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According to the definition by Birchfield (
1998
), the transformation from the
coordinate system
I
i
of camera
i
into the sensor coordinate system
S
i
is given by
the matrix
⎡
⎤
α
u
α
u
cot
θ
0
⎣
⎦
,
A
i
=
0
α
v
/
sin
θ
0
(1.11)
0
0
1
with
α
u
,
α
v
,
θ
,
u
0
, and
v
0
as the intrinsic parameters of camera
i
.In(
1.11
), the scale
parameters
α
u
and
α
v
are defined according to
α
u
=−
bk
v
.
The coordinates of an image point in the image coordinate system
I
i
correspond-
ing to a scene point
bk
u
and
α
v
=−
C
i
x
defined in a world coordinate system
W
corresponding to
the coordinate system
C
i
of camera
i
are obtained by
˜
⎡
⎣
⎤
⎦
−
b
000
I
i
C
i
˜
−
˜
x
=
0
b
00
0010
x
,
(1.12)
which may be regarded as the projective variant of (
1.1
).
The complete image formation process can be described in terms of the projective
3
4matrix
P
i
which is composed of the intrinsic and extrinsic camera parameters
according to
×
S
i
P
i
W
W
x
˜
=
x
˜
=
A
i
[
R
i
|
t
i
]
x
,
˜
(1.13)
such that
P
i
=
. For each camera
i
, the linear projective transformation
P
i
describes the image formation process in projective space.
A
i
[
R
i
|
t
i
]
1.2.2.2 The Essential Matrix
At this point it is illustrative to regard the derivation of the epipolar constraint in
the framework of projective geometry. Birchfield (
1998
) describes two cameras re-
garding a scene point
W
x
defined in
the two image coordinate systems. Since these vectors are projective vectors,
W
x
which is projected into the vectors
I
1
x
and
I
2
˜
˜
˜
x
is
˜
1 while
I
1
x
and
I
2
x
are of size 3
of size 4
1. The cameras are assumed to be
pinhole cameras with the same principal distance
b
, and
×
˜
˜
×
x
are given in
normalised coordinates; i.e. the vectors are scaled such that their last (third) coordi-
nates are 1. Hence, their first two coordinates represent the position of the projected
scene point in the image with respect to the principal point, measured in units of
the principal distance
b
, respectively. As a result, the three-dimensional vectors
I
1
I
1
x
I
2
˜
and
˜
x
˜
and
I
2
x
correspond to the Euclidean vectors from the optical centres to the projected
points in the image planes.
Following the derivation by Birchfield (
1998
), the normalised projective vector
˜
I
1
x
W
from the optical centre of camera 1 to the image point of
x
in image 1, the
I
2
x
normalised projective vector
˜
from the optical centre of camera 2 to the image
W
point of
x
in image 2, and the vector
t
connecting the two optical centres are
coplanar. This condition can be written as
I
1
˜
x
T
t
x
=
R
I
2
˜
˜
×
0
(1.14)
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