Graphics Reference
In-Depth Information
b)
Show that the image projections are unchanged by this similarity trans-
formation — that is, that
x
∼
P
X
. (Note: this is just a special case
∼
P
X
of Equation (
6.30
)).
6.12 Assuming a calibrated stereo rig in the canonical form of Equation (
6.32
),
determine the solution to the triangulation problem for a correspondence
between the points
x
,
y
)
(
)
(
in the second
image implied by Figure
6.9
. Hint: the 3D location of the midpoint can
be computed as the solution to a simple linear least-squares problem.
6.13 Determine
P
and
P
in the canonical form of Equation (
6.31
)ifthe
fundamental matrix for an image pair is given by
x
,
y
in the first image and
10
−
6
10
−
5
10
−
2
1.7699
×
1.0889
×
−
1.6599
×
10
−
6
10
−
10
10
−
4
=
F
−
3.3788
×
7.1503
×
8.0432
×
(6.81)
10
−
2
10
−
3
1.4372
×
−
2.9790
×
1
6.14 Determine the transformations that must be applied to the estimated
{
in the Sturm-Triggs projective factorization algorithm to
account for initially normalizing the image feature locations by appropriate
similarity transformations
T
i
.
6.15 Provide a sketch to show that if we only know the fundamental matrices
F
12
,
F
13
,
F
23
relating three views, then knowing the projections
x
1
and
x
2
of a
point
X
in the first and second images entirely determines its projection
x
3
in the third view. This is one of several three-view properties encapsulated
by the
trifocal tensor
.
P
i
}
and
{
X
j
}
6.16
Suppose that we estimate the fundmental matrix
F
12
between images 1
and 2 and then use it to determine
P
1
and
P
2
in canonical form. That is,
P
1
. Then we estimate the fundmental
matrix
F
23
between images 2 and 3 and then use it to determine
P
2
and
P
3
in canonical form. That is,
P
2
=[
I
3
×
3
|
0
3
×
1
]
and
P
2
=[
A
2
|
a
2
]
. It might seem
like we can obtain
P
1
,
P
2
, and
P
3
in a consistent projective frame based on
=[
I
3
×
3
|
0
3
×
1
]
and
P
3
=[
A
3
|
a
3
]
A
2
a
2
P
1
=[
I
3
×
3
|
0
3
×
1
]
P
2
=[
A
2
|
a
2
]
P
3
=[
A
3
|
a
3
]
(6.82)
0
3
×
1
1
and continue onward through the sequence. What's the problem with this
approach? (Again, this is an issue that can be addressed with the trifocal
tensor; see, e.g., [
145
].)
6.17 Verify that combining Equations (
6.42
)-(
6.44
) results in the form of
H
given
by Equation (
6.45
).
6.18 Verify the steps of Equation (
6.48
) that relate
ω
i
to
Q
. Hint: first show that
a
i
v
)
ω
−
1
i
K
i
R
i
)
since
R
i
is a rotation
(
A
i
−
K
1
=
K
i
R
i
. Thennote that
=
(
K
i
R
i
)(
matrix.
6.19 Explicitly determine the first row of the 4
5 linear system for the elements
of
Q
in terms of the elements of
P
i
corresponding to Equation (
6.51
).
×
6.20
Show that the matrix
Q
in Equation (
6.48
) is related to the projective-to-
Euclidean upgrade matrix
H
by
