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and then rescale each column of
M
to have unit norm. This process can be iterated
until the measurement matrix stops changing significantly.
To start the process, we need reasonable initial estimates of the projective depths.
One possibility is simply to initialize
factorization algorithm to obtain a candidate collection of
P
's and
X
's, and compute
the homogeneous reprojections
λ
ij
=
ˆ
λ
ij
is obtained as the third
x
ij
. A new estimate of
ˆ
element of
x
ij
. We then iterate the factorization algorithmuntil the reprojection error
stops changing significantly.
A key problemwith the approach is that in practice, all of the 3Dpoints are unlikely
to be seen in all of the images. In this case, we eliminate cameras and points until
we have a “nucleus” of 3D points that are seen in all of the images from a subset
of the cameras. When the factorization algorithm has converged, we can use the
resectioning algorithm described in Section
6.3.1
to estimate new camera matrices
that see some of the 3D nucleus points based on several of their feature locations.
We also use the triangulation algorithm described in Section
6.4.2
to estimate new
processes are sketched in Figure
6.11
.
Therefore, the overall projective factorization algorithm is:
1. Determine a subset of scene points and cameras so that the measurement
matrix is completely filled.
2. Normalize the set of points in each image by an appropriate translation and
scale.
3.
1.
4. Form the measurement matrix
M
.
5. Alternate rescaling the rows of
M
to have unit norm and the columns of
M
to
have unit norm until
M
stops changing significantly.
6. Determine the SVD
M
Initialize all
λ
ij
=
UDV
.
=
(a)
(b)
Figure 6.11.
Interpolating projective camera matrices and scene structure using resectioning
and triangulation. (a) If the projective cameras and scene points given by white planes/circles are
known, the shaded camera matrix can be computed by resectioning, since the world coordinates
and image projections are both known. (b) If the projective cameras and scene points given by
white planes/circles are known, the shaded points can be computed by triangulation, since the
camera matrices and image projections are both known.
16
Sturmand Triggs also described amore complicated approach to initializing the projective depths
based on the estimated fundamental matrices between image pairs.
17
Since this is only a projective reconstruction, Hartley's algorithm [
189
] should be used.
