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Fig. 1.2
Definition of epipolar geometry according to Horn (
1986
). The
epipolar lines
of the
image points
I
1
x
and
I
2
x
are drawn as
dotted lines
)into
I
2
q
1
(cf. Fig.
1.2
). The point
I
2
x
in im-
age 2 is located on the line connecting
I
2
c
1
and
I
2
q
1
(drawn as a dotted line in
Fig.
1.2
), which is the 'epipolar line' corresponding to the point
I
1
x
in image 1.
For image 1, an analogous geometrical construction yields the line connecting the
points
I
1
c
2
and
I
1
q
2
(where
I
1
c
2
is the optical centre of camera 2 projected into
image 1) as the epipolar line corresponding to the point
I
2
x
in image 2. Alterna-
tively, the epipolar lines can be obtained by determining the intersection lines be-
tween the image planes and the 'epipolar plane' defined by the scene point
C
1
x
and
the optical centres
C
1
c
1
and
C
2
c
2
(cf. Fig.
1.2
). From the fact that each epipolar
line in image 1 contains the image
I
1
c
2
of the optical centre of camera 2 it fol-
lows that all epipolar lines intersect in the point
I
1
c
2
, and analogously for image 2.
Hence, the points
point on the ray at infinity (
s
→∞
I
1
c
2
=
I
2
c
1
=
e
2
are termed epipoles, and the restriction
on the image positions of corresponding image points is termed the epipolar con-
straint.
Horn (
1986
) shows that as long as the extrinsic relative camera orientation given
by the rotation matrix
R
and the translation vector
t
are known, it is straightforward
to compute the three-dimensional position of a scene point
e
1
and
W
x
with image coor-
I
1
x
v
1
)
T
I
2
x
v
2
)
T
, expressed as
C
1
x
and
C
2
x
in the two
dinates
=
(
u
1
,
ˆ
ˆ
and
=
(
u
2
,
ˆ
ˆ
camera coordinate systems. Inserting (
1.8
)into(
1.7
) yields
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