Image Processing Reference
In-Depth Information
T
L
M
L
p
=
0
T
R
M
R
p
=
0
(13.71)
This is equivalent to solving for
p
in the homogeneous equation
A
=
T
L
M
L
T
R
M
R
,
Ap
=
0
,
with
(13.72)
which can be done by minimizing
2
. The solution is, however, in the
world coordinate system, which may not be desirable if there is a series of calibra-
tion parameter matrices of a fixed stereo system that amenate from different world
coordinates. One might then wish to compute the position of
P
in the left camera
coordinates. We discuss this below.
By using Eq. (13.39) for (
−−−→
p
T
A
T
Ap
O
L
P
)
LC
, the 3D position vector of the point
P
in the
left camera coordinates, one can obtain
(
−−−→
O
L
P
)
LC
=
M
E
(
−−−→
O
W
P
)
WH
=
M
E
p
(13.73)
where
R
L
,
t
L
are the rotation and the translation matrices constituting
M
E
, respec-
tively, being the extrinsic matrix of the left camera w.r.t. the world frame. A substi-
tution in the first line of Eq. (13.71) then affords a formulation in the left camera
coordinates:
T
L
M
L
p
=
T
L
M
I
M
E
p
=
T
L
M
I
(
−−−→
(13.74)
O
L
P
)
LC
=
0
where we have used
M
L
=
M
I
M
E
. Defing the unknown
p
as the homogenized
(
−−−→
O
L
P
)
LC
:
O
L
P
)
LCH
=
(
−−−→
p
=(
−−−→
O
L
P
)
LC
:
1
(13.75)
and appending a null column to
M
I
via
M
I
[
I
,
0
], we can thus rewrite Eq. (13.74)
and thereby express the left camera equation as
T
L
M
I
[
I
,
0
]
p
=
0
(13.76)
The vector
p
is, as
p
is, homogenized and 4D except that it is in the left camera
coordinates.
We now turn to the right camera equation in Eq. (13.71) with the purpose
of rewriting the unknown in the left camera coordinates. Remembering from Eq.
(13.39) that
O
W
P
)
WH
=
(
−−−→
,
p
=(
−−−→
O
W
P
)
W
1
M
E
=[
R
L
,
t
L
]
,
and
(13.77)
we can rewrite Eq. (13.73):
(
−−−→
O
L
P
)
LC
=
M
E
p
=
R
L
(
−−−→
O
W
P
)
W
+
t
L
(13.78)
