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on signal processing[14], geometric transform[10], co-occurrence matrix[15],
efficient algorithms for Fourier transforms[9], and FIR filter banks[11] has been
reported. Moreover, image processing on hexagonal lattices, ranging from hard-
ware to application algorithms [3, 4, 5, 12, 22, 24, 26, 27, 28] has also been widely
researched. Moreover, a book that collects researches about image processing on
hexagonal lattices has been published[16]. It is also known that the human eye re-
ceptors follow a hexagonal alignment, there are works related to human perception
and image processing on hexagonal lattice. Overington[17] proposed lightweight
image processing methods on hexagonal lattices. Gabor filters on hexagonal lattice,
believed to be performed in the lower levels of the human vision system, have also
been proposed[25].
In this chapter, consistent gradient filters on hexagonal lattices are described,
adding to our paper [20]. The filters are derived on the basis of a previously proposed
on square lattices [2]. The relationship between the derived filters and existing filters
on a hexagonal lattice designed in another way is then discussed. After that, the
derived filters are compared with conventional optimized filters on square lattices.
2
Preliminaries
First,
F
(
u
,
v
)
is taken as the Fourier transform of
f
(
x
,
y
)
:
∞
∞
e
−
2π
i
(
ux
+
vy
)
dxdy
F
(
u
,
v
)=
f
(
x
,
y
)
(1)
−
∞
−
∞
In this chapter, The pixel placement on hexagonal lattices and square lattices is
defined as Figure 1.
It is assumed that the input image contains frequency components only in the
hexagonal region which is described in Figure 2. According to the sampling theo-
rem, the impulse response of an image sampled by an hexagonal lattice is repeated
and the repeating unit region is illustrated in Figure 2 ([7]).
3
Least Inconsistent Image
Let the x-axis and the y-axis correspond to the horizontal axis (0
◦
) and the verti-
cal axis (90
◦
) respectively. Let
h
a
(
x
,
y
)
,
h
b
(
x
,
y
)
,and
h
c
(
x
,
y
)
be elements of discrete
gradient filters in the directions 0
◦
,
60
◦
, and 120
◦
respectively, on hexagonal lat-
tices, while the arguments
(
x
,
y
)
of them indicate a point in a traditional orthogonal
coordinate system.
The gradients of image
f
in the directions of the filters, which are derived
by the convolution, are denoted by
f
a
(
x
,
y
)
(
x
,
y
)
,
f
b
(
x
,
y
)
,
f
c
(
x
,
y
)
,
)=
∑
x
1
,
f
a
(
x
,
y
)=
h
a
(
x
,
y
)
∗
f
(
x
,
y
h
a
(
x
1
,
y
1
)
f
(
x
−
x
1
,
y
−
y
1
)
(2)
y
1
∈
RH
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