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is difficult. The chaining process is therefore hierarchical, rather than sequential,
as illustrated in Figure
6.12
b. For example, Pollefeys et al. [
368
] suggested using
Torr et al.'s approach to select the next keyframe in a sequence at the point where
the epipolar geometry model explains the correspondences better than a projective
transformation. Nistér [
349
] described another example of this idea, using triples of
images insteadof pairs. Thebasic approach is to scan through the sequence of images,
choosing triples that balance the number of good featurematches with a qualitymea-
sure of the estimated trifocal tensor. In either case, when the camera moves quickly,
the keyframes are taken close together; when the camera moves slowly, we may be
able to robustly estimate relationships between keyframes hundreds of frames apart.
As in the previous section, after estimating the camera matrices for keyframes, the
other cameramatrices canbe estimatedusing resectioning (Figure
6.11
a), and refined
with an overall pass of bundle adjustment (see Section
6.5.3
).
6.5.2
Euclidean Reconstruction
Just as for a stereo rig, if we know each camera's calibration matrix
K
i
, we can undo
the projective ambiguity in Equation (
6.40
). However, in practice this information
is difficult to obtain for image sequences in which the camera may be constantly
zooming in an unknown way. The process of obtaining a
Euclidean reconstruction
of the cameras and environment using general images (i.e., without a calibration
pattern) is alsoknownas
self-
or
auto-calibration
. The idea is touse knownproperties
of the camera calibration matrices
K
i
(for example, that the skew is 0 and/or the
principal point is at the center of the images) to put constraints on the unknown
entries of
H
, the 4
×
4 matrix in Equation (
6.40
). Our goal is to determine the entries
of
H
so that
P
i
=
P
i
H
(6.41)
=
K
i
[
R
i
|
t
i
]
P
i
where
P
i
is the result of a projective reconstruction, and
is a Euclidean
reconstruction where all the
K
i
's are in the desired form.
Similarly to Equation (
6.32
), let's assume that the first projective camera matrix
has the form
P
1
=[
I
3
×
3
|
0
3
×
1
]
(6.42)
which after applying
H
becomes the Euclidean camera matrix
P
1
=
K
1
[
I
3
×
3
|
0
3
×
1
]
(6.43)
If we split the 4
×
4
H
into submatrices in the form
Ab
c
H
=
(6.44)
d
where
A
is 3
×
3,
b
and
c
are 3
×
1, and
d
is a scalar, then combining Equations (
6.41
)-
(
6.44
) gives
A
=
K
1
and
b
=
0; since
H
is nonsingular, we can choose
d
=
1 so that we
have the simple form
K
1
0
H
=
(6.45)
c
1
