Graphics Reference
In-Depth Information
camera calibration parameters in the
{
K
i
}
given by
2
m
K
i
K
i
P
i
QP
i
F
−
(6.55)
K
i
K
i
P
i
QP
i
F
i
=
1
F
where
F
is the Frobenius norm.
Later, Pollefeys et al. [
368
] suggested pre-normalizing the projective camera
matrices
P
i
using
·
−
1
w
2
w
+
h
0
h
2
P
i
=
P
i
(6.56)
0
w
+
h
0
0
1
prior to self-calibration, where
w
and
h
are the width and height of the images. They
also recommended weighting Equations (
6.51
)-(
6.54
) to reflect reasonable estimates
about the cameras' unknown internal parameters. For example, we are usually much
more certain that
ω
i
(
1, 2
)
=
0 (i.e., the cameras have zero skew) than we are that the
principal points
, so Equation (
6.52
) should have much more
weight in the linear system than Equations (
6.53
)-(
6.54
).
We should keep in mind that self-calibration from an image sequence fails in
certain
critical configurations
, enumeratedby Sturm[
471
,
472
]. Unfortunately, some
of these critical configurations are not unusual when it comes to video shot for visual
effects. For the variable focal length case described earlier, critical configurations
include a camera that translates but does not rotate, as well as a camera that moves
along an elliptical path, pointing straight ahead (e.g., a camera pointing straight out
of a car's windshield as it takes a curve in the road).
As we discuss next, we always follow up a Euclidean reconstruction with the non-
linear process of bundle adjustment over all internal and external parameters, so
it's not always necessary to obtain a highly accurate Euclidean reconstruction at
this stage.
(ω
i
(
1, 3
)
,
ω
i
(
2, 3
))
=
(
0, 0
)
6.5.3
Bundle Adjustment
After upgrading a projective reconstruction to a Euclidean one, we have a good initial
estimate of the cameras and structure. However, this reconstruction is typically not
ready to use in a high-quality application for twomain reasons. First, the various pro-
jective and Euclidean reconstruction algorithms don't always minimize an intuitive,
geometrically sensible quantity — the natural one being the reprojection error. Sec-
ond, the Euclidean upgrade step doesn't usually strictly enforce constraints on the
camera calibrationmatrices (such as zero skew) since it's the result of a least-squares
problem over noisy data.
We address both of these problems in a final step of
bundle adjustment
, the joint
minimization of the sum of reprojection errors
m
n
2
1
χ
ij
d
(
x
ij
,
P
i
X
j
)
(6.57)
i
=
1
j
=
