Optimal Differential Filter on Hexagonal Lattice - Computational Intelligence Paradigms in Advanced Pattern Classification - page 78

h d for the 90 ◦ direction is composed from the filters for the 60 ◦ and 120 ◦ directions
as follows:

1
√ 3 ( h d ≡ h b + h c ) . (111)

The filters for the 30 ◦ and 150 ◦ directions are the obtained by rotating h d (Figure 6).
In the same way, for square lattices, gradient filters for the 30 ◦ , 120 ◦ and 150 ◦

directions were prepared by composing Ando's filters in the 0 ◦ and 90 ◦ directions.

60 ◦ ,

a - - -

0 0

-a a - a-

0 0 0 0 0

-a a- a- a-

0 0

a = 0.2886751

h e h d h f

Fig. 6 Derived differential filters for 30, 90 and 150 degree from filters for 0, 60 and 120
degree

8.4 Relationship between Derived Filter and Staunton Filter

The relationship between the filters derived here (whose radius is 1) and Staunton
filters[23] is investigated as follows.
Staunton designed a set of edge detecting filters as illustrated in Figure 7. He
mentioned that the element values in these filters, which are 1 or 1, are nearly

optimal according to Davies' design principle[6]. This principle is based on a super
sampling of a disc whose center is the center of a filter in the image domain. On the
contrary, the derivation of the present filters is in frequency domain.
The equations for detecting intensity and orientation from convolution values
with the Staunton filters are described in the following.
Intensity Int hex

−

Staunton is given as

fp a +

1
√ 3

Int hex
Staunton

fp c +

,

≡ fp a fp c (112)

where fp a and fp c are convolution values with filters p a and p c , respectively 1
Orientation is given as

1
√ 3 (

Ori hex
Staunton

≡ arctan ( fp a + fp c , fp a − fp c )) . (113)

1

Since we use opposite signed p c , the sign of the third term differs from that in [23]. The
constant

1
√ 3

2
√ 3

also differs because the present grid distance is 1, but it is in [23].