Graphics Reference
In-Depth Information
A set of
n
point correspondences then yields a system of equations for the matrix
elements of
F
according to
⎡
⎤
u
(
1
)
1
u
(
1
)
2
u
(
1
)
2
v
(
1
)
1
u
(
1
)
2
u
(
1
)
1
v
(
1
)
2
v
(
1
)
1
v
(
1
)
2
v
(
1
)
2
u
(
1
)
1
v
(
1
)
1
1
⎣
⎦
.
.
.
.
.
.
.
.
.
G
f
=
f
u
(n)
1
u
(n)
2
u
(n)
2
v
(n)
1
u
(n)
2
u
(n)
1
v
(n)
2
v
(n)
1
v
(n)
2
v
(n)
2
u
(n)
1
v
(n)
1
1
=
0
.
(1.52)
The scale factor of the matrix
F
remains undetermined by (
1.52
). A unique solution
(of unknown scale) is directly obtained if the coefficient matrix
G
is of rank 8.
However, if is it assumed that the established point correspondences are not exact
due to measurement noise, the rank of the coefficient matrix
G
is 9 even if only eight
point correspondences are taken into account, and the accuracy of the solution for
F
generally increases if still more point correspondences are regarded. In this case, the
least-squares solution for
f
is given by the singular vector of
G
which corresponds
to its smallest singular value, for which
=
1.
Hartley and Zisserman (
2003
) point out that a problem with this approach is the
fact that the fundamental matrix obtained from (
1.52
) is generally not of rank 2 due
to measurement noise, while the epipoles of the image pair are given by the left and
right null-vectors of
F
, i.e. the eigenvectors belonging to the zero eigenvalues of
F
T
and
F
, respectively. These do not exist if the rank of
F
is higher than 2. The
constraint that
F
is of rank 2 can be taken into account by replacing the solution
obtained based on the singular value decomposition (SVD) of the coefficient matrix
G
as defined in (
1.52
) by the matrix
G
f
becomes minimal with
f
F
which minimises
−
F
F
F
F
with det
=
0.
The term
A
F
is the Frobenius norm of a matrix
A
with elements
a
ij
given by
m
n
min
(m,n)
trace
A
∗
A
=
2
2
σ
i
A
F
=
1
|
a
ij
|
=
(1.53)
i
=
1
j
=
i
=
1
with
A
∗
as the conjugate transpose of
A
and
σ
i
as its singular values. If it is assumed
that
F
=
UDV
T
is the SVD of
F
with
D
as a diagonal matrix
D
=
diag
(r,s,t)
,
F
which minimises the Frobenius norm
−
F
where
r
≥
s
≥
t
, the matrix
F
F
is
F
U
diag
(r, s,
0
)V
T
.
At this point, Hartley and Zisserman (
2003
) propose an extension of this “eight-
point algorithm” to determine the fundamental matrix
F
. Since the elements of
the fundamental matrix may be of strongly different orders of magnitude, it is
favourable to normalise by a translation and scaling transformation in each image
the sensor (pixel) coordinates of the image points, given by
(u
(j)
i
given by
=
,v
(j)
i
,
1
)
T
with
i
indicating the image index and
j
denoting the index of the pair of corre-
sponding points. Hence, the average of the image points, which are given in nor-
malised homogeneous coordinates, is shifted to the origin of the sensor coordinate
∈{
1
,
2
}
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