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different orientations. This problem of estimating the internal parameters is often
simply called
camera calibration
, and is discussed in Section
6.3.2
.
5
In both cases, the algorithms require that we have some control over the cameras
and their environment, whichmay be a reasonable assumption in a lab setting or on a
closedmovie set. Section
6.5
addresses algorithms better suited tomatchmoving from
a video sequence inwhichno information is available about the imaged environment.
6.3.1
Resectioning
Let's assume we're given a set of 3D points with known world coordinates
{
(
X
1
,
Y
1
,
Z
1
)
,
...
,
(
X
n
,
Y
n
,
Z
n
)
}
and their corresponding pixel locations in an image
{
(
. As mentioned earlier, these known 3D locations typically arise
from some type of external survey of the environment. The corresponding 2D loca-
tions may be hand-picked, or possibly automatically located (e.g., the points could
correspond to the centers of unique ARTag markers [
140
] affixed to the walls of a set).
We assume that the 3D points don't all lie on the same plane and that six or more
correspondences are available.
The relationship in Equation (
6.11
) between a 3D point
x
1
,
y
1
)
,
...
,
(
x
n
,
y
n
)
}
(
X
i
,
Y
i
,
Z
i
)
, the corre-
sponding 2D pixel
(
x
i
,
y
i
)
, and the twelve elements of the camera matrix P is:
∼
X
i
Y
i
Z
i
1
x
i
y
i
1
P
11
P
12
P
13
P
14
P
21
P
22
P
23
P
24
P
31
P
32
P
33
P
34
(6.12)
That is, the vectors on the left-hand side and right-hand side of Equation (
6.12
)
are scalar multiples of each other; thus, their cross-product is zero. This observation
leads to two linearly independent equations in the elements of
P
(see Problem
6.5
):
]
=
A
i
[
P
11
P
12
P
13
P
14
P
21
P
22
P
23
P
24
P
31
P
32
P
33
P
34
0
(6.13)
where
0000
X
i
Y
i
Z
i
1
−
y
i
X
i
−
y
i
Y
i
−
y
i
Z
i
−
y
i
A
i
=
(6.14)
X
i
Y
i
Z
i
10000
−
x
i
X
i
−
x
i
Y
i
−
x
i
Z
i
−
x
i
Collecting these equations for all the correspondences results in a 2
n
×
12 linear
system
Ap
=
0. Since the camera matrix is homogeneous, we can minimize
Ap
2
subject to the constraint that
1 using the Direct Linear Transform described
in Section
1
. Just as described in that section, we normalize the 2D and 3D sets of
points prior to solving the estimation problem, and afterward transform the resulting
P
matrix to relate the original image and world coordinates. We can use the solution
as the starting point for minimizing a more geometrically meaningful error (such
as the sum of squared reprojection errors in the image plane), but this may not be
necessary.
At this point, we have an estimate of the camera matrix
P
, but often we really
want explicit values for the internal parameters
K
and the external parameters
p
2
=
(
R
,
t
)
.
5
However, we slightly abuse this terminology in this chapter, using “calibration” to refer to the
process of estimating both internal and external parameters.
