Graphics Reference
In-Depth Information
This makes sense; we've removed seven degrees of freedom (a similarity transfor-
mation) to fix the first camera, leaving eight degrees of freedom: the unknown five
entries of
K
1
and the 3
1 vector
c
, which is related to the 3D projective distortion
×
K
−
1
=−
c
,
so that
K
1
0
H
=
(6.46)
v
K
1
−
1
Now, if we denote
P
i
=[
A
i
|
a
i
]
(6.47)
K
i
K
i
)
−
1
from Equation (
6.23
), it's straightforward
and recall the definition of
ω
=
(
i
to show (see Problem
6.18
) that
ω
−
1
i
a
i
v
)ω
−
1
a
i
v
)
=
(
A
i
−
(
A
i
−
1
P
i
P
i
ω
−
1
1
−
ω
−
1
1
v
=
(6.48)
v
ω
−
1
1
v
ω
−
1
1
−
v
P
i
QP
i
∼
Here, we have introduced a 4
4 symmetric matrix
Q
that depends only on the
they relate the projective camera matrices (which we know) to the elements of the
projective transformation
H
and the camera calibration matrices via
×
ω
i
.
i
, at this point we can impose constraints on
the form we wish it to take based on our knowledge about the camera calibration
matrices
K
i
. These in turn impose constraints on
Q
through Equation (
6.48
).
For example, let's suppose the principal point is known to be
While we don't know the values of
ω
(
0, 0
)
and the aspect
ratio
α
y
/α
x
is known to be 1 — reasonable assumptions for a good camera. In this
case, only the focal length
f
i
of each camera is unknown, and each
ω
i
has an extremely
simple form:
f
−
2
i
00
0
f
−
2
i
0
001
ω
=
(6.49)
i
So does the matrix
Q
:
q
1
00
q
2
0
q
1
0
q
3
001
q
4
q
2
q
3
q
4
q
5
Q
=
(6.50)
If we denote the rows of
P
i
as
P
i
,
P
i
,
P
i
and expand Equation (
6.48
) in this special
situation, we obtain four linear equations in the five unknowns of
Q
, corresponding
18
It is also related to what is called the
plane at infinity
for the projective reconstruction.
19
Q
is also known as the
absolute dual quadric
[
499
].
