Image Processing Reference
In-Depth Information
because its validity is required if two points in stereo view correspond to the same
3D scene point,
P
.
We show next that the identity Eq. (13.103) can also be interpreted as two equa-
tions of two lines lying in the left and the right (analog) image frames, respectively.
First, we assume that we have a world point
P
that moves along the line repre-
sented by
−−−−→
O
R
P
R
. The right image of such a a point is always
P
R
, to the effect that
(
−−−−−→
O
R
P
R
)
T
remains unchanged, and one obtains
⎛
⎞
x
y
1
0=(
−−−−−→
O
R
P
R
)
T
E
(
−−−−−→
O
L
P
L
)
L
=(
a, b, c
)
⎝
⎠
(13.104)
=
ax
+
by
+
c
=0
where
−−−−−→
O
L
P
L
=(
x, y,
1)
T
has been assumed. Naturally,
a, b,
and
c
are the scalars
that must be obtained via
(
a, b, c
)=(
−−−−−→
O
R
P
R
)
T
E
(13.105)
which is a vector that has a constant direction,
10
as long as
P
is a point along the
infinite line represented by
−−−−→
O
R
P
R
. Accordingly, the direction of (
a, b, c
)
T
remains
unchanged, even for the point
O
R
. In conclusion,
E
L
must lie on the line
ax
+
by
+
c
=0which passes through the point
P
L
. This line is sometimes called the (left)
epipolar line
(of point
P
). An important byproduct of this reasoning is that
E
L
must
lie on all (left) epipolar lines (of all points
P
in the 3D space) as long as the extrinsic
parameters,
E
, encoding the relative position and gaze of the stereo cameras are
unchanged. In other words, all (left) epipolar lines intersect at the (left) epipole,
E
L
.
Second, assuming that the world point
P
now moves along the line that corre-
sponds to
−−−−→
O
L
P
L
, we can find analogous results for the right image frame. Accord-
ingly, it can be shown that the line
a
x
+
b
y
+
c
=0
(13.106)
with
=
E
(
−−−−→
O
L
P
L
)
L
and
−−−−→
O
R
P
R
=(
x
,y
,
1)
T
(
a
,b
,c
)
T
(13.107)
is the right epipolar line with the Right epipole
E
R
, both defined in analogy with
their left counterparts.
The vector (
−−−→
O
L
E
L
)
L
is not in the image plane of the left camera, and it is not
homogenized, i.e., it is an ordinary 3D vector. Yet, it is in the null space of the matrix
E
:
10
The vector
−−−−−→
O
R
P
R
is represented in homogeneous coordinates, i.e., it can be known only
up to a scale factor. In turn this causes (
a, b, c
)
T
to change only up to a scale factor when
−−−−−→
O
R
P
R
is scaled.
