Graphics Reference
In-Depth Information
cameras
observed
feature locations
scene points
Figure 6.10.
Bundle adjustment for projective reconstruction. We want to adjust the “bundles”
of rays emanating fromeach camera to the scene points in order to bring the estimated projections
onto the image plane as close as possible to the observed feature locations. The reprojection error
corresponds to the sum of squared distances between every observation and the intersection of
the corresponding line and image plane.
equations (
6.36
) into a large matrix equation:
λ
11
x
11
λ
12
x
12
···
λ
1
n
x
1
n
P
1
P
2
.
P
m
=
X
1
X
2
λ
21
x
21
λ
22
x
22
···
λ
2
n
x
2
n
···
(6.38)
X
n
.
.
.
.
.
.
λ
m
1
x
m
1
λ
m
2
x
m
2
···
λ
mn
x
mn
Here, the
λ
ij
are the unknown scalar multiples such that the
∼
in Equation (
6.36
)is
an equality:
P
i
X
j
. These are called the
projective depths
, and the matrix of
feature locations on the left-hand side of Equation (
6.38
) is called the
measurement
matrix
.
Since Equation (
6.38
) expresses the 3
m
λ
ij
x
ij
=
×
n
measurement matrix as the product of
a3
m
×
4 matrix containing the cameras
{
P
i
}
anda4
×
n
matrix containing the scene
points
, we can see this large matrix has rank at most four. This suggests a natural
factorization algorithm based on the SVD. That is, given a guess for the projective
depths, we form the measurement matrix on the left-hand side of Equation (
6.38
)
(call it
M
) and determine the SVD
M
{
X
j
}
UDV
.
U
is 3
m
=
×
n
,
V
is
n
×
n
, and
D
is a
n
n
diagonal matrix of singular values, which from this reasoning should ideally
only have four nonzero elements. Therefore, we define
D
4
as the left-hand
n
×
×
4
matrix of
D
. We estimate the 3
m
×
4 matrix of cameras on the right-hand side of
Equation (
6.38
)as
UD
4
and the 4
n
matrix of scene points on the right-hand side of
Equation (
6.38
) as the first four rows of
V
.
As for previous algorithms, normalization of the data prior to applying the algo-
rithm is critical to get good results if the data is noisy. A simple approach is to first
normalize the feature locations in each image in the usual way (i.e., apply a similarity
transformation to each image plane so that the features have zero mean and average
×
distance from the origin of
√
2). Next, we rescale each row of
M
to have unit norm,
