Graphics Reference
In-Depth Information
with
R
and
t
as the rotational and translational parts of the coordinate transformation
from the first into the second camera coordinate system. Now
[
]
×
t
is defined as the
×
[
]
×
y
=
×
×
3
3 matrix for which it is
t
t
y
for an arbitrary 3
1 vector
y
. The matrix
(d,e,f)
T
,itis
[
]
×
=
t
is called the 'cross product matrix' of the vector
t
.For
t
⎡
⎤
0
−
fe
⎣
⎦
.
[
]
×
=
−
d
−
ed
0
f
0
t
(1.15)
Equation (
1.14
) then becomes
I
1
x
T
[
x
=
x
T
x
=
]
×
R
I
2
I
1
E
I
2
t
0
,
(1.16)
with
E
=[
t
]
×
R
(1.17)
as the 'essential matrix' describing the transformation from the coordinate system of
one camera into the coordinate system of the other camera. Equation (
1.16
)shows
that the epipolar constraint can be written as a linear equation in homogeneous co-
ordinates. Birchfield (
1998
) states that
E
provides a complete description of how
corresponding points are geometrically related in a pair of stereo images. Five pa-
rameters need to be known to compute the essential matrix; three correspond to
the rotation angles describing the relative rotation between the cameras, while the
other two denote the direction of translation. It is not possible to recover the abso-
lute magnitude of translation, as increasing the distance between the cameras can be
compensated by increasing the depth of the scene point by the same amount, thus
leaving the coordinates of the image points unchanged. The essential matrix
E
is
of size 3
3 but has rank 2, such that one of its eigenvalues (and therefore also its
determinant) is zero. The other two eigenvalues of
E
are equal (Birchfield,
1998
).
×
1.2.2.3 The Fundamental Matrix
It is now assumed that the image points are not given in normalised coordinates but
in sensor pixel coordinates by the projective 3
1 vectors
S
1
x
and
S
2
x
. According to
Birchfield (
1998
), distortion-free lenses yield a transformation from the normalised
camera coordinate system into the sensor coordinate system as given by (
1.11
),
leading to the linear relations
×
˜
˜
x
S
1
A
1
I
1
x
˜
=
˜
(1.18)
S
2
=
A
2
I
2
x
.
x
˜
˜
The matrices
A
1
and
A
2
contain the pixel size, pixel skew, and pixel coordinates
of the principal point of the cameras, respectively. If lens distortion has to be taken
into account, e.g. according to (
1.3
) and (
1.4
), the corresponding transformations
may become nonlinear. Birchfield (
1998
) shows that (
1.16
) and (
1.18
) yield the
expressions
Search WWH ::
Custom Search