Image Processing Reference
In-Depth Information
n
and
C
n
is a real constant, estimates the direction parameter
tan
−
1
(
a, b
)
as well as the error via
I
20
and
I
11
according to theorem 11.1. For
n<
0
the follow-
ing scheme yields the analogous estimates:
where
0
≤
2
)=
C
n
Γ
∗{n,σ
2
}
(
Γ
{
1
,σ
1
}
k
f
k
)
2
I
20
(
|
F
(
ω
ξ
,ω
η
)
|
∗
∗
(11.105)
k
Γ
∗{n,σ
2
}
k
Γ
{
1
,σ
1
}
k
|
2
)=
C
n
|
f
k
|
2
I
11
(
|
F
(
ω
ξ
,ω
η
)
|∗|
∗
(11.106)
where
Γ
∗{n,σ
2
}
=(
Γ
{n,σ
2
}
)
∗
.
We note that the parameter
C
n
is constant w.r.t. to (
x
l
,y
l
) and has no implications
to applications because it can be assumed to have been incorporated to the image the
filter is applied to. In turn, this amounts to a uniform scaling of the gray-value gamut
of the original image. There are two parameters employed by the suggested scheme
that control filter sizes:
σ
1
, which is the same as in the ordinary structure tensor,
determining how much of the high frequencies are assumed to be noise; and
σ
2
,
representing the size of the neighborhood.
Equations (11.103) and (11.105) can be implemented via separable convolutions
since the filters
Γ
{n,σ
2
k
are separable for all
n
. The same goes for Eqs. (11.104) and
(11.106), provided that
n
is even. Consequently, for even
n
, both
I
20
and
I
11
can
|Γ
{n,σ
2
}
k
be computed with 1D filters. For odd
n
, only
is not separable. For such
patterns, while
I
20
can be computed by the use of 1D filters, the computation of
I
11
will need one true 2D convolution or an inexact approximation of it obtainable, e.g.,
by the singular value decomposition of the 2D filter (Sect. 15.3). Alternatively, the
computational costs can still be kept small by working with small
σ
2
|
and Gaussian
pyramids.
Lemma 11.6 assumes that there is a window function whose purpose is to limit
the estimation of
I
20
and
I
11
to a neighborhood around the current image point. Apart
from
n
=0, straight line patterns, the local gradient direction arg
d
dz
=arg(
z
n
2
)
is not well-defined in the origin for patterns generated by Eq. (11.87). This is visible
in Fig. 11.6. The factor
n
in the window function is consequently justified,
since it suppresses the origin as information provider for
n
|
x
+
iy
|
=0. Figure 11.6 shows
that the filters that are suggested by the lemma for various patterns vanish at the
origin except for the (Gaussian) one used for straight line extraction.
As mentioned, for
n
=0the origins of the target patterns are singular. Because of
this, with increased
, the continuous image of such a pattern becomes increasingly
difficult to discretize in the vicinity of the origin, to the effect that their discrete
images will have an appearance less faithful to the underlying continuous image
near the origin. As a consequence, approximating such patterns with band-limited
or other regular functions, a necessity for accurate approximation of the integrals
representing
I
20
and
I
11
, will be problematic because the singularity at the origin
is barely or not at all accounted for already at the original discrete image. This can
be achieved by signal theoretically correct sampling [63], e.g., when the square of
an image on a discrete grid is needed, then the discrete image must be assured to
|
n
|
