Graphics Reference
In-Depth Information
single plane. In general, the fundamental matrix for an image pair taken by a pair of
5.4.2
Estimating the Fundamental Matrix
Just like a projective transformation, the fundamental matrix can be estimated from
a set of feature matches at locations
{
(
x
1
,
y
1
)
,
...
,
(
x
n
,
y
n
)
}
in the first image plane and
x
1
,
y
1
)
x
n
,
y
n
)
}
{
(
in the second image plane. The simplest estimation algorithm
is very similar to the DLT algorithm described in Section
5.1
. Each correspondence
defines a linear equation in the nine unknowns of
F
via Equation (
5.34
), namely
,
...
,
(
x
i
x
i
x
i
y
i
x
i
y
i
x
i
y
i
y
i
y
i
x
i
y
i
1
f
11
f
12
f
13
f
21
f
22
f
23
f
31
f
32
f
33
]
=
[
][
0
(5.40)
Collecting the linear equations for each point yields an
n
×
9 linear system
Af
0, which can be solved similarly to the method we discussed for a projec-
tive transformation. The basic algorithm, also called the normalized
eight-point
=
1. The input is two sets of features
{
(
x
1
,
y
1
)
,
...
,
(
x
n
,
y
n
)
}
in the first image plane
x
1
,
y
1
)
x
n
,
y
n
)
}
and
in the second image plane. Normalize each set of
feature matches to have zero mean and average distance from the origin
√
2.
This can be accomplished by a pair of similarity transformations, represented
as 3
{
(
,
...
,
(
3 matrices
T
and
T
applied to the homogeneous coordinates of the
points.
2. Construct the
n
×
9 matrix
A
, where each feature match generates a row given
by Equation (
5.40
).
3. Compute the singular value decomposition of
A
,
A
×
UDV
.
D
will be a 9
9
diagonal matrix with positive entries that decrease from upper left to lower
right. Let
f
be the last column of
V
(a 9
=
×
×
1 vector).
3matrix
F
, filling in the elements from left to right in each
4. Reshape
f
into a 3
×
row.
5. Compute the singular value decomposition of
F
,
F
UDV
.
D
will be a 3
3
diagonal matrix with positive entries that decrease from upper left to lower
right. Set the lower right (3,3) entry of
D
equal to zero to create a new diagonal
matrix
D
and replace
F
with
U DV
.
6. Recover the final fundamental matrix estimate as
F
=
×
T
FT
.
=
The new Step 5 is required since the previous steps don't guarantee that the esti-
mated
F
is rank two (a requirement for a fundamental matrix). Step 5 has the effect
of replacing the
F
from Step 4 with the nearest rank-2 matrix.
14
Figure
5.12
illustrates the result of estimating the fundamental matrix (and thus,
the epipolar geometry) for a real image pair using the normalized eight-point algo-
rithm. We can see that the resulting epipolar lines are well estimated — that is,
the original feature matches and other corresponding points clearly lie along the
estimated conjugate epipolar lines.
12
For details on degenerate scene configurations where the fundamental matrix is not uniquely
defined, see Maybank [
317
].
13
So named since at least eight points are required to obtain a unique solution.
14
In the sense of minimizing the Frobenius, or sum-of-squares, norm.
