Graphics Reference
In-Depth Information
The equation of the epipolar line in
I
2
for a fixed
(
x
,
y
)
∈
I
1
is easily obtained from
Equation (
5.34
):
%
&
x
y
1
x
y
1
F
=
0
(5.35)
x
y
1
%
&
That is, the coefficients on
x
,
y
, and 1 are given by the three values of
F
.
x
,
y
)
The equations for epipolar lines in
I
1
can be similarly obtained by fixing
(
in
Equation (
5.34
).
The fundamental matrix is only defined up to scale, since any scalar multiple of
F
also satisfies Equation (
5.34
). Furthermore, the 3
3 fundamental matrix only has
rank 2; that is, it has one zero eigenvalue. We can see why this is true froma geometric
argument. As mentioned earlier, all the epipolar lines in
I
1
intersect at the epipole
e
×
x
,
y
)
∈
=
(
x
e
,
y
e
)
. Therefore, for any
(
I
2
,
e
lies on the corresponding epipolar line;
that is,
%
x
y
1
x
e
y
e
1
&
F
=
0
(5.36)
x
,
y
)
holds for every
(
. This means that
=
x
e
y
e
1
F
0
(5.37)
]
is an
eigenvector of
F
with eigenvalue 0. Therefore, the epipoles in both images can easily
be obtained from the fundamental matrix by extracting its eigenvectors.
A useful way of representing
F
is a factorization based on the epipole in the second
image:
11
]
is an eigenvector of
F
with eigenvalue 0. Similarly,
x
e
,
y
e
,1
That is,
[
x
e
,
y
e
,1
[
x
e
y
e
1
F
=
M
(5.38)
×
where
M
is a full-rank 3
×
3 matrix, and we use the notation
0
−
e
3
e
2
[
]
×
=
e
e
3
0
−
e
1
(5.39)
−
e
2
e
1
0
In this form,
F
is clearly rank-2 since the skew-symmetric matrix
]
×
is rank-2.
The fundamental matrix is not defined for an image pair that shares the same cam-
era center; recall our assumption was that the two cameras are in different positions.
In this case, the images are related by a stronger constraint: a projective transfor-
mation that directly specifies each pair of corresponding points, as discussed in
Section
5.1
. The same type of relationship holds when the scene contains only a
[
e
11
See Problem
6.10
for a derivation of this factorization.
