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IEEE TRANS. SIGNAL PROC. 11
Fig. 9.
Left: Original (hexagonal) image; Right: Decomposed image with lowpass filter p
Fig. 10.
Decomposed images with highpass filters q (1) (left) and q (2) (right)
A. 6-fold axial symmetry
Let
and denote
[ ] [ ] [ 0 1
1 0
1 −1
L 0 = , L
1 0 1 =
, L
1 −1 2 =
0 −1
L 3 = −L 0 , L 4 = −L 1 , L 5 = −L 2 ,
R 1 =
[ 0 1
−1 1
Then for a j, 0 ≤ j ≤ 5, {p k } is symmetric around the symmetry axis S j in Fig. 7 if and only if p Lj k = p k ; and
{p R1 k} is the π 3 (anticlockwise) rotation of {p k}.
Observe that
L j = (R 1 ) j L 0 , 0 ≤ j ≤ 5.
Thus, instead of considering all L j , 0 ≤ j ≤ 5, we need only consider L 0 , R 1 when we discuss the 6-fold axial
symmetry of a filter bank. First we have the following proposition.
Proposition 1: A filter bank {p, q (1) , q (2) } has 6-fold axial symmetry if and only if it satisfies
]
.
]
,
(25)
[p, q (1) , q (2) ] T (R1 −T ω) = N 1 (ω)[p, q (1) , q (2) ] T (ω), (26)
[p, q (1) , q (2) ] T (L 0 ω) = N 2 (ω)[p, q (1) , q (2) ] T (ω), (27)
where
⎡
N 1 (ω) = ⎣
1 0 0
0 0 e −i(2ω 1+ω 2 )
0 e i(2ω 1+ω 2 )
0
⎤
⎡
⎦ , N 2 (ω) = ⎣
1 0 0
0 e i(ω 1−ω 2 )
0 0 e −i(ω 1−ω 2 )
⎤
⎦ . (28)
Proof. For a filter bank {p, q (1) , q (2) }, let h (1) (ω) = e iω 1 q(1) (ω), h (2) (ω) = e −iω 1 q(2) (ω). Then with the fact
L j = R j 1 L 0, 0 ≤ j ≤ 5, we know {p, q (1) , q (2) } has 6-fold axial symmetry if and only if
p(R1 −T ω) = p(L 0 ω) = p(ω), (29)
h (1) ((R1 −T ) 2 ω) = h (1) ((R1 −T ) 4 ω) = h (1) (L 0 ω) = h (1) (ω), (30)
h (2) (−ω) = h (1) (ω). (31)