Graphics Reference
In-Depth Information
6.4.2
Recovering Calibrated Cameras
The situation ismuch improved if we already know the camera calibrationmatrices
K
and
K
. For example, if we have access to the cameras prior to mounting them on the
rig, we canuse the plane-based technique described in Section
6.3.2
to independently
estimate these internal parameters.
In this case, we consider the camera matrices
P
=[
P
=[
I
3
×
3
|
0
3
×
1
]
R
|
t
]
(6.32)
K
−
1
x
ˆ
=
These correspond to transforming the coordinates of the two images by
x
K
−
1
x
, which are called
normalized coordinates
. FromEquation (
6.29
), the
fundamental matrix takes the simple form
x
=
ˆ
and
F
=[
t
]
×
R
(6.33)
and in this situation we refer to it as the
essential matrix
. Longuet-Higgins [
300
]
introduced the essential matrix in a classic paper and proved that the rotation and
translation parameters could be uniquely extracted up to a scale ambiguity. In par-
ticular, any essential matrix can be decomposed via singular value decomposition as
F
V
. Then there are exactly four possibilities for the camera matrix
P
in Equation (
6.32
):
=
U
diag
(
1, 1, 0
)
P
∈{[
UWV
|
UWV
|−
UW
V
|
UW
V
|−
u
3
]
,
[
u
3
]
,
[
u
3
]
,
[
u
3
]}
(6.34)
where
u
3
is the last column of
U
and
0
10
100
001
−
=
W
(6.35)
Only one of the four candidates is physically possible (i.e., corresponds to scene
points that are in front of both image planes in the stereo rig). This can be tested by
triangulating
one of the feature matches — that is, projecting lines from the camera
centers through the corresponding image locations and finding their intersection in
three-dimensional space. In practice, the feature matches are noisy, so the two rays
will probably not actually intersect. In this case, we estimate the point of intersection
as the midpoint of the shortest line segment connecting the two rays, as illustrated in
Figure
6.9
.
14
When using noisy featurematches to estimate the relative rotation and translation
between a pair of calibrated cameras within a RANSAC approach, the minimal five-
point algorithm of Nistér [
350
] should be used instead of sequentially estimating and
factoring
F
.
14
This method only makes sense when we have calibrated cameras; the midpoint of the segment
has no meaning in a projective coordinate frame. In the projective case, the triangulation method
described by Hartley and Sturm [
189
] should be applied. This method is based on minimizing the
error between a featurematch and the closest pair of conjugate epipolar lines, and involves finding
the roots of a sixth-degree polynomial. See also Section
7.2
.
