Graphics Reference
In-Depth Information
Show how the rotation and translation parameters
r
1
,
r
2
,
r
3
,
t
i
in
Equation (
6.20
) corresponding to the camera position for each view of the
stationary plane can be obtained once the camera is calibrated (i.e.,
K
and
H
i
are known). What is a possible problemwith this techniquewhen dealing
with noisy data?
6.9
6.10
In this problem, we'll derive the form of the fundamental matrix given in
Equation (
6.29
). Remember that the equation of the epipolar line in the
second image for a fixed
(
x
,
y
)
in the first image is given by Equation (
5.34
):
%
&
x
y
1
x
y
1
F
=
0
(6.77)
a) To determine a line in an image, we must know two points on it. Verify
that the parameters of a line
connecting two points in homogeneous
]
and
x
,
y
,1
]
are givenby
]
×
[
[
=[
coordinates givenby
x
,
y
,1
x
,
y
,1
x
,
y
,1
]
, where
[
×
denotes the cross product of the vectors.
b) Verify that for two vectors
t
and
v
in
3
,
t
R
×
v
=[
t
]
×
v
, where
0
−
t
3
t
2
[
t
]
×
=
t
3
0
−
t
1
(6.78)
−
t
2
t
1
0
c) Verify that mathematically, the point in homogeneous world coordi-
x
y
1
K
−
1
nates given by
projects to
(
x
,
y
)
in the first image. This
0
point corresponds to traveling infinitely far along the ray from the first
camera center through
(
x
,
y
)
on the image plane.
d)
Show that the homogeneous image coordinates of the projection of this
world point in the second image are given by
K
RK
−
1
]
.
e) Nowwe know two points on the epipolar line: the epipole in the second
image (since all epipolar lines intersect at the epipole) and the point in
(d). Thus, conclude from parts (a) and (b) that
[
x
,
y
,1
K
t
]
×
K
RK
−
1
F
=[
(6.79)
6.11 Consider the similarity transformation of world coordinates given by the
4
×
4 homogeneous matrix
Rt
0
H
=
(6.80)
λ
a) Determine the effects of this similarity transformation on a camera
matrix
P
and a scene point
X
— that is, compute
P
and
X
in the new
coordinate system.