Image Processing Reference
In-Depth Information
the role of the world frame in analogy with Sect. 13.2, and consequently, the inter-
pretation of
t
,
R
remains as listed above. We express the cross-product operation
−−−−→
O
R
O
L
×
q
as a matrix multiplication with a vector
q
, that is:
−−−−→
O
R
O
L
×
q
=
T
·
q
(13.96)
where, given that
−−−−→
O
R
O
L
=(
X, Y, Z
)
T
(13.97)
the matrix
T
is determined by the elements of
−−−−→
O
R
O
L
as
⎛
⎞
0
−
ZY
⎝
⎠
T
=
Z
0
−
X
(13.98)
−
YX
0
The coplanarity equation, (13.95), for points
O
L
PO
R
then yields:
(
−−−→
O
R
P
)
T
TR
(
−−−→
O
L
P
)
L
=
0
(13.99)
Consequently, the coplanarity equation can be expressed as
(
−−−→
O
R
P
)
T
E
(
−−−→
O
L
P
)
L
=0
,
(13.100)
where
E
=
TR
(13.101)
Calling the third components of (
−−−→
O
L
P
)
L
,
−−−→
O
R
P
as
Z
L
,Z
R
, and the focus length of
the two camera frames as
f
L
,f
R
, respectively, we can, from Eq. (13.100) obtain
Z
R
−−−→
Z
L
−−−→
(
f
R
O
R
P
)
T
E
(
f
L
O
L
P
)
L
=0
(13.102)
so that
(
−−−−−→
O
R
P
R
)
T
E
(
−−−−→
O
L
P
L
)
L
=0
(13.103)
Note that both
−−−−−→
O
R
P
R
and (
−−−−→
O
L
P
L
)
L
are now image points expressed in homoge-
neous coordinates in the
right
and the
left image planes
, respectively. In what follows
we will not mark the homogenized position vectors in the two image planes, because
all such vectors are homogenized.
The matrix
E
, often named the
essential matrix
for the stereo system,
9
is not
affected when the point
P
moves in the space. It changes only when the relative
position and gaze of the cameras change by a translation (via
T
), and/or by a rotation
(via
R
), respectively. The equation can be utilized when determining correspondence
9
Note that
E
=
TR
and the stereo extrinsic matrix,
M
E
=[
R
,
t
], discussed in Sect. 13.4
are obtained from the same information.
