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IEEE TRANS. SIGNAL PROC. 6
for 1 ≤ l, l ′ ≤ 2, ω ∈ IR 2 , where η j , 0 ≤ j ≤ 2 are the representatives of the group Z 2 /(M T Z 2 ), δ k is the
kronecker-delta sequence: δ k = 1 if k = 0, and δ k = 0 if k ≠ 0. When M is the dilation matrix M 1 or M 2 in (2),
we may choose η j , 0 ≤ j ≤ 2 to be
η 0 = (0, 0), η 1 = (1, 0), η 2 = (−1, 0). (8)
Filter banks {p, q (1) , q (2) } and {˜p, ˜q (1) , ˜q (2) } are also said to be biorthogonal if they satisfy (5)-(7); and a filter
bank {p, q (1) , q (2) } is said to be orthogonal if it satisfies (5)-(7) with ˜p = p, ˜q (l) = q (l) , 1 ≤ l ≤ 2.
Let {p, q (1) , q (2) } and {˜p, ˜q (1) , ˜q (2) } be a pair of FIR filter banks. Let φ and ˜φ be the scaling functions (with
dilation matrix M) associated with lowpass filters p(ω) and ˜p(ω) respectively, namely, φ, ˜φ satisfy the refinement
equations:
φ(x) = ∑ k∈Z 2 p k φ(Mx − k), ˜φ(x) = ∑ k∈Z 2 ˜p k ˜φ(Mx − k), (9)
and ψ (l) , ˜ψ (l) , 1 ≤ l ≤ 2 are given by
ψ (l) (x) = ∑ k∈Z 2 q(l) k
φ(Mx − k),
˜ψ (l) (x) = ∑ k∈Z 2 ˜q(l) ˜φ(Mx k
− k),
where p k , ˜p k , q (l)
k , ˜q(l) k
are the impulse response coefficients of p(ω), ˜p(ω), q(l) (ω), ˜q (l) (ω), respectively
If {p, q (1) , q (2) } and {˜p, ˜q (1) , ˜q (2) } are biorthogonal to each other (with dilation M), then under certain mild
conditions (see e.g. [44], [45], [43], [23]), φ and ˜φ are biorthogonal duals: ∫ IR φ(x)˜φ(x − k) dx
2
= δ k , k ∈ Z 2 , where δ k = δ k1 δ k2 . In this case, ψ (l) , ˜ψ (l) , l = 1, 2, are biorthogonal wavelets, namely, {ψ (l)
j,k : l =
1, 2, j ∈ Z, k ∈ Z 2 (l)
} and { ˜ψ
j,k : l = 1, 2, j ∈ Z, k ∈ Z2 } are Riesz bases of L 2 (IR 2 ) and they are biorthogonal to
each other:
∫
ψ (l)
j,k (x) ˜ψ (l′ )
j ′ ,k
(x)dx = δ ′ j−j ′δ l−l ′δ k−k ′,
IR 2
for j, j ′ ∈ Z, 1 ≤ l, l ′ ≤ 2, k, k ′ ∈ Z 2 , where
ψ (l)
j,k (x) = 3 j 2 ψ (l) (M j x − k), ˜ψ(l) j,k (x) = 3 j 2 ˜ψ(l) (M j x − k), j ∈ Z, k ∈ Z 2 .
Remark 1: One can verify that {M1 −T η j : j = 0, 1, 2} = {M2 −T η j : j = 0, 1, 2}, where η j , j = 0, 1, 2 are
the representatives for both Z 2 /M1 T Z2 and Z 2 /M2 T Z2 given in (8). Thus, {p, q (1) , q (2) } and {˜p, ˜q (1) , ˜q (2) } are
biorthogonal with one of M 1 , M 2 , say M 1 , then they are also biorthogonal to each other with the other dilation
matrix, M 2 .
φ and ˜φ are refinable functions along Z 2 . φ, ˜φ and ψ (l) , ˜ψ (l) , l = 1, 2 are the conventional scaling functions and
wavelets. Let U be the matrix defined by [ √ ]
1 3
Then U transforms the regular unit hexagonal lattice G onto the square lattice Z 2 . Define
U =
0
3
2 √ 3
3
Φ(x) = φ(Ux), Ψ (l) (x) = ψ (l) (Ux),
˜Φ(x) = ˜φ(Ux), ˜Ψ (l) (x) = ˜ψ (l) (Ux), l = 1, 2.
Then Φ and ˜Φ are refinable along G with the same coefficients p k and ˜p k for φ and ˜φ, and Ψ (l) , l = 1, 2 and
˜Ψ (l) , l = 1, 2 are hexagonal biorthogonal wavelets (along the hexagonal lattice G). The reader refers to [46] for
refinable functions along a general lattice.
In the rest of this section, we give the definitions of the symmetries of filter banks considered in this paper.
Definition 1: A hexagonal filter bank {p, q (1) , q (2) } is said to have 2-fold rotational symmetry if its lowpass filter
p(ω) is invariant under π rotation, and its second highpass filter q (2) is the π rotation of its first highpass filter q (1) .
Definition 2: Let S j , 0 ≤ j ≤ 5 be the axes in the left part of Fig. 7. A hexagonal filter bank {p, q (1) , q (2) }
is said to have 6-fold axial symmetry or 6-fold line symmetry if (i) its lowpass filter p(ω) is symmetric around
.
(10)
(11)