Information Technology Reference
In-Depth Information
H
v
[2]
Fig. 1.6
Multiple-view
geometry for the cameras
c
1
and
c
2
1
v
[1]
1
v
[1
1
H
v
[1]
H
v
[2]
2
v
[1]
1
2
v
[2]
1
v
[2
2
v
[1]
2
H
v
[2]
2
v
[1]
2
v
[2
1
n
1
n
2
c
2
H
R
c
1
From an inspection of Figure 1.6, it is easy to verify that the following equalities
hold true:
H
R
=
H
v
[1]
=
H
v
[2]
2
v
[1]
1
2
v
[2]
1
(1.16)
i.e.
, the rigid motion between
c
1
and
c
2
is equal to the rigid motion between
v
[1]
1
v
[1]
2
v
[2]
1
v
[2]
2
and
, and between
and
. Using (1.16) and (1.14) into (1.15),
we obtain
H
v
[2]
H
R
=
H
v
[2]
H
v
[2]
H
v
[2]
=
H
v
[2]
H
v
[1]
H
v
[2]
H
R
H
v
[2]
=
H
v
[2]
H
−
1
R
=
H
−
1
R
v
[1
2
,
1
v
[1]
1
1
v
[2]
2
1
v
[1]
1
2
v
[2]
1
1
v
[2]
2
2
v
[1]
1
2
v
[1]
2
2
v
[1]
2
2
and (1.13) is thus proved.
Remark 1.2.
Note that Equation 1.13 allows one to estimate the rigid motion
H
R
also when the epipolar geometry between
is not well-defined (
small
baseline
case). In fact the epipolar geometry between the virtual cameras is always
well-defined by construction.
c
1
and
c
2
1.4
Mirror Calibration and Image-based Camera Localization
In this section we address the mirror calibration and image-based camera localiza-
tion problems using the PCS properties presented in Sections 1.2 and 1.3. Proposi-
tion 1.5 in the next section will be instrumental for Proposition 1.8 in Section 1.4.2.
1.4.1
Mirror Calibration
Consider the setup in Figure 1.4 and assume that at least
n
≥
2 points can be directly
observed by
c
at
u
i
,
i
∈{
1
,...,
n
}
. If the same set of points is also reflected by
Search WWH ::
Custom Search