Graphics Reference
In-Depth Information
6.3
Show that theworld-to-camera coordinate transformation in Equation (
6.8
)
can also be represented in terms of the camera center
C
by:
X
c
Y
c
Z
c
X
Y
Z
=
%
−
&
R
C
(6.73)
RC
.
6.4 Now that we have defined the process of image formation, we can verify the
validity of the view interpolation process from Section
5.8
. That is, suppose
that two cameras are given by
Hence, show that
t
=−
f
00
f
C
X
f
000
0
f
00
0010
−
P
=
f
C
Y
001 0
f
=
P
0
0
−
(6.74)
These correspond to two cameras whose centers are on the world plane
Z
0 and whose image planes are parallel to this plane (rectified images
are a special case of this situation).
a)
=
]
is the homogeneous coordinate of an arbitrary scene
point and
x
and
x
are the homogeneous coordinates of its projections
in the resulting images, show that for any fixed value of
s
If
X
=[
X
,
Y
,
Z
,1
∈[
0, 1
]
,
s
x
∼
(
1
−
s
)
x
+
P
s
X
(6.75)
sP
. That is, linearly
interpolating image correspondences produces a physically correct
result corresponding to projecting the scene using a new camera.
where
P
s
is a new camera matrix given by
(
1
−
s
)
P
+
b)
Show that the image plane of
P
s
is also parallel to
Z
=
0, that the camera
sf
. Thus,
is centered at
(
sC
x
,
sC
y
,0
)
, and that it has focal length
(
1
−
s
)
f
+
P
s
is “in between” the two original cameras.
6.5 Show that:
a) The cross-product of two 3D vectors in the same direction is 0.
b) The cross-product of the vectors on the left- and right-hand sides of
Equation (
6.12
) produces three linear equations in the elements of
P
,
two of which are given by Equation (
6.14
).
c) The unused linear equation is linearly dependent on the other two.
6.6 Determine
K
,
R
, and
t
for the camera matrix given by
−
1.2051
−
1.0028
−
1.9474
−
5
P
=
−
1.0056
0.9363
1.1671
40
(6.76)
−
0.1037
0.0583
−
0.9929
−
10
6.7 Determine the 2
implied by one
planar projective transformation (i.e., determine
A
i
in Equation (
6.25
)asa
function of the elements of
H
i
).
×
5 linear system for the elements of
ω
6.8
Show how Equation (
6.25
) can be simplified (that is, the estimation can be
taken over fewer parameters) if either:
a)
α
y
/α
x
is known, or
the aspect ratio
b)
the principal point is known.
