Image Processing Reference
In-Depth Information
ω
y
|F|
2
ω
ω
x
Fig. 10.9. The line-fitting process is illustrated by the linear symmetry direction vector
k
,
the angular frequency vector
|F
(
ω
)
|
2
are
ω
, and the distance vector
d
. The function values
represented by color. The frequency coordinate vector is
ω
e
(
k
)=
ω
d
2
(
ω
,
k
)
(10.13)
where
d
(
and a candidate axis
k
.
This is the TLS error function for a discrete data set. Noting that
ω
,
k
) is the shortest distance between a point
ω
k
=1, then the
t
k
)
k
. As illustrated by Fig. 10.9, the vector
d
projection of
ω
on the vector
k
is (
ω
represents the difference between
ω
and the projection of
ω
. This difference vector
is orthogonal to
k
t
k
)
k
d
=
ω
−
(
ω
(10.14)
with its norm being equal to the shortest distance, i.e.,
d
=
d
(
ω
,
k
). Consequently,
the square of the norm of
d
provides:
d
2
(
t
k
)
k
2
ω
,
k
)=
ω
−
(
ω
(10.15)
=
ω
−
t
k
)
k
T
ω
−
t
k
)
k
(
ω
(
ω
(10.16)
Since we have a Fourier transform function,
F
, defined on dense angular fre-
quency coordinates in
E
2
, instead of a sparse point set, Eq. (10.13) needs to be
modified. The following error function is a generalization of Eq. (10.13) to dense
point sets. It weights the squared distance contribution at an angular frequency point
ω
|
2
and integrates all error contributions
e
(
k
)=
with the energy
|
F
(
ω
)
d
2
(
2
d
ω
,
k
)
|
F
(
ω
)
|
ω
(10.17)
