Graphics Reference
In-Depth Information
x
I
0
I
1
Epipolar plane
p
0
p
1
c
0
c
1
e
0
e
1
Epipolar lines
Figure 5.3
Epipolar geometry for two images of a scene. A point
p
0
on either image plane lies on a
unique epipolar plane through the two viewpoints
c
0
and
c
1
. The intersection of this plane
with the image planes
I
0
and
I
1
are epipolar lines. Any point in the scene that projects to
p
0
must project to a point
p
1
on the epipolar line in
I
1
.
is smallest. The neighborhood is a rectangular region of width 2
w
x
+
1by2
w
y
+
1
centered at
(
x
,
y
)
in
I
0
and
(
x
+
dx
,
y
+
dy
)
in
I
1
. The minimum is taken over all
(
under the restriction that the search points stay inside image
I
1
.This
neighborhood search assumes that the scale and orientation of the images are
essentially the same. If one image is rotated with respect to the other, the search
also has to include rotations of the neighborhood.
The neighborhood search can be made more efficient by employing
epipolar
geometry
. Normally the film plane of a camera is behind the focal point (i.e.,
the aperture) of the camera. The image can, however, be regarded as lying on a
virtual image plane in front of the focal point. This arrangement is essentially the
same as the virtual camera setup used in computer graphics rendering: the focal
point is the “eye” point, the size and position of the image rectangle controls
the field of view, and the perpendicular to the virtual image plane provides the
“gaze” direction. Figure 5.3 illustrates this for two separate virtual cameras, each
of which is aimed at the point labeled
x
. In this context, the camera focal point is
often called the “viewpoint,” the “eye point,” or the “center of projection;” they
are labeled
c
0
and
c
1
in Figure 5.3. Each point on the ray from the viewpoint
c
0
through
x
projects to the point
p
0
in image
I
0
, but the same points project to a line
in
I
1
from the viewpoint
c
1
of the other camera. This line is an
epipolar line
.The
ray from
c
1
to
x
likewise projects to a single point
p
1
in
I
1
, and to an epipolar line
in
I
0
. The plane containing the point
x
and the focal points
c
0
and
c
1
is known as
dx
,
dy
)
