O
R
P
R
)
R
(13.115)
The epipolar equation (13.112) can then be denoted as
(
−−−−−→
O
R
P
R
)
RD
F
(
−−−−−→
O
L
P
L
)
LD
=0
,
(13.116)
where
F
=(
M
I
)
−T
E
(
M
I
)
−
1
(13.117)
The matrix
F
is often called the
fundamental matrix
, which can be obtained via a set
of correspondences in digital image coordinates [156].
The coordinates of a vector
q
represented in the left analog image coordinates
can be transformed back and forth to left digital image coordinates. Evidently, the
same can be done for the left camera image. Because of this, identical reasoning
can be followed to reach analogous conclusions as those in lemma 13.10, but for the
fundamental matrix. This is given precision in the following lemma.
Lemma 13.11.
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
(13.118)
−
YX
0
Then, two points
P
L
and
P
R
in the left and the right (digital) image coordinates
are images of the same world point
P
if and only if
(
−−−−→
C
R
P
R
)
RD
F
(
−−−−→
C
L
P
L
)
LD
=0
,
with
F
=(
M
I
)
−T
E
(
M
I
)
−
1
.
(13.119)
Furthermore, the homogenized (digital) image coordinates of the left and right
epipoles
,
E
L
,
E
R
, are in the null spaces of the
fundamental matrix
,
F
, and its trans-
pose,
F
T
, respectively:
F
T
(
−−−−→
F
(
C
L
E
L
)
LD
=
0
,
C
R
F
R
)
RD
=
0
(13.120)
The main points of interest with epipoles and the epipolar lines include the fol-
lowing:
