Image Processing Reference
In-Depth Information
Table 9.1. Gaussians approximating an ideal characteristic function
Cutoff frequency:
2
Norm
σ
0
[
x
-domain]
σ
0
[
ω
-domain]
L
1
0.98
1.0
L
2
0.82
1.2
L
3
0.78
1.3
L
∞
0.75
1.3
the spatial domain filters. Drawn in the frequency domain, Fig. 9.4 illustrates the
corresponding Gaussians and the characteristic function they approximate.
It can be shown that for cases when
B
has values other than
π
, corresponding to
a size reduction of
2
B
, we obtain the critical frequency domain standard deviation
parameter as
0
.
75
0
.
65
B
B
log(256)
,
B
log(256)
2
,
0
.
85
B
0
.
85
B
2
σ
0
∈
≈
⇒
σ
0
2
The interval above is an upper bound for
σ
0
, for the loosest sampling afforded by the
bandwidth
B
. The critical upper bound on the right originates from the upper bound
of the interval, which is obtained by using the
L
∞
norm and observing that one has
the option to sample denser than the loosest allowable sampling. Accordingly, the
lower bound for the spatial filter parameter for the same size reduction factor yields:
0
.
37
2
π
=
0
.
75
T
B
,
0
.
5
2
π
2
,
T
0
.
75
T
σ
0
∈
⇒
2
σ
0
B
2
Here
T
=
2
B
is the critical sampling period, which is also the critical size reduction
factor. This is because, throughout, we assumed that the original sampling period is
1, implying that the maximum frequency (Nyquist) is normalized to
π
.
9.4 Extending Gaussians to Higher Dimensions
Gaussians are valuable to image analysis for a number of reasons. In the previous
sections, we studied how well it is possible to fit the Gaussian to the characteristic
function in the frequency domain and why the Gaussian family should be used in-
stead of the sinc function family. Although in 1D we suggested Gaussians, there are
many other function families that are used in practice for a myriad of reasons,
4
.To
appreciate the use of the Gaussian family of functions in signal analysis, it is more
appropriate to study them in higher dimensions than 1. We first extend their definition
to include
G
aussians in
N
dimensions
:
4
Some frequently encountered 1D filter families include Chebychev, Butterworth, and co-
sine functions.
