Graphics Reference
In-Depth Information
25
system, and their root-mean-square distance to the origin obtains the value
√
2. This
transformation is performed according to
⎛
⎞
⎛
⎞
u
(j)
i
u
(j)
i
v
(j)
i
1
ˇ
⎝
⎠
=
⎝
⎠
v
(j)
i
1
T
i
,
(1.54)
ˇ
where the transformation matrices
T
i
are given by
⎡
⎤
u
(j)
i
s
i
0
−
s
i
j
⎣
⎦
v
(j)
i
T
i
=
0
s
i
−
s
i
j
00
1
√
2
with
s
i
=
.
(1.55)
(u
(j)
i
u
(j)
i
(v
(j)
i
v
(j)
i
−
j
)
2
+
−
j
)
2
j
F
is then obtained based on (
1.52
), where
A normalised fundamental matrix
the image points
(u
(j)
i
,v
(j)
i
,
1
)
T
are replaced by the normalised image points
u
(j)
i
v
(j)
i
,
1
)
T
, followed by enforcing the singularity constraint on
F
using the
(
ˇ
,
ˇ
F
according to
F
SVD-based procedure described above. Denormalisation of
=
T
2
ˇ
FT
1
then yields the fundamental matrix
F
for the original image points.
The linear methods to determine the fundamental matrix
F
all rely on error mea-
sures which are purely algebraic rather than physically motivated. In contrast, the
'gold standard method' described by Hartley and Zisserman (
2003
) yields a fun-
damental matrix which is optimal in terms of the Euclidean distance in the im-
age plane between the measured point correspondences
S
1
S
2
˜
˜
x
i
and
x
i
and the es-
x
(e)
i
x
(e)
i
S
1
S
2
timated point correspondences
˜
and
˜
, which exactly satisfy the relation
x
(e)T
i
x
(e)
i
S
2
F
S
1
˜
˜
=
0. It is necessary to minimise the error term
d
2
S
1
x
(e
i
d
2
S
2
x
(e
i
.
x
i
,
S
1
x
i
,
S
2
E
G
=
+
(1.56)
i
In (
1.56
), the distance measure
d(
S
x
,
S
x
(e)
)
describes the Euclidean distance in the
˜
˜
x
(e)
, i.e. the reprojection error. To
minimise the error term (
1.56
), Hartley and Zisserman (
2003
) suggest defining the
camera projection matrices as
P
1
=[
S
S
image plane between the image points
x
and
˜
˜
I
|
0
]
(called the canonical form) and
P
2
=
[
and the set of three-dimensional scene points that belong to the measured
point correspondences
S
1
M
|
t
]
x
(e)
i
x
i
and
S
2
x
i
as
W
x
i
. It then follows that
S
1
P
1
W
˜
˜
˜
=
x
i
and
˜
x
(e)
i
S
2
P
2
W
x
i
. The Euclidean error term
E
G
according to (
1.56
) is minimised with
respect to the projection matrix
P
2
, defined by the matrix
M
and the vector
t
, and the
scene points
W
˜
=
˜
˜
x
i
. Due to the special form of camera matrix
P
1
, the matrix
F
follows
x
(e)
i
x
(e)
i
M
. The correspondingly estimated image points
S
1
and
S
2
as
F
=[
t
]
×
˜
˜
exactly
x
(e)T
i
x
(e)
i
satisfy the relation
S
1
F
S
2
0. The Euclidean error term (
1.56
) is minimised
with a nonlinear optimisation algorithm such as the Levenberg-Marquardt method
(Press et al.,
2007
).
=
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