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Ta b l e 2
Localization and SNR
Filter
Num. of Pixels
Loc
SNR
Sqr 3
×
3
9
1.1522
4.51
Sqr 4
×
4
16
1.2698
6.76
Sqr 5
×
5
25
1.5081
9.12
Hex 1
7
1
4.51
Hex
√
3
13
1.0833
7.41
Hex 2
19
1.2553
10.75
12
10
8
6
4
2
Fig. 5
Filters on hexagonal
lattices have better SNR de-
spite localization increases
in comparison to those on
square lattices.
0
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Loc
Hex
Sqr
is a trade-off between localization
Loc
and
SNR
. However, for a given size, filters
derived on hexagonal lattices achieve better
SNR
-to-
Loc
ratio than filters derived on
square lattices, as shown in Figure 5.
8
Experimental Evaluation
8.1
Construction of Artificial Images
The consistent gradient filters on hexagonal and square lattices are compared as fol-
lows. Specifically, to get both gradient intensity and orientation at any point on the
image analytically, artificial images defined by mathematical functions were con-
structed. The error between the ideal value and the value obtained from the filtered
image were the measured.
Since it was assumed in the derivation of the element values of the gradient filters
that the frequency characteristics of the input image are close to white noise, images
that present other frequency profiles for evaluation were constructed.
The artificial input image is taken as
f
,and
f
int
and
f
ori
(
,
)
(
,
)
(
x
,
y
)
x
y
x
y
are taken
as its ideal gradient intensity and its ideal orientation, respectively.
f
ex
1
,
f
ex
2
and
f
ex
3
are defined as follows.
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