Image Processing Reference
In-Depth Information
−
R
·
(
−−−−→
O
W
O
C
)
W
is constant for all points
P
as soon as
This is because the term
R
,
(
−−−−→
O
W
O
C
)
W
are fixed. The parameters
R
,
(
−−−−→
O
W
O
C
)
W
are called extrinsic param-
eters because they encode the direction and the position of the camera frame relative
to the world frame. Classically, we can make the relationship between the two po-
sition vectors linear by representing (
−−−→
O
W
P
)
W
in its homogeneous coordinates. By
augmenting the matrix
R
, we thus obtain
(
−−−→
O
C
P
)
C
=
M
E
(
−−−→
O
W
P
)
WH
(13.36)
where
4
O
W
P
)
WH
=
(
−−−→
R
(
−−−→
O
W
O
C
)
W
]=[
R
,
(
−−−−→
(
−−−→
O
W
P
)
W
1
O
C
O
W
)
C
]
,
M
E
=[
R
,
−
(13.37)
represent the
extrinsic matrix
and the homogeneous coordinates of the point
P
w.r.t.
the world frame, respectively. Here we have used lemma 13.4 to obtain (
−−−−→
O
C
O
W
)
C
=
R
(
−−−→
O
W
O
C
)
W
, which is the relationship between the two representations of the dis-
placement vector between the camera and the world frames. The extrinsic matrix is
accordingly a 3
−
4 matrix obtained by juxtaposing
R
with the column vector of the
interframe displacement, which we define as follows for convenience:
×
=(
−−−−→
t
=(
t
X
,t
Y
,t
Z
)
T
O
C
O
W
)
C
=
R
(
O
W
O
C
)
W
−
(13.38)
We summarize these results as follows:
Lemma 13.6.
Given the camera and the world frames
C
,
W
at the different origins
O
C
,O
W
,
and that they are related to each other as
W
R
T
,
then the coordinates
of a point in one frame can be transformed linearly to those in the other as
(
−−−→
C
=
O
C
P
)
C
=
M
E
(
−−−→
O
W
P
)
WH
(13.39)
where
O
W
P
)
WH
=
(
−−−→
(
−−−→
O
W
P
)
W
1
M
E
=[
R
,
t
]
,
(13.40)
with
t
=(
−−−−→
O
C
O
W
)
C
(13.41)
4
Note that we construct matrices from other matrices by use of their symbols and brackets
[
·, ·
], and then multiply (!) these following the rules of block matrix operations in linear
algebra. A summary is given in the Appendix, Sect. 13.8.
