Image Processing Reference
In-Depth Information
Using this norm to measure the amount of dissimilarity, we obtain
e
∞
=max
ω
|
G
(
ω
)
−
χ
(
ω
)
|
=max(
a,
1
−
a
)
(9.21)
where
a
is in the interval ]0
,
1[ and is defined geometrically as in Fig. 9.3. The figure
illustrates a Gaussian with a specific
σ
, demonstrating that the
||
e
||
∞
is determined
by the intersection of the line
ω
=
π
2
with the graph of
G
(
ω
) because this is where the
highest absolute values of the difference will occur. The norm equals the maximum
of the pair
a
and 1
−a
, as depicted in the figure. By varying
σ
, the intersection point
can be varied with the expectation that we will find a low value for max(
a,
1
− a
).
The lowest possible norm is evidently obtained for
σ
0
, producing
a
=0
.
5 i.e.,
2
)=exp
=
1
2
)
2
2(
σ
)
2
(
π
2
G
(
π
−
(9.22)
so that the analytic solution,
π
log(256)
≈
σ
0
=
1
.
3
(9.23)
is the solution for the nonlinear minimization problem given in Eq. (9.19) when
k
approaches
∞
. In the spatial domain this corresponds to a Gaussian filter with
σ
0
=
log(256)
π
≈
0
.
75
(9.24)
Accordingly, the Gaussian with maximum 1 that approximates the constant function
1, the best in the
L
∞
norm, is the one placed in the middle and that attenuates 50%
at the characteristic function boundaries.
The used norm affects the optimal parameter selection. Had we chosen another
value for
k
in the
k
norm, we would obtain another value for
σ
0
, yielding a different
Gaussian. This is one of the reasons why there is not a unique down-sampling (or up-
sampling) filter: because the outcome depends on what norm one chooses to measure
the dissimilarity with.
We study here just how much the choice of the error metrics influences the
σ
0
that determines the width of the Gaussian. The minimization of Eq. (9.19) can be
achieved for finite
k
numerically. The values of
σ
0
that yield minimal dissimilari-
ties measured in different norms are listed in Table 9.1 when the cutoff frequency is
B
2
L
π
2
.
Column
σ
0
represents the corresponding standard deviations of the filters
in the spatial domain, with
σ
0
=1
/σ
0
. The table shows that
σ
0
varies in the interval
[1
,
1
.
3], where the upper bound is given by Eq. (9.24). This suggests that there is
a reasonably large interval from which
σ
0
can be picked up to depreciate the high
frequencies sufficiently while not depreciating the low frequencies too severely for
down-sampling. The
σ
0
determines the filter
G
in the frequency domain. The cor-
responding filter in the spatial domain is given by the
g
(
x
) in Eq. (9.11), where the
optimal
σ
0
=
is the inverse of the
σ
0
. Table 9.1 also lists the
σ
0
values that determine
