Graphics Reference
In-Depth Information
5. Compute
H
2
as
1
0 0
H
2
=
0
1 0
RT
(5.42)
x
∗
01
−
1
/
The first matrix in Equation (
5.42
) moves the epipole to infinity along the
x
-axis, while the overall transformation resembles a rigid motion in the
neighborhood of
x
0
,
y
0
)
.
6. Apply the projective transformation
H
2
M
(where
M
was determined in Step
2) to the features in
I
1
and the projective transformation
H
2
to the features
in
I
2
to get a transformed set of feature matches
(
{
(
ˆ
x
1
,
y
1
ˆ
)
,
...
,
(
ˆ
x
n
,
y
n
ˆ
)
}
and
x
1
,
y
1
)
x
n
,
y
n
)
}
respectively.
7. At this point, the two images are rectified, but applying
H
2
M
to
I
1
may result
in an unacceptably distorted image. The next step is to find a horizontal shear
and translation that bring the feature matches as close together as possible.
We compute this transformation by minimizing the function
{
(
ˆ
ˆ
,
...
,
(
ˆ
ˆ
n
x
i
)
2
1
(
x
i
+
ˆ
y
i
+
ˆ
− ˆ
a
b
c
(5.43)
i
=
to obtain values for
a
,
b
, and
c
. This is a simple linear least-squares problem.
8. Compute
H
1
as
abc
010
001
H
1
=
H
2
M
(5.44)
Figure
5.14
illustrates the result of applying Hartley's rectification algorithm to the
real images from Figure
5.12
. We can see that the new epipolar lines are horizontal
and aligned, and that the inevitable warping of the two images is not too severe.
An alternate approach that does not require an initial estimate and factorization of
the fundamental matrix was proposed by Isgrò and Trucco [
213
]. Seitz and Dyer [
434
]
also proposed a rectification method particularly well suited to the view morphing
application discussed in Section
5.8
.
(a)
(b)
Figure 5.14.
Rectifying the two images from Figure
5.12
results in horizontal and aligned
epipolar lines.
