Graphics Reference
In-Depth Information
7.
Let the the 3
m
4 matrix of cameras on the right-hand side of Equation (
6.38
)
be
UD
4
and the 4
×
n
matrix of points on the right-hand side of Equation (
6.38
)
be the first four rows of
V
, where
D
4
is the left-hand
n
×
×
4 matrix of
D
.
ˆ
8. Compute the reprojection
x
ij
for each camera and point.
9.
If the average reprojection error has converged, stop. Otherwise, let
λ
ij
be the
ˆ
x
ij
and go to Step 4.
10. Un-normalize the camera matrices and world coordinates.
11. Resection and triangulate the non-nucleus cameras and scene points.
third element of
An alternate approach to projective factorization was proposed by Mahamud
et al. [
311
], who noted that the minimization of
m
n
2
1
χ
λ
ij
x
ij
−
P
i
X
j
(6.39)
ij
i
=
1
j
=
is linear in the elements of
P
if the
X
's are known and vice versa. This suggests a
natural algorithm of alternating the estimation of one set of quantities while the
other is fixed, and has the advantage that not all points must be seen in all images.
Along the same lines, Hung and Tang [
210
] proposed to cycle through updating the
cameras, scene points, and inverse projective depths; fixing two of the quantities and
estimating the third is a linear least-squares problem. Both algorithms were shown
to provably converge.
Clearly, minimizing Equation (
6.37
) or Equation (
6.39
) can only result in a recon-
structionof the cameras andworldpoints up to a 3Dprojective transformation, by the
same argument as in Equation (
6.30
). That is, we can replace all the camera matrices
and world points by
P
i
X
j
H
−
1
X
j
=
P
i
H
=
(6.40)
for any 4
4 non-singular matrix
H
, and still obtain the same measurement matrix.
Thismeans that at the end of projective reconstruction, wewill most likely have a very
strange set of cameramatrices and 3Dpoints (e.g., resembling Figure
6.8
d), which are
not immediately useful. Section
6.5.2
addresses the estimation of an
H
that upgrades
the projective reconstruction to a Euclidean one (e.g., resembling Figure
6.8
b).
×
6.5.1.2 Sequential and Hierarchical Updating
The factorization approach doesn't take advantage of the fact that in matchmoving
applications, the images come froma temporal sequence, inwhich successive images
are similar. Projective reconstruction algorithms for long image sequences frequently
use a
chaining
approach, in which the projective cameras are successively estimated
based on pairs or triples of images, as illustrated in Figure
6.12
a.
For example, let's suppose we're given three successive images from a sequence,
with a set of feature matches tracked through the triple. From the first two images,
we can estimate the fundamental matrix
F
12
from feature matches and obtain a
pair of projective cameras
P
1
and
P
2
in the form of Equation (
6.31
), as well as the 3D
projective points
X
j
that correspond to the features. In the third image, we can use the
estimated
X
j
and their projections
x
3
j
to solve the resectioning problem as described
in Section
6.3.1
, obtaining an estimate for
P
3
. It's easy to iterate this process for more
