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.
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