Optimal Differential Filter on Hexagonal Lattice - Computational Intelligence Paradigms in Advanced Pattern Classification - page 62

In the frequency domain, by using Parseval's theorem, and by considering that the
same shape of spectra appears repeatedly due to the sampling theorem, the problem
is transformed into the minimization of the following on a single unit of spectra:

i 1 v G

√ 3
2

2

2

| 2 π iuG ( u , v ) − F a ( u , v ) | + 2 π

2 u

+ ( u , v ) − F b ( u , v )

D

i v G

dudv

√ 3
2

2

1
2 u

+ 2 π − + ( u , v ) − F c ( u , v ) , (9)

where

− 1 / 2 √ 3

1 / 2

D •

dudv ≡

/ √ 3 •

dudv

/ √ 3 0 v − 1

− v / √ 3 + 1 / √ 3 − 1 / 2 √ 3

1 / 2 0

+ • dudv +

/ √ 3 •

dudv

2 √ 3 / √ 3 0 1 / − 1 / 2 − v − 1

v / √ 3 + 1 / √ 3 1 / 2 √ 3

1 / 2

0

+ • dudv +

2 √ 3 •

dudv . (10)

2 √ 3 − 1 / 2 1 / − 1 / 2 − 1 /

This area D is represented in Figure 2. The integrand of (9) is expanded as follows
(where

• ∗ denotes the complex conjugate of

• ):

Fig. 2 Hexagonal region to
be integrated