SUBMISSION TO IEEE TRANS. IMAGE PROC.
1
FIR Filter Banks for Hexagonal Data Processing
Qingtang Jiang
Abstract—Images are conventionally sampled on a rectangular
lattice, and they are also commonly stored as such a lattice. Thus,
traditional image processing is carried out on the rectangular
lattice. The hexagonal lattice was proposed more than four
decades ago as an alternative method for sampling. Compared
with the rectangular lattice, the hexagonal lattice has certain
advantages which include that it needs less sampling points; it
has better consistent connectivity and higher symmetry; and that
the hexagonal structure is pertinent to the vision process. In this
paper we investigate the construction of symmetric FIR hexago-
Fig. 1.
Rectangular lattice (left) and hexagonal lattice (right)
nal lter banks for multiresolution hexagonal image processing.
We obtain block structures of FIR hexagonal lter banks with 3fold rotational symmetry and 3-fold axial symmetry. These block
structures yield families of orthogonal and biorthogonal FIR
visual system. Hence, the hexagonal lattice has been used
hexagonal lter banks with 3-fold rotational symmetry and 3-fold
in many areas such as edge detection [10], [11] and pattern
axial symmetry. In this paper, we also discuss the construction
of orthogonal and biorthogonal FIR lter banks with scaling
recognition [12]-[16]. The hexagonal lattice has also been
functions and wavelets having optimal smoothness. In addition,
applied in Geoscience and other elds. For example, in the
we present some of such orthogonal and biorthogonal FIR lters
Soil Moisture and Ocean Salinity (SMOS) space mission led
banks.
by the European Space Agency, the data collected by the Y-
Index Terms—Hexagonal lattice, hexagonal data, 3-fold ro-
shaped antenna of the SMOS space mission is hexagonal data
FIR
(sampled on a hexagonal lattice) [17], [18]. A hexagon-based
hexagonal lter bank, biorthogonal FIR hexagonal lter bank,
grid has been adopted by the U.S. Environmental Protection
tational
symmetry,
3-fold
axial
symmetry,
orthogonal
compactly supported orthogonal and biorthogonal hexagonal
wavelets.
Agency for global sampling problems [19], [20].
Research on hexagonal image processing goes back to
EDICS Category: MRP-FBNK
Petersen and Middleton [1] over 40 years ago. Since then,
researchers in different areas have made many contributions in
I. I NTRODUCTION
the study of hexagonal image processing on various topics, see
Traditional 2-D data (image) processing is carried out on the
[7]-[9] and the references therein. Multiscale (multiresolution)
rectangular lattice since 2-D data is conventionally sampled
image processing has been one of the most popular areas
at the sites (points) on a square or rectangular lattice and
of research investigation in various scientic and engineering
it is also commonly stored as such a lattice. See a square
disciplines during the decade of the 1990's, and it has been
lattice in the left part of Fig. 1. The hexagonal lattice (in the
used in many applications. However, as it was pointed out in
right part of Fig. 1) was proposed more than four decades
[7], multiresolution hexagonal image processing is a research
ago as an alternative method for sampling. Compared with a
area with a slow pace of activity. One probable reason for this
rectangular lattice, a hexagonal lattice has certain advantages
could be that most researchers in the “Wavelets” community
[1]-[9]. It was shown in [1] that, for functions band-limited in a
are accustomed to the traditional rectangular lattice. Another
circular region in the frequency domain, the hexagonal lattice
reason is probably that current approaches have encoun-
needs a smaller number (about 13.4% smaller) of sampling
tered difculties in the construction and design of desirable
points to maintain equally high frequency information than
hexagonal lter banks to be used for multiresolution image
the square lattice. The hexagonal structure has better consistent
processing. To the author's best knowledge, [21]-[25], [4],
connectivity: each elementary cell of a hexagonal lattice has
[26] are the papers available on the construction/design of
six neighbors of the same type while an elementary cell of a
hexagonal lter banks with both lowpass and highpass lters
square lattice has two different types of neighbors. The hexag-
constructed. [21] presented a few FIR (nite impulse response)
onal structure possesses higher symmetry: a regular hexagonal
hexagonal lter banks which achieved near orthogonality. [22]
lattice has 12-fold symmetry while a square lattice has 8-
provided one 7-channel (
fold symmetry. Other advantages of the hexagonal structure
bank for image coding. The authors in [23] designed FIR
over the square structure include that it offers greater angular
hexagonal lter banks by minimizing the lter bank error
resolution of images and it is closely related to the human
and intra-band aliasing error function and applied their lters
Q. Jiang is with the Department of Mathematics and Computer Science,
√
7-renement)
FIR hexagonal lter
to image compression and orientation analysis. The highpass
University of Missouri–St. Louis, St. Louis, MO 63121, USA, e-mail:
lters used in [23] are suitable spatial shifting and frequency
[email protected] , web: http://www.cs.umsl.edu/∼jiang .
modulations of the lowpass lter. FIR hexagonal lter banks
Research supported by UM Research Board 10/05 and UMSL Research
Award 10/06.
was also designed in [24] by the same method as in [23]
but with a different lter bank error and intra-band aliasing
2
SUBMISSION TO IEEE TRANS. IMAGE PROC.
error function. The FIR lter banks designed in [21], [23],
v2
[24] are not perfect reconstruction lter banks. Construction
U
of biorthogonal hexagonal lter banks was fully investigated in
v1
[25] and a few biorthogonal FIR lter banks were constructed
there. However, it is difcult to construct biorthogonal lter
−1
banks with smooth wavelets by their approach which also
U
results in lters with large lter lengths. The author in [4]
(also in [26]) proposed a novel block structure of orthogonal/biorthogonal FIR hexagonal lter banks with certain
Fig. 2.
symmetry. However, a rigorously mathematical proof of the
(right)
Regular unit hexagonal lattice (left) and square lattice
Z2
symmetry and the (bi)orthogonality of the lter banks in [4]
is desired, and the issue of how to select the free parameters
in these lter banks needs to be addressed.
Though the hexagonal lter banks designed in [23] are
not perfect reconstruction lter banks, experimental results
on their applications to image compression and orientation
analysis carried out in [23] and their applications to digital
mammographic feature enhancement and the recognition of
(see e.g. [27] for the shift-invariant theory along general
lattices). To design hexagonal lter banks, here we transform
the hexagonal lattice to the square lattice so that we can use
the well-developed integer-shift multiscale analysis theory and
methods.
G
Let
complex annotations in [12], [13] are appealing. Therefore, the
G = {n1 v1 + n2 v2 :
construction/design of hexagonal lter banks deserves further
investigation. The main objective of this paper is to construct
where
T
v1 = [1, 0] ,
orthogonal and biorthogonal FIR hexagonal lter banks with
certain symmetry which is pertinent to the symmetry structure
Let
of the hexagonal lattice.
be the regular unit hexagonal lattice dened by
U
"
U=
briey show that the problem of lter construction along
the hexagonal lattice can be transformed into that along the
Z2 .
After that we discuss the symmetry
of lter banks and review some basic results on orthogonal/biorthogonal lter banks. In Section III, we present block
structures of orthogonal and biorthogonal FIR lter banks with
3-fold rotational symmetry. In Section IV, we provide block
structures of orthogonal and biorthogonal FIR lter banks with
3-fold axial symmetry. These structures include that in [4].
In both Sections III and IV, we also discuss the construction
of orthogonal and biorthogonal FIR lter banks with scaling
functions and wavelets having optimal smoothness.
In this paper we use the following notations. For x =
[x1 , x2 ]T , y = [y1 , y2 ]T , x · y denotes their dot (inner) product
xT y. For Ra function f on IR2 , fˆ denotes its Fourier transform:
fˆ(ω) = IR2 f (x)e−ix·ω dx. For a matrix M , we use M ∗
to denote its conjugate transpose M T , and for a nonsingular
−T
−1 T
matrix M , M
denotes (M
) . For ω = [ω1 , ω2 ]T , let
z1 = e−iω1 , z2 = e−iω2 .
(1)
Then
U
√
3
3
√
2 3
3
#
.
(3)
transforms the regular unit hexagonal lattice into the
k, k ∈ Z2 . See
P Fig. 2. −ig·ω
1
For a hexagonal lter H(ω) =
with its
g∈G Hg e
4
1
(real) impulse response Hg (in this paper a factor
is added
4
for the convenience), by the transformation with the matrix U ,
P
1
−ik·ω
for
we have a corresponding lter h(ω) =
k∈Z2 hk e
4
square data (squarely sampled data) with its impulse response
hk = HU −1 k .
Conversely, corresponding to a square lter
P
h(ω)P= 14 k∈Z2 hk e−ik·ω , we have a
1
−ig·ω
hexagonal lter H(ω) =
. In the frequency
g∈G hU g e
4
domain, the relationship between H(ω) and h(ω) (with hk =
HU −1 k ) are given by
(lter for square data)
H(ω) = h(U −T ω).
The matrix
U
also transforms the scaling functions and
wavelets along the hexagonal lattice to those along the square
Z2 . For example, if Φ is the scaling function
associated
P
1
−ig·ω
a lowpass hexagonal lter H(ω) =
H
,
ge
g∈G
4
lattice
namely, it satises
Φ(x) =
In this section, after showing that the problem of lter
X
Hg Φ(2x − g),
x ∈ IR2 ,
g∈G
construction along the hexagonal lattice can be transformed
into that along the square lattice, we provide some basic results
1
0
square lattice with sites
with
II. P RELIMINARIES
then
on the symmetry and the orthogonality/biorthogonality of lter
φ
dened by
φ(x) = Φ(U −1 x)
banks.
is the scaling function associated with square lter
A. Transforming the hexagonal lattice to the square lattice
Z2
Most multiscale analysis theory and algorithms for image
processing are developed along the square lattice with sites
k ∈ Z2 , though they could be established along general lattices
(2)
√
1
3T
] .
v2 = [− ,
2 2
be the matrix dened by
This paper is organized as follows. In Section II, we rst
square lattice of
n1 , n2 ∈ Z},
P
h(ω) =
1
−ik·ω
, where hk = HU −1 k . Therefore, to design
k∈Z2 hk e
4
hexagonal lters, we need only to construct lters along the
2
traditional lattice Z . Then the matrix U will transform the
2
lters, scaling functions and wavelets along the lattice Z into
those along the hexagonal lattice.
Q. JIANG: FILTER BANKS FOR HEXAGONAL DATA PROCESSING
3
S1
Observe
{P, Q
(1)
,Q
e1
that R
(2)
(3)
,Q
}
fN
ee , R
e2
= W
ee .
= See N
Thus
if
satises (5), then it satises (4). There-
fore, if a hexagonal lter bank has 3-fold axial symmetry, then
v2
it has 3-fold rotational symmetry.
v1
S2
The
3-fold rotational symmetry is considered in [25], where
it is called the hexagonal symmetry. Both the 3-fold rotational
symmetry and the 3-fold axial symmetry are closely related to
the symmetry structure of the hexagonal lattice. In this paper
we consider lter banks with these two types of symmetry.
Compared with lter banks with 3-fold axial symmetry, lter
S3
Lines
Fig. 3.
S1 , S2 , S3
banks with 3-fold rotational symmetry have less symmetry but
they provide more exibility for the construction of lters. It
in hexagonal lattice
should be up to one's specic application to choose lter banks
with 3-fold rotational or axial symmetry.
For
B. Symmetry of lter banks
Since the hexagonal lattice has the highest degree of symmetry, it is desirable that hexagonal lter banks designed also
have certain symmetry pertinent to the symmetric structure of
the hexagonal lattice. In this paper, we consider two types
of symmetry: 3-fold rotational symmetry and 3-fold axial
symmetry which are dened below.
Denition 1: A hexagonal lter bank
{P, Q(1) , Q(2) , Q(3) }
3-fold rotational symmetry if its lowpass lter
P (ω) is invariant under the 23 π and 43 π rotations, and its high2
4
(2)
(3)
pass lters Q
and Q
are the π and π (anticlockwise)
3
3
(1)
rotations of highpass lter Q
, respectively.
Let S1 , S2 and S3 be the lines in the regular unit hexagonal
lattice shown in Fig. 3. Namely, S1 , S2 and S3 are the lines
√
√
passing through the origin with slopes
3, 0 and − 3,
is said to have
respectively.
{P, Q(1) , Q(2) , Q(3) }
is said to have 3-fold axial symmetry (or 3-fold line symmetry)
if its lowpass lter P (ω) is symmetric around S1 , S2 and S3 ,
(1)
and its highpass lter Q
is symmetric around the axis S1
2
(2)
(3)
and other two highpass lters Q
and Q
are the
3 π and
4
(1)
, respectively.
3 π (anticlockwise) rotations of Q
e1 , R
e2 , N
ee , W
f and See be the matrices dened by
Let R
"
#
"
√
√ #
1
3
−√12
−
− 23
2
2
e
e
√
R1 =
, R2 =
,
3
− 23 − 21
− 12
2
"
√ #
¸
·
1
3
−
2
2
e
f= 1 0
√
,
Ne =
,
W
1
3
0 −1
2
2
"
√ #
1
3
−
−
2
√2
See =
.
3
1
− 2
2
Denition 2: A hexagonal lter bank
{P, Q(1) , Q(2) , Q(3) } has 3-fold
only if for all g ∈ G ,
{p, q
(1)
(1)
hexagonal lter bank {P, Q
, Q(2) , Q(3) }, let
(2) (3)
, q , q } be the corresponding square lter bank
a
U in (3). Namely, the
p(ω), q (`) (ω), 1 ≤ ` ≤ 3 are
(`)
PU −1 k , QU −1 k , respectively. Let R1 , R2 , Ne , W and Se denote
e1 U −1 , U R
e2 U −1 , U N
ee U −1 , U W
f U −1 and
the matrices U R
−1
e
U Se U
respectively, namely,
·
¸
·
¸
−1 1
0 −1
R1 =
, R2 =
,
(6)
−1 0
1 −1
after the transformation by the matrix
impulse responses
·
Ne =
0 1
1 0
(`)
pk , qk
¸
·
,W =
Then one can show that
of
1 −1
0 −1
¸
·
, Se =
P, Q(1) , Q(2) , Q(3)
−1 0
−1 1
¸
.
(7)
satisfy (4) if and
only if
(2)
(1)
(3)
(1)
pR1 k = pR2 k = pk , qk = qR1 k , qk = qR2 k , k ∈ Z2 ;
and that
½
P, Q(1) , Q(2) , Q(3)
(8)
satisfy (5) if and only if
pNe k = pW k = pSe k = pk ,
(1)
(1)
(2)
(1)
(3)
(1)
qNe k = qk , qk = qR1 k , qk = qR2 k , k ∈ Z2 .
(9)
To summarize, we have the following proposition.
{P, Q(1) , Q(2) , Q(3) } be a hexagonal
lter bank and {p, q
, q (2) , q (3) } be its corresponding square
(1)
lter bank. Then {P, Q
, Q(2) , Q(3) } has 3-fold rotational
(1) (2) (3)
symmetry if and only if {p, q
, q , q } satises (8); and
(1)
(2)
(3)
{P, Q , Q , Q } has 3-fold axial symmetry if and only if
{p, q (1) , q (2) , q (3) } satises (9).
Proposition 1: Let
(1)
In the following, for the convenience, we say a square lter
bank
{p, q (1) , q (2) , q (3) } has 3-fold rotational symmetry (3-fold
axial symmetry resp.) if it satises (8) ( (9) resp.).
Then one can easily show that
rotational symmetry if and
PR
e
1g
and that
(1)
= Pg , Q(2)
g =Q
= PR
e
e1 g
R
2g
{P, Q
(1)
(2)
,Q ,Q
g ∈ G,
(3)
}
has
(1)
, Q(3)
g =Q
3-fold
e2 g
R
;
C. Biorthogonality, sum rule order and smoothness
(4)
In this subsection, we review some results on orthogo-
axial symmetry if
and only if for all
(
PNe
= PW
e g = PSee g = Pg ,
(1)
(2)
(1)
(3)
(1)
Q
= Qg , Qg = Q
, Qg = Q
.
ee g
e1 g
e2 g
N
R
R
eg
(1)
nal/biorthogonal (square) lter banks. Denote
η0 = [0, 0]T , η1 = [π, π]T , η2 = [π, 0]T , η3 = [0, π]T .
(5)
FIR lter banks
(10)
{p, q (1) , q (2) , q (3) } and {p̃, q̃ (1) , q̃ (2) , q̃ (3) } are
said to be biorthogonal or they are perfect reconstruction lter
4
SUBMISSION TO IEEE TRANS. IMAGE PROC.
banks if
for all nonnegative integers
X
p(ω + ηk )p̃(ω + ηk ) = 1,
(11)
0≤k≤3
X
1 ≤ ` ≤ 3, (12)
0≤k≤3
X
0
q (` ) (ω + ηk )q̃ (`) (ω + ηk ) = δ`0 −` ,
(13)
0≤k≤3
1 ≤ `, `0 ≤ 3, ω ∈ IR2 , where δ` is the kronecker-delta
(1) (2) (3)
sequence. A lter bank {p, q
, q , q } is said to be orthog(`)
onal if it satises (11)-(13) with p̃ = p, q̃
= q (`) , 1 ≤ ` ≤ 3.
Let φ and φ̃ be the scaling function associated with p and
p̃ respectively. Then (11) is the necessary condition for φ and
φe to be biorthogonal duals:
Z
e − k) dx = δk δk ,
φ(x)φ(x
(14)
1
2
for
IR2
k = [k1 , k2 ]T ∈ Z2 . We say φ is orthogonal if it
e = φ. Under certain mild conditions, the
satises (14) with φ
condition (11) is also sufcient for the biorthogonality of φ
and φ̃. For example, if the cascade algorithms associated with
p and p̃ are convergent, or under a stronger condition that
both φ and φ̃ are stable, then the biorthogonality of p and p̃
(orthogonality of p resp.) imply that φ and φ̃ are biorthogonal
duals (φ is orthogonal resp.) (see e.g. [28], [29]). One can
for all
check the convergence of the cascade algorithm associated
with a given
p
by calculating the eigenvalues of the so-called
p,
transition operator associated with
see e.g. [30] for the
{p, q (1) , q (2) , q (3) }
(1) (2) (3)
{p̃, q̃ , q̃ , q̃ }, if the associated scaling functions φ
φ̃ are biorthogonal duals, then ψ (`) , ψ̃ (`) dened by
and
ω b ω
d
(`) (ω) = q (`) ( )φ(
ψ
),
2
2
ω b̃ ω
d̃
ψ (`) (ω) = q̃ (`) ( )φ( ),
2
2
(`)
{ψj,k
(`)
Z, k ∈ Z2 } and {ψ̃j,k : 1 ≤ ` ≤ 3, j
2
2
biorthogonal bases of L (IR ), where
are biorthogonal wavelets, namely,
(`)
ψj,k (x)
j
=2 ψ
(`)
j
(2 x − k),
(`)
ψ̃j,k (x)
: 1 ≤ ` ≤ 3, j ∈
∈ Z, k ∈ Z2 } are
= 2 ψ̃
(`)
j
(2 x − k).
{p, q (1) , q (2) , q (3) },
(`)
if the associated scaling function φ is orthogonal, then ψ
(`)
dened above are orthogonal wavelets, namely, {ψj,k : 1 ≤
` ≤ 3, j ∈ Z, k ∈ Z2 } is an orthogonal basis of L2 (IR2 ). The
reader is referred to [31] and [32] for the multiscale analysis
For a (lowpass) lter
that
p(ω)
p(ω) =
has sum rules of order
1
4
P
m
if
p(ω). The reader sees [33]
p(ω) is also a necessary
condition for the high smoothness order of φ under certain
conditions such as the stability of φ. For example, for a stable
φ, if it is in the Sobolev space W n (IR2 ) (see the denition
=
=
X
k
X
−ik·ω
pk e
p
k k = 4,
, we say
and
(2k1 + 1)α1 (2k2 )α2 p(2k1 +1,2k2 )
p(ω)
X
k
must have sum rules of order at least
n + 1.
In the following two sections we obtain block structures
of orthogonal/biorthogonal FIR lter banks with 3-fold rotational symmetry and with 3-fold axial symmetry. These
orthogonal/biorthogonal lter banks are given by some free
parameters. When a family of lter banks is available (given by
free parameters), one can design the lters with desirable properties for one's specic applications. For example, one may
consider to design lters with the optimum time-frequency
localization (see [34] and [35] for the design of 1-D matrixvalued lters with the optimum time-frequency localization).
In this paper we consider the lters based on the smoothness
φ. The smoothness of the
ψ (1) , ψ (2) , ψ (3) is the same as φ since they
are nite linear combinations of φ and its integer shifts. The
smoothness of φ and its associated wavelets is important for
of the associated scaling functions
associated wavelets
some applications. For example, certain smoothness of them
is required for the image reconstruction. In addition, a scaling
function and its associated wavelets with poor smoothness
result in poor frequency localization of them.
In the consideration of smoothness, we will compute the
Sobolev smoothness of scaling functions. For
by
on
W s (IR2 )
IR2 with
s ≥ 0,
denote
the Sobolev space consisting of functions
f (x)
Z
(1 + |ω|2 )s |fˆ(ω)|2 dω < ∞.
IR2
f ∈ W s (IR2 ) with s > k + 1 for some
2
k
then f ∈ C (IR ). We use the smoothness
If
positive integer
k,
formula in [36] to
See [37] for the detailed formulas for the Sobolev smoothness of scaling functions/vectors and [38] for algorithms and
Matlab routines to nd the Sobolev smoothness order.
For an FIR lowpass lter
α2
(2k1 ) (2k2 + 1) p(2k1 ,2k2 +1)
p(ω)
given by some free pa-
rameters, the procedures to construct the scaling function
φ
with the (locally) optimal Sobolev smoothness are described as
follows: (1) Solve the linear equations for the sum rule orders
p(ω)
p(ω)
has the desired sum rule order. The resulting
is still given by some (but less) free parameters. (2)
Adjust the free parameters for the resulting
p(ω)
by applying
the algorithms/software in [37]/[38] to achieve the optimal
Sobolev smoothness for
φ.
III. O RTHOGONAL AND BIORTHOGONAL FIR FILTER
BANKS WITH
α1
3- FOLD ROTATIONAL SYMMETRY
In this section we consider the construction of orthogonal
and biorthogonal lter banks with 3-fold rotational symmetry.
k
=
associated with
of the Sobolev space below), then its associated lowpass lter
such that
k∈Z
P2
X
(2k1 )α1 (2k2 )α2 p(2k1 ,2k2 )
k
φ
compute the Sobolev smoothness order of scaling functions.
j
Similarly, for an orthogonal lter bank
theory and its applications.
0 ≤ α1 + α2 < m.
p(ω) is
for the details. High sum rule order of
details. For biorthogonal FIR lter banks
and
with
equivalent to the approximation order and accuracy of the
scaling function
p(ω + ηk )q̃ (`) (ω + ηk ) = 0,
α1 , α2
Under certain mild conditions, sum rule order of
(2k1 + 1)α1 (2k2 + 1)α2 p(2k1 +1,2k2 +1) ,
The 3-fold rotational symmetry of lter banks is discussed in
§III.
A, and block structures of orthogonal and biorthogonal
Q. JIANG: FILTER BANKS FOR HEXAGONAL DATA PROCESSING
5
FIR lter banks with 3-fold rotational symmetry are presented
in
§III.
B and
§III.
Based on the above discussion, we reach the following result
on the lter banks with 3-fold rotational symmetry.
C, respectively.
Theorem 1: If

A. FIR lter banks with 3-fold rotational symmetry
p(ω), q (1) (ω), q (2) (ω), q (3) (ω) are FIR lters with
(1) (2) (3)
response coefcients pk , qk , qk , qk . Then l{p, q (1) , q (2) , q (3) } has 3-fold rotational symmetry,
Suppose
impulse
ter bank

{p, q (1) , q (2) , q (3) }
is given by

p(ω)
1
 q (1) (ω)  1
 ei(ω1 +ω2 )



 q (2) (ω)  = 2 An (2ω) · · · A1 (2ω)A0  e−iω1
e−iω2
q (3) (ω)
namely it satises (8), if and only if
½
p(ω) =
q (2) (ω)
the following proposition.
= q (1) (R2−T ω).
R2 = R12 , R13 = I2 ,
leads to

1
 0
M0 = 
 0
0
Next,
we
be
given
to
consider
by
the
the
0
0
0
1
lter
product
B. Orthogonal lter banks with 3-fold rotational symmetry
In this subsection, we provide a block structure of orthogonal lter banks with 3-fold rotational symmetry. For an FIR
0
0 
.
1 
0
bank
of
lter bank
(15)
that
we
can
block
matrices.
As-
write
A(R1−T ω)
where
M0
=
M0 A(ω)M0−1 ,
where
Write
(16)
has 3-fold rotational symmetry and it could be used as the
initial symmetric lter bank, while both
and
diag(1, e
, e−iω1 , e−iω2 )
−i(ω1 +ω2 )
, eiω1 , eiω2 )
Next we use
A(ω) = A diag(1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 )
and
A(ω) = A diag(1, e−i(ω1 +ω2 ) , eiω1 , eiω2 )
(18)
A is a 4×4 (real) constant matrix.
A(ω) dened by (17) or by (18), A(ω)
if and only if A has the form:


a11 a12 a12 a12
 a21 a22 a23 a24 

A=
(19)
 a21 a24 a22 a23  .
a21 a23 a24 a22
as the block matrices, where
One can verify that for
satises (16)
(17)
are given in (10). Then
U (ω)

p(ω + η3 )
q (1) (ω + η3 ) 
,
q (2) (ω + η3 ) 
q (3) (ω + η3 )
{p, q (1) , q (2) , q (3) }
is
satises
ω ∈ IR2 .
(21)
p(ω), q (1) (ω), q (2) (ω), q (3) (ω) as
µ
1
p(ω) =
pee (2ω) + poo (2ω)ei(ω1 +ω2 ) +
2
¶
poe (2ω)e−iω1 + peo (2ω)e−iω2 ,
µ
1 (`)
(`)
q (`) (ω) =
q (2ω) + qoo
(2ω)ei(ω1 +ω2 ) +
2 ee
¶
−iω2
−iω1
(`)
(`)
qoe (2ω)e
+ qeo (2ω)e
, 1 ≤ ` ≤ 3.
V (ω) denote the polyphase
q (ω), q (3) (ω)}:

pee (ω) poo (ω)
 q (1) (ω) q (1) (ω)
oo
 ee
V (ω) =  (2)
(2)
 qee (ω) qoo (ω)
(3)
(3)
qee (ω) qoo (ω)
Let
satisfy (16) and they could be used to build the block matrices.
p(ω + η2 )
q (1) (ω + η2 )
q (2) (ω + η2 )
q (3) (ω + η2 )
U (ω)U (ω)∗ = I4 ,
{1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 }
i(ω1 +ω2 )
η1 , η2 , η3
denote
p(ω + η1 )
q (1) (ω + η1 )
q (2) (ω + η1 )
q (3) (ω + η1 )
orthogonal if and only if
is the matrix dened by (15). Clearly
diag(1, e
{p, q (1) , q (2) , q (3) },
U (ω) =

p(ω)
 q (1) (ω)

 q (2) (ω)
q (3) (ω)
{p, q (1) , q (2) , q (3) }
[p(ω), q (1) (ω), q (2) (ω), q (3) (ω)]T
(1)
(2)
(3)
T
as A(2ω)[p0 (ω), q0 (ω), q0 (ω), q0 (ω)] , where A(ω) is
a 4 × 4 matrix with trigonometric polynomial entries,
(1) (2) (3)
and {p0 , q0 , q0 , q0 } is another FIR lter bank. If both
(1) (2) (3)
(1) (2) (3)
{p, q , q , q } and {p0 , q0 , q0 , q0 } have 3-fold rotational symmetry, then Proposition 2 leads to that A(ω) satises
sume
In the next two subsections, we show that the block structure
with 3-fold rotational symmetry.

0
1
0
0
where
in (20) will yield orthogonal and biorthogonal FIR lter banks
iT
p, q (1) , q (2) , q (3) (R1−T ω) =
h
iT
M0 p(ω), q (1) (ω), q (2) (ω), q (3) (ω) ,
where
n ∈ Z+ ,
fold rotational symmetry.
has 3-fold
rotational symmetry if and only if it satises
h
A0 is a constant matrix of
the form (19), and each Ak (ω) is given by (17) or (18)
with A being a constant matrix Ak of the form (19), then
{p(ω), q (1) (ω), q (2) (ω), q (3) (ω)} is an FIR lter bank with 3for some
{p, q (1) , q (2) , q (3) }
Proposition 2: A lter bank



(20)
p(R1−T ω) = p(R2−T ω),
= q (1) (R1−T ω), q (3) (ω)
This, together with the facts that

matrix of
{p(ω), q (1) (ω),
(2)
poe (ω)
(1)
qoe (ω)
(2)
qoe (ω)
(3)
qoe (ω)
peo (ω)
(1)
qeo (ω)
(2)
qeo (ω)
(3)
qeo (ω)



.

(22)
Clearly,
[p(ω), q (1) (ω), q (2) (ω), q (3) (ω)]T =
1
V (2ω)[1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 ]T ,
2
and one can show that (21) is equivalent to
V (ω)V (ω)∗ = I4 ,
ω ∈ IR2 .
(23)
6
SUBMISSION TO IEEE TRANS. IMAGE PROC.
Therefore, to construct orthogonal
{p, q (1) , q (2) , q (3) }, we need
V (ω) such that it satises (23).
(1) (2) (3)
If {p, q
, q , q } is given by (20), then V (ω) =
An (ω)An−1 (ω) · · · A1 (ω)A0 . Furthermore, if the constant matrices Ak , 0 ≤ k ≤ n, are orthogonal, then V (ω) satises (23). One can obtain that if a constant matrix Ak
only to construct
where
ηk
with
+
in
±
for
t0 = 2.22285908185090, ζ0 = 0.02319874938139,
t1 = 0.18471231877448, ζ1 = 0.60189976981183,
t2 = 0.04160159358460
(ζ2 is a free parameter),
W 1.1388 (IR2 ).
we get the smoothest
φ
with
φ ∈
We have considered orthogonal lters with more non-zero
impulse response coefcients by using more blocks
Ak (ω)
in (20). Unfortunately, in term of the smoothness of the
scaling functions, using a few more blocks
Ak (ω)
does not
yield orthogonal scaling functions with signicantly higher
2tk
3t2k − 1
, βk =
, γk = −αk − ηk − ζk ,
1 + 3t2k
1 + 3t2k
q
1
ηk = (−ζk − αk ± αk2 + 4βk2 − 3ζk2 − 2ζk αk ). (25)
2
Thus an orthogonal matrix Ck of the form (19) is given by
two free parameters tk and ζk .
(1) (2) (3)
Theorem 2: If {p, q
, q , q } is given by (20), where
each Ak (ω) is given by (17) or (18) with A being
diag(s1 , s2 , s2 , s2 )Ck diag(s3 , s4 , s4 , s4 ) for some Ck given in
(1) (2) (3)
(24), then {p, q
, q , q } is an orthogonal FIR lter bank
αk =
with 3-fold rotational symmetry.
Transforming
the matrix
U
{p, q (1) , q (2) , q (3) }
rotational symmetric biorthogonal lter banks, which give us
more exility for the construction of perfect reconstruction
lter banks.
C. Biorthogonal lter banks with 3-fold rotational symmetry
{p, q (1) , q (2) , q (3) } and {p̃, q̃ (1) , q̃ (2) , q̃ (3) } be two lter
e (ω) be their polyphase matrices dened by
banks, V (ω) and V
(1) (2) (3)
(22). Then one can shows as in §III.B that {p, q
,q ,q }
(1) (2) (3)
and {p̃, q̃
, q̃ , q̃ } are biorthogonal to each other if and
e (ω) satisfy
only if V (ω) and V
Let
given in Theorem 2 with
rotational symmetry given by a block structure. For this family
of orthogonal lter banks given by free parameters
k ≤ n,
smoothness order. In the next section, we consider 3-fold
V (ω)Ve (ω)∗ = I4 , ω ∈ IR2 .
to lter banks on the hexagonal lattice, we have
a family of orthogonal FIR hexagonal lter banks with 3-fold
t k , ζk , 0 ≤
one can design the lters with desirable properties for
one's specic applications. Here we consider the lters based
φ.
on the smoothness of the associated scaling functions
Next
{p, q (1) , q (2) , q (3) } is the FIR lter bank given by (20)
with Ak (ω) given by (17) or (18) for A to be some 4 × 4
real matrix Ak , then V (ω) = An (ω)An−1 (ω) · · · A1 (ω)A0 .
∗ −1
Suppose Ak , 0 ≤ k ≤ n are nonsingular. Then (V (ω) )
=
−T
Ãn (ω)Ãn−1 (ω) · · · Ã1 (ω)Ã0 with Ã0 = A0 and
If
we consider two examples based on this structure.
Example 1: Let
{p, q
(1)
,q
(2)
,q
(3)
}
be the orthogonal lter
bank with 3-fold rotational symmetry given by

1
C0
C1 are given by (24) for some t0 , ζ0 , t1 , ζ1 . The
lowpass lter p(ω) is given by three free parameters t0 , ζ0 and
t1 . With the choice of + in ± for η0 in (25), and
√
√
√
2 + 13
5 − 13
4 + 13
t0 =
, ζ0 =
, t1 = −
,
3
24
3
the corresponding p(ω) has sum rule order 2, and the scaling
0.9425
function φ is in W
(IR2 ). For p(ω) given by (26), the
maximum order of sum rules it can have is 2. From the
the highest Sobolev smoothness order
Example 2: Let
{p, q (1) , q (2) , q (3) }
φ
0.94254
is almost
can gain.
be the orthogonal lter
bank with 3-fold rotational symmetry given by
C2 diag(1, e2i(ω1 +ω2 ) , e−2iω1 , e−2iω2 ) ·
C1 diag(1, e−2i(ω1 +ω2 ) , e2iω1 , e2iω2 ) ·
C0 [1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 ]T ,
i(ω1 +ω2 ) −iω1 −iω2
Ãk (ω) = A−T
,e
,e
)
k (1, e
(27)
−i(ω1 +ω2 ) iω1 iω2
Ãk (ω) = A−T
, e , e ).
k (1, e
(28)
One can easily show that if
(26)
and
numerical calculations, we also nd that
or

 ei(ω1 +ω2 ) 

C1 diag(1, e−2i(ω1 +ω2 ) , e2iω1 , e2iω2 )C0 
 e−iω1  ,
e−iω2
where
by (17) with free
choice
in (25), and
of the form (19) is orthogonal, then it can be written as
diag(s1 , s2 , s2 , s2 )Ck diag(s3 , s4 , s4 , s4 ), where s` = ±1, 1 ≤
` ≤ 4, and


αk βk βk βk
 βk γk ηk ζk 

Ck = 
(24)
 βk ζk γk ηk  ,
βk ηk ζk γk
C0 , C1 , C2 are matrices dened
tk , ζk , k = 0, 1, 2. With the
parameters
dose
A−T
k .
Ak
has the form of (19), then so
{p̃, q̃ (1) , q̃ (2) , q̃ (3) } with
e
V (ω) = (V (ω)∗ )−1 also has 3-fold
Thus, by Proposition 1,
its polyphase matrix
rotational symmetry. Therefore, we have the following result.
Theorem 3: Let
{p, q (1) , q (2) , q (3) }
be the FIR lter bank
given by (20) with
or
where
Ak (ω) = Ak (1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 )
(29)
Ak (ω) = Ak (1, e−i(ω1 +ω2 ) , eiω1 , eiω2 ),
(30)
Ak , 0 ≤ k ≤ n
real matrices of
given by

p̃(ω)
1
 ei(ω1 +ω2 ) 
 q̃ (1) (ω)  1




 q̃ (2) (ω)  = 2 Ãn (2ω) · · · Ã1 (2ω)Ã0  e−iω1  ,
e−iω2
q̃ (3) (ω)

where

4 × 4 nonsingular
{p̃, q̃ (1) , q̃ (2) , q̃ (3) } is

are
the form (19). Suppose
Ã0 = A−T
0 , Ãk (ω)
(31)
is given by (27) (if
given by (29)) or by (28) (if
Ak (ω)
Ak (ω)
is
is given by (30)).
Q. JIANG: FILTER BANKS FOR HEXAGONAL DATA PROCESSING
7
{p̃, q̃ (1) , q̃ (2) , q̃ (3) } is an FIR lter bank biorthogonal
{p, q , q (2) , q (3) } and it has 3-fold rotational symmetry.
Then
to
(1)
banks with 3-fold axial symmetry are presented in
§VI.C,
§VI.B
and
respectively.
Theorem 3 provides a family of biorthogonal FIR lter
banks with 3-fold rotational symmetry. Compared with the
orthogonal lter banks given in Theorem 2, this family of
biorthogonal lter banks has more exibility for the design of
A. FIR lter banks with 3-fold axial symmetry
{p, q (1) , q (2) , q (3) }
Let
desired lters.
Example 3: Let
{p, q (1) , q (2) , q (3) }
and
{p̃, q̃ (1) , q̃ (2) , q̃ (3) }
½
be the biorthogonal lter banks with 3-fold rotational symmetry given by Theorem 3 with
n = 1,
namely,
i(ω1 +ω2 )
−iω1
p(ω) = p(Ne ω) = p(W −T ω) = p(Se −T ω),
q (1) (ω) = q (1) (Ne ω) = q (2) (R1T ω) = q (3) (R2T ω).
axial symmetry, we rst have the following proposition about
the 3-fold axial symmetry property of a lter bank.
−iω2 T
Proposition 3: A lter bank
A0 [1, e
,e
,e
] ,
(1)
(2)
(3)
T
[p̃(ω), q̃ (ω), q̃ (ω), q̃ (ω)] =
−2i(ω1 +ω2 ) 2iω1 2iω2
A−T
,e
,e
)·
1 diag(1, e
and
A1
W 0.5212 (IR2 )
iT
p, q (1) , q (2) , q (3) (Ne ω) =
h
iT
M1 p(ω), q (1) (ω), q (2) (ω), q (3) (ω) ,
h
iT
p, q (1) , q (2) , q (3) (W −T ω) =
h
iT
M2 p(ω), q (1) (ω), q (2) (ω), q (3) (ω) ,
h
iT
p, q (1) , q (2) , q (3) (Se −T ω) =
h
iT
M3 p(ω), q (1) (ω), q (2) (ω), q (3) (ω) ,
are nonsingular matrices of the form (19).
and the lowpass lters
has 3-fold
h
A0 and A1 such
φ ∈ W 1.1860 (IR2 ), φ̃ ∈
p(ω) and p̃(ω) have sum
We can choose the free parameters for
that the resulting scaling functions
{p, q (1) , q (2) , q (3) }
axial symmetry if and only if it satises
i(ω1 +ω2 ) −iω1 −iω2 T
A−T
,e
,e
] ,
0 [1, e
A0
(32)
Before we derive a block structure of lter banks with 3-fold
[p(ω), q (1) (ω), q (2) (ω), q (3) (ω)]T =
A1 diag(1, e−2i(ω1 +ω2 ) , e2iω1 , e2iω2 ) ·
where
be an FIR lter bank. Then it has
3-fold axial symmetry, that is it satises (9), if and only if
rules of order 2 and order 1 respectively. Since smoothness order of the scaling functions is still low, the selected parameters
are not provided here.
Here and in the next example, we intently construct the
(33)
(34)
(35)
scaling functions with one smoother than the other so that
one lter bank can be used as the analysis lter bank and the
where
smoother scaling function and wavelets.
Example 4: Let
{p, q (1) , q (2) , q (3) }
and
{p̃, q̃ (1) , q̃ (2) , q̃ (3) }
be the biorthogonal lter banks with 3-fold rotational symmetry given by Theorem 3 with
(1)
(2)
n = 2:
(3)
T
[p(ω), q (ω), q (ω), q (ω)] =
A2 diag(1, e2i(ω1 +ω2 ) , e−2iω1 , e−2iω2 ) ·
A1 diag(1, e−2i(ω1 +ω2 ) , e2iω1 , e2iω2 ) ·
A0 [1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 ]T ,
[p̃(ω), q̃ (1) (ω), q̃ (2) (ω), q̃ (3) (ω)]T =
2i(ω1 +ω2 ) −2iω1 −2iω2
A−T
,e
,e
)·
2 diag(1, e

1
 0
M1 = 
 0
0

1
 0
M3 = 
 0
0
other can be used as the synthesis lter bank which requires
The
proof
of
0
1
0
0
0
0
0
1
0
0
1
0
0
1
0
0


0
1
 0
0 
 , M2 = 
 0
1 
0
0

0
0 
.
0 
1
Proposition
3
is
given
0
0
0
1
0
0
1
0

0
1 
,
0 
0
(36)
in
Appendix
Next, we consider the lter bank
{p, q (1) , q (2) , q (3) }
can
of
be
given
by
the
product
block
B.
which
matrices.
As-
A0 ,
A1 and A2 such that the resulting scaling functions φ ∈
W 1.5294 (IR2 ), φ̃ ∈ W 0.3859 (IR2 ) and the lowpass lters p(ω)
and p̃(ω) have sum rules of order 2 and order 1 respectively.
[p(ω), q (1) (ω), q (2) (ω), q (3) (ω)]T can be writ(1)
(2)
(3)
T
ten as B(2ω)[p0 (ω), q0 (ω), q0 (ω), q0 (ω)] , where B(ω)
is a 4 × 4 matrix with trigonometric polynomial entries
(1) (2) (3)
and {p0 , q0 , q0 , q0 } is another FIR lter bank. If both
(1) (2) (3)
{p, q (1) , q (2) , q (3) } and {p0 , q0 , q0 , q0 } have 3-fold axial
symmetry, then Proposition 3 implies that B(ω) satises
½
B(Ne ω) = M1 B(ω)M1 , B(W −T ω) = M2 B(ω)M2 ,
(37)
B(Se −T ω) = M3 B(ω)M3 ,
The selected parameters are provided in Appendix A.
where
M1 , M2
(36).
Since
−2i(ω1 +ω2 ) 2iω1 2iω2
A−T
,e
,e
)·
1 diag(1, e
i(ω1 +ω2 ) −iω1 −iω2 T
A−T
,e
,e
] ,
0 [1, e
where
A0 , A1
and
A2
are nonsingular matrices of the form
(19). In this case, we can select the free parameters for
IV. O RTHOGONAL AND BIORTHOGONAL FIR FILTER
BANKS WITH
3- FOLD AXIAL SYMMETRY
sume
diag(1, e
and
both
−i(ω1 +ω2 )
M3
are
diag(1, e
, eiω1 , eiω2 )
the matrices
i(ω1 +ω2 ) −iω1
,e
satisfy
(37),
dened
, e−iω2 )
they
by
and
could
be used to build the block matrices. Next we will use
In this section we study the construction of orthogonal and
B(ω) = B
biorthogonal lter banks with 3-fold axial symmetry. The 3fold axial symmetry of lter banks is discussed in
that
§VI.A,
and block structures of orthogonal and biorthogonal FIR lter
diag(1, e
i(ω1 +ω2 )
, e−iω1 , e−iω2 )
(38)
and
B(ω) = B
diag(1, e
−i(ω1 +ω2 )
, eiω1 , eiω2 )
(39)
8
SUBMISSION TO IEEE TRANS. IMAGE PROC.
B is a 4×4 (real) constant matrix.
B(ω) dened by (38) or by (39), B(ω)
if and only if B has the form:


b11 b12 b12 b12
 b21 b22 b23 b23 

B=
(40)
 b21 b23 b22 b23  .
b21 b23 b23 b22
as the block matrices, where
orthogonal hexagonal lter banks with 3-fold axial symmetry
One can verify that for
given by a block structure. This structure is the one given by
satises (37)
Allen in [4], and it is referred here as Allen's structure. [4]
and [26] constructed several orthogonal lter banks based on
the compaction of lters. Here we consider the lters based
on the smoothness of the associated scaling functions.
Example 5: Let
{p(ω), q (1) (ω), q (2) (ω), q (3) (ω)} be the or-
thogonal lter bank given by

Based on the above discussion, we reach the following
theorem on the lter banks with 3-fold axial symmetry.
Theorem 4: If
{p, q (1) , q (2) , q (3) }
is given by



p(ω)
1
 q (1) (ω)  1
 ei(ω1 +ω2 )



 q (2) (ω)  = 2 Bn (2ω) · · · B1 (2ω)B0  e−iω1
e−iω2
q (3) (ω)





1
 ei(ω1 +ω2 ) 

B1 diag(1, e−2i(ω1 +ω2 ) , e2iω1 , e2iω2 )B0 
 e−iω1  ,
e−iω2
(44)
B0 , B1 given by (42) for some b0 , b1 ∈ IR. The hexagonal
with
lter bank corresponding to this lter bank is called the L(41)
n ∈ Z+ , where B0 is a 4 × 4 constant matrix of the
form (40), each Bk (ω) is given by (38) or (39) for some 4 × 4
(1) (2) (3)
constant matrix Bk of the form (40), then {p, q
,q ,q }
for some
Trigon of the R-Trigon in [4]. One can show that with the
choices of
b0 =
has 3-fold axial symmetry.
√
1
(2 − 13),
9
b1 =
√
1
(4 − 13),
3
p(ω) has sum rule order 2, and the
W 0.9425 (IR2 ). Actually, this resulting
the resulting lowpass lter
scaling function
B. Allen's orthogonal lter banks
In this subsection, we show that the block structure in
(41) will yield 3-fold axial symmetric orthogonal FIR lter
banks, which were studied in [4], [26]. For an FIR lter
p(ω)
p11 = p0−1 = p−10
p22 = p0−2 = p−20
of the form (40) is
orthogonal, then it can be written as (refer to [4])
3b2k − 1

1
 2bk
Bk =
1 + 3b2k  2bk
2bk
2bk
1 + b2k
−2b2k
−2b2k

2bk
−2b2k 

−2b2k 
1 + b2k
2bk
−2b2k
1 + b2k
−2b2k
(42)
with
bk ∈ IR.
Theorem 5: If
(43)
2bk
−(1 + b2k )
2b2k
2b2k
{p, q (1) , q (2) , q (3) }
Bk (ω) = Bk diag(1, e

2bk
2bk

2b2k
2b2k

2
2

2bk
−(1 + bk )
2
2
−(1 + bk )
2bk
is given by (41) with
i(ω1 +ω2 )
,e
−iω1
,e
−iω2
),
p10 = p01 = p−1−1
√
19 + 13
,
=
48
√
1 − 13
=
,
16
(`)
pk , q k
of the
√
13 − 13
=
,
16
√
−5 + 13
=
,
48
p23 = p32 = p1−2 = p−21 = p−1−3 = p−3−1
√
√
55 + 9 13
3 + 13 (1)
(1)
, q11 =
,
q00 = −
16
√ 48
1 − 13
(1)
(1)
(1)
q10 = q01 = q−1−1 =
,
√ 16
√
5 + 7 13
−21 + 5 13 (1)
(1)
(1)
q0−1 = q−10 =
, q22 = −
,
48 √
48
−17 + 5 13
(1)
(1)
q0−2 = q−20 =
,
√48
−9 + 13
(1)
(1)
q32 = q23 =
,
48
√
11 − 3 13
(1)
(1)
(1)
(1)
q1−2 = q−21 = q−1−3 = q−3−1 =
,
48
or it can be written as
1
Bk =
·
1 + 3b2k

±(1 − 3b2k )

±2bk


±2bk
±2bk
√
13 + 3 13
=
,
16
p00
hence, (41) gives a family of orthogonal lter banks. One

is the lowpass lter of sum rule order 2 in Example
lters are
{p, q (1) , q (2) , q (3) }, let V (ω) denote its polyphase matrix
(1) (2) (3)
dened in (22). If {p, q
, q , q } is given by (41), then
V (ω) = Bn (ω)Bn−1 (ω) · · · B1 (ω)B0 . Thus, if the constant
matrices Bk are orthogonal, then V (ω) satises (23), and
Bk
is in
1. The non-zero impulse response coefcients
bank
can check that if a constant matrix
φ
and
(2)
qk
,
(3)
qk
are given by (8).
We have also considered orthogonal lters with more nonzero impulse response coefcients by using more blocks
or
Bk (ω) = Bk diag(1, e
−i(ω1 +ω2 )
Bk , 0 ≤ k ≤ n are given
{p, q (1) , q (2) , q (3) } is an orthogonal
where
,e
iω1
,e
iω2
),
by (42) or (43), then
lter bank with 3-fold
axial symmetry.
Transforming
the matrix
U
{p, q (1) , q (2) , q (3) }
given in Theorem 5 with
to hexagonal lter banks, we have a family of
Bk (ω)
Bk (ω)
in (41). Again, we nd that using a few more blocks
does not yield orthogonal scaling functions with sig-
nicantly higher smoothness order.
{p, q (1) , q (2) , q (3) } be the FIR lter bank given by
−1 −1
(41) with Bk (ω) = Bk diag(1, z1 z2 , z1 , z2 ) (or Bk (ω) =
−1 −1
Bk diag(1, z1 z2 , z1 , z2 )), where Bk , 0 ≤ k ≤ n are 4 × 4
nonsingular constant real matrices of the form (40), and z1 , z2
Let
Q. JIANG: FILTER BANKS FOR HEXAGONAL DATA PROCESSING
9
3ct1 ).
Fig. 4.
With
Non-zero impulse response coefcients of lters with Allen's
dened
by
(1).
Then
{p̃(ω), q̃ (1) (ω), q̃ (2) (ω), q̃ (3) (ω)}
given by
B̃k (ω) = Bk−T diag(1, z1−1 z2−1 , z1 , z2 ) (or B̃k (ω) =
−T
Bk diag(1, z1 z2 , z1−1 , z2−1 ) correspondingly), is biorthogonal
(1) (2) (3)
to {p, q
, q , q } and it has 3-fold axial symmetry. There-
while if
free parameters. Compared with the orthogonal lter banks of
3-fold axial symmetry given in Theorem 5, these biorthogonal
lter banks give us more exibility for the design of desired
lters. However, in terms of the smoothness of the scaling
φ
and
φ̃,
they do not yield smooth
φ, φ̃
with
reasonable supports. Because of this, we introduce in the next
t3 , t4
are given by (47), then
b̃
z2
d˜
z2
d˜
z2
c̃
z2


,


−1
e
D(ω)
= (D(ω)∗ )
2
e
·
D(ω)
=
(t5 − t2 )(at5 + 2at2 − 3bt1 )

ã1 − cab̃1 (z1−1 z2−1 + z1 + z2 )

 ẽ1 − ac (c̃1 z1−1 z2−1 + d˜1 z1 + d˜1 z2 )

 ẽ1 − ac (d˜1 z1−1 z2−1 + c̃1 z1 + d˜1 z2 )
ẽ1 − ac (d˜1 z1−1 z2−1 + d˜1 z1 + c̃1 z2 )
of the form (40),
one has a family of biorthogonal FIR lter banks given by
functions
b̃
z1
d˜
z1
c̃
z1
d˜
z1
ã = (t3 + 2t4 )(t3 − t4 ), b̃ = −t1 (t3 − t4 ),
c̃ = at3 + at4 − 2ct1 , d˜ = ct1 − at4 , ẽ = −c(t3 − t4 ),
where
Bk
(48)

where
1
B̃n (2ω) · · · B̃1 (2ω)B0−T [1, z1−1 z2−1 , z1 , z2 ]T ,
2
fore, by choosing nonsingular matrices
given in (46),
2
e
D(ω)
=
·
(t3 − t4 )(at3 + 2at4 − 3ct1 )

ã − bab̃ (z1 z2 + z11 + z12 ) b̃z1 z2

 ẽ − ab (c̃z1 z2 + zd˜ + zd˜ ) c̃z1 z2

1
2
 ẽ − b (dz
˜ 1 z2 + c̃ + d˜ ) dz
˜ 1 z2

a
z1
z2
b ˜
d˜
c̃
˜ 1 z2
ẽ − a (dz1 z2 + z1 + z2 ) dz
structure (left) and those with new structure (right)
are
t2 , t5
1
16 (t3
− t4 )2 (at3 + 2at4 −
−1
e
D(ω)
= (D(ω)∗ ) is
In these two cases, det(D(ω)) is
is
(49)

b̃1
c̃1
d˜1
d˜1
b̃1
d˜1
c̃1
d˜1
b̃1
d˜1
d˜1


,

c̃1
where
subsection another family of biorthogonal lter banks with 3-
ã1 = (t5 + 2t2 )(t5 − t2 ), c̃1 = at2 + at5 − 2bt1 ,
b̃1 = −t1 (t5 − t2 ), d˜1 = bt1 − at2 , ẽ1 = −b(t5 − t2 ).
fold axial symmetry.
C. Biorthogonal lter banks with 3-fold axial symmetry
Theorem 6: Let
In this subsection we introduce another family of biorthogonal lter banks with 3-fold axial symmetry which is based
on the following block matrix:

a
1
t
1
D(ω) = 
2  t1
t1
b + cz1 z2
t5 + t3 z 1 z 2
t2 + t4 z 1 z 2
t2 + t4 z 1 z 2
b+
t2 +
t5 +
t2 +
cz1−1
t4 z1−1
t3 z1−1
t4 z1−1
b+
t2 +
t2 +
t5 +
cz2−1
t4 z2−1
t4 z2−1
t3 z2−1


,

z1 , z2
a, b, c, tj , 1
j
≤
5,
are
real
numbers,
and
D(ω) satis{p, q (1) , q (2) , q (3) }
are given by (1). One can verify that
es (37).
with
≤
D(ω)
denser
yields primal lter banks
non-zero
impulse
response
coefcients
be the lter bank given
by



p(ω)
1
 q (1) (ω)  1
 ei(ω1 +ω2 )



 q (2) (ω)  = 2 Dn (±2ω) · · · D0 (±2ω)  e−iω1
e−iω2
q (3) (ω)




(50)
(45)
where
{p, q (1) , q (2) , q (3) }
(and
hence, they will produce smoother scaling functions and
wavelets). For example, the black dots in the left part of
Fig. 4 indicate the non-zero impulse response coefcients
for some
n ∈ Z+ ,
with parameters
Dk (ω), 0 ≤ k ≤ n are given by (45)
(1) (2) (3)
satisfying (46) or (47). If {p̃, q̃
, q̃ , q̃ }
where
is the lter bank given by


p̃(ω)
1
 ei(ω1 +ω2 )
 q̃ (1) (ω)  1
e
e



 q̃ (2) (ω)  = 2 Dn (±2ω) · · · D0 (±2ω)  e−iω1
e−iω2
q̃ (3) (ω)



,

B1 (2ω)B0 [1, eiω1 eiω2 , e−iω1 , e−iω2 ]T , while
the non-zero impulse response coefcients of p(ω) from
D2 (2ω)D1 (2ω)[1, eiω1 eiω2 , e−iω1 , e−iω2 ]T are shown in the
where
right part of Fig. 4 as the black dots.
FIR lter banks with 3-fold axial symmetry and they are
tj such that det(D(ω)) is a
−1
e
constant and hence D(ω)
= (D(ω)∗ ) is a matrix with each
entry being a (Laurent) polynomial of z1 , z2 . The possible
biorthogonal to each other.
choices are
lter banks for one's particular applications. Here we provide
of
p(ω)
(51)
from
We can choose some special
t2 = t5 =
or
t3 = t4 =
bt1
,
a
ct1
.
a
(46)
or by
e k (ω) = (Dk (ω)∗ )−1 , 0 ≤ k ≤ n, are given by (48)
D
(1) (2) (3)
(49), then {p, q
, q , q } and {p̃, q̃ (1) , q̃ (2) , q̃ (3) } are
Again, with a family of symmetric biorthogonal FIR lter
banks available in Theorem 6, one can design biorthogonal
two lter banks based on the smoothness of the associated
scaling functions
φ and φ̃. In the following two examples, as in
Examples 3 and 4, we intently construct the scaling functions
(47)
with one smoother than the other.
10
SUBMISSION TO IEEE TRANS. IMAGE PROC.
Example 6: Let
select parameters
[p(ω), q (1) (ω), q (2) (ω), q (3) (ω)]T =
1
D1 (−2ω)D0 (2ω)[1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 ]T ,
2
[p̃(ω), q̃ (1) (ω), q̃ (2) (ω), q̃ (3) (ω)]T =
1e
e 0 (2ω)[1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 ]T ,
D1 (−2ω)D
2
for
A1
are
a11 = 1.74210812926244, a12 = 0.41721745109326,
a21 = −2.18192416765921, a22 = 2.51403080377387,
a23 = 0.55679974918931, a24 = 0.55661861238169;
and select parameters
be the biorthogonal FIR lter banks given in Theorem 6 with
n = 1
aij
aij
for
A2
are
a11 = 0.51226668946537, a12 = −0.00417155767962,
a21 = −0.83390119661419, a22 = −0.33354163795606,
a23 = 1.76565003212551, a24 = 0.28836976159098.
and the parameters satisfying (46). Then we can
choose free parameters such that the resulting scaling functions
φ ∈ W 1.5105 (IR2 ), φ̃ ∈ W 0.4067 (IR2 ) and the lowpass lters
p(ω), p̃(ω) have sum rules of order 2 and order 1 respectively.
The impulse response coefcients of the resulting lter banks
are provided in Appendix C.
A PPENDIX B
Example 7: Let
[p(ω), q (1) (ω), q (2) (ω), q (3) (ω)]T =
1
D2 (2ω)D1 (−2ω)D0 (2ω)[1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 ]T ,
2
[p̃(ω), q̃ (1) (ω), q̃ (2) (ω), q̃ (3) (ω)]T =
1e
e 1 (−2ω)D
e 0 (2ω)[1, ei(ω1 +ω2 ) , e−iω1 , e−iω2 ]T ,
D2 (2ω)D
2
Proof of Proposition 3. Using
using the following facts (i)-(v) about relationship among the
(i).
(iii).
(v).
free parameters such that the corresponding scaling functions
banks are provided in provided in Appendix D.
and
R2 ,
one can show that (32) leads
to (33)-(35):
n = 2 and the parameters satisfying (46). Then we can choose
The impulse response coefcients of these two resulting lter
Ne , W, Se , R1
matrices
be the biorthogonal FIR lter banks given in Theorem 6 with
φ ∈ W 1.7055 (IR2 ), φ̃ ∈ W 0.5869 (IR2 ), and the lowpass lters
p(ω), p̃(ω) have sum rules of order 2 and order 1 respectively.
R1 = W Ne , R2 = Se Ne , one
can easily show that (33)-(35) imply (32). On the other hand,
Ne = R1T Ne R2−T ; (ii). R2−T = Ne W −T ;
R1−T = Ne Se −T ; (iv). R1−T W −T = Ne R1−T ;
R2−T W −T = Ne R2−T .
Here we just show the formulas in (33)-(35) for
p, q
of other formulas in (33)-(35) for
For
q (2) ,
(1)
and
q (2) . The proof
q (3) is similar.
we have
q (2) (Ne ω) = q (1) (R1−T Ne ω)
V. C ONCLUSION
= q (1) (Ne R2−T ω)(by
In this paper we obtain block structures of FIR lter
q
banks with 3-fold rotational symmetry and FIR lter banks
=
with 3-fold axial symmetry. These block structures yield
= q (1) (R2−T ω) = q (3) (ω);
−T
(W ω) = q (R1 W −T ω)
q (1) (Ne R1−T ω)(by (iv)) = q (1) (R1−T ω) = q (2) (ω);
(2)
−T
(i))
(1)
q (2) (Se −T ω) = q (1) (R1−T Se −T ω)
orthogonal/biorthogonal hexagonal lter banks with 3-fold
= q (1) (Ne R2−T ω)(by
rotational symmetry and orthogonal/biorthogonal hexagonal
(iii))
= q (1) (R2−T ω) = q (3) (ω),
lter banks with 3-fold axial symmetry. The construction of
orthogonal and biorthogonal FIR lters with scaling functions
as desired.
having optimal smoothness is discussed and several orthogonal/biorthogonal lters banks are presented. Our future work
is to apply these hexagonal lter banks for hexagonal image
A PPENDIX C
processing applications such as image enhancement and edge
detection. In this paper we just consider the hexagonal lters
with the dyadic renement. In the future, we will also consider
the hexagonal lters with the
√
3
and
√
7
renements and
The
biorthogonal
lters
in
W 1.5105 (IR2 ), φ̃ ∈ W 0.4067 (IR2 ):
sponses of p(ω) are
Example
5
with
φ
∈
the non-zero impulse re-
study the construction of idealized hexagonal tight frame lter
banks which contain the “idealized” highpass lters with nice
p00 = 0.98832654285769,
frequency localizations.
p01 = p10 = p−1−1 = 0.49156433996017,
p−10 = p0−1 = p11 = 0.50750780089008,
A PPENDIX A
Selected parameters in Example 4: selected
aij
for
A0
are
p−2−2 = p02 = p20 = 0.00389115238077,
p−3−2 = p−2−3 = p13 = p31 =
a11 = 0.53418431122656, a12 = 0.26151738104791,
a21 = 0.01459363514388, a22 = 1.24160756693777,
p−12 = p2−1 = −0.00250260029669,
p−3−3 = p30 = p03 = p−2−1 = p−1−2 = p−11 = p1−1
a23 = 0.03247147746833, a24 = 0.03247589636386;
= p21 = p12 = 0.00197768658105;
Q. JIANG: FILTER BANKS FOR HEXAGONAL DATA PROCESSING
q (1) (ω)
the non-zero impulse responses of
11
are
sponse coefcients of
p01 = p10 = p−1−1 = 0.49156433996017,
p−10 = p0−1 = p11 = 0.50750780089008,
p−2−2 = p02 = p20 = 0.00389115238077,
p−3−2 = p−2−3 = p13 = p31 =
p−12 = p2−1 = −0.00250260029669,
p−3−3 = p30 = p03 = p−2−1 = p−1−2 = p−11 = p1−1
= p21 = p12 = 0.00197768658105;
(1)
q01 = q10 = 0.29863493249023,
(1)
q−1−1 = −0.80239194082786,
(1)
(1)
(1)
(1)
(1)
q11 = q−10 = q0−1 = 0.01803315183284,
q02 = q20 = −0.10159057223583,
(1)
q−2−2 = 0.29676935480645,
(1)
(1)
(1)
(1)
(1)
(1)
q1−1 = q12 = q21 = q−11
= q03 = q30 = −0.05163362721657,
(1)
q31 =
(1)
q−2−1
(1)
q−3−2
of
(2)
(3)
qk , qk
p̃(ω) are
and
the non-zero impulse response coefcients of
(1)
(1)
(1)
q13 = q2−1 = q−12 = 0.06533812386146,
(1)
(1)
= q−1−2 = q−3−3 = 0.15083366397237,
(1)
= q−2−3 = −0.19086764092259;
p̃00 = 2.07524327772965,
p̃01 = p̃10 = p̃−1−1 = 0.57048205301141,
p̃−10 = p̃0−1 = p̃11 = 0.72714621546633,
p̃20 = p̃02 = p̃−2−2 = −0.14011100395303,
p̃22 = p̃−20 = p̃0−2 = −0.36957363065319,
p̃13 = p̃31 = p̃2−1 = p̃−12 =
p̃−3−2 = p̃−2−3 = −0.14881413423887,
p̃2−2 = p̃−22 = p̃24 = p̃42 =
p̃−4−2 = p̃−2−4 = 0.07563510434817;
q̃ (1) (ω)
and
(1)
(1)
q̃00 = 8.03759108770683, q̃−1−1 = −7.85514652959852,
(1)
(1)
q̃01 = q̃10 = −4.09997773834547,
(1)
(1)
(1)
q̃−10 = q̃0−1 = q̃11 = −0.03023283293225,
(1)
(1)
q̃02 = q̃20 = 1.00695892898345,
(1)
q̃−2−2 = 1.92923241081903,
(1)
(1)
(1)
q̃22 = q̃−20 = q̃0−2 = 0.0153659024747,
(2)
q̃k
,
(3)
q̃k
are given by (8).
A PPENDIX D
The
biorthogonal
lters
in
W 1.5105 (IR2 ), φ̃ ∈ W 0.4067 (IR2 ):
Example
6
,
(3)
qk
with
φ
∈
the non-zero impulse re-
are
are given by (8); the non-zero impulse response
p̃(ω)
are
p̃00 = 2.07524327772965,
p̃01 = p̃10 = p̃−1−1 = 0.57048205301141,
p̃−10 = p̃0−1 = p̃11 = 0.72714621546633,
p̃20 = p̃02 = p̃−2−2 = −0.14011100395303,
p̃22 = p̃−20 = p̃0−2 = −0.36957363065319,
p̃13 = p̃31 = p̃2−1 = p̃−12 =
p̃−3−2 = p̃−2−3 = −0.14881413423887,
p̃2−2 = p̃−22 = p̃24 = p̃42 =
p̃−4−2 = p̃−2−4 = 0.07563510434817;
the non-zero impulse response coefcients of
(1)
(1)
(1)
(1)
q̃31 = q̃13 = q̃2−1 = q̃−12 = 1.06950715506265,
(1)
(1)
(1)
(1)
q̃2−2 = q̃−22 = q̃24 = q̃42 = −0.54357931582244,
(1)
(1)
q̃−3−2 = q̃−2−3 = 2.04906854466516,
(1)
(1)
q̃−2−4 = q̃−4−2 = −1.04144350256089;
and
(2)
qk
coefcients of
are
q (1) (ω)
(1)
q00 = 0.03511796779580,
(1)
(1)
q01 = q10 = 0.29863493249023,
(1)
q−1−1 = −0.80239194082786,
(1)
(1)
(1)
q11 = q−10 = q0−1 = 0.01803315183284,
(1)
(1)
q02 = q20 = −0.10159057223583,
(1)
q−2−2 = 0.29676935480645,
(1)
(1)
(1)
(1)
q1−1 = q12 = q21 = q−11
(1)
(1)
= q03 = q30 = −0.05163362721657,
(1)
(1)
(1)
(1)
q31 = q13 = q2−1 = q−12 = 0.06533812386146,
(1)
(1)
(1)
q−2−1 = q−1−2 = q−3−3 = 0.15083366397237,
(1)
(1)
q−3−2 = q−2−3 = −0.19086764092259,
are given by (8); the non-zero impulse responses
the non-zero impulse responses of
are
p00 = 0.98832654285769,
(1)
q00 = 0.03511796779580,
(1)
p(ω)
q̃ (1) (ω)
are
(1)
(1)
q̃00 = 8.03759108770683, q̃−1−1 = −7.85514652959852,
(1)
(1)
q̃01 = q̃10 = −4.09997773834547,
(1)
(1)
(1)
q̃−10 = q̃0−1 = q̃11 = −0.03023283293225,
(1)
(1)
q̃02 = q̃20 = 1.00695892898345,
(1)
q̃−2−2 = 1.92923241081903,
(1)
(1)
(1)
q̃22 = q̃−20 = q̃0−2 = 0.0153659024747,
(1)
(1)
(1)
(1)
q̃31 = q̃13 = q̃2−1 = q̃−12 = 1.06950715506265,
(1)
(1)
(1)
(1)
q̃2−2 = q̃−22 = q̃24 = q̃42 = −0.54357931582244,
(1)
(1)
q̃−3−2 = q̃−2−3 = 2.04906854466516,
(1)
(1)
q̃−2−4 = q̃−4−2 = −1.04144350256089,
12
SUBMISSION TO IEEE TRANS. IMAGE PROC.
and
(2)
(1)
(3)
(1)
q̃k = q̃R1 k , q̃k = q̃R2 k .
ACKNOWLEDGMENT
The author thanks James D. Allen and Xiqiang Zheng for
providing preprints [4] and [9]. The author is very grateful to
the anonymous reviewers for making many valuable suggestions that signicantly improve the presentation of the paper.
[21] E.A. Adelson, E. Simoncelli, and R. Hingorani, “Orthogonal pyramid
transforms for image coding”, In SPIE Visual Communications and Image
Processing II (1987), vol. 845, 1987, pp. 50–58.
[22] A.B. Watson and A. Ahumada, Jr., “A hexagonal orthogonal-oriented
pyramid as a model of image presentation in visual cortex”, IEEE Trans.
Biomed. Eng., vol. 36, no. 1, pp. 97–106, Jan. 1989.
[23] E. Simoncelli and E. Adelson, “Non-separable extensions of quadrature
mirror lters to multiple dimensions”, Proceedings of the IEEE, vol. 78,
no. 4, pp. 652–664, Apr. 1990.
[24] H. Xu, W.-S. Lu, and A. Antoniou, “A new design of 2-D non-separable
hexagonal quadrature-mirror-lter banks”, in Proc. CCECE, Vancouver,
R EFERENCES
[1] D.P. Petersen and D. Middleton, “Sampling and reconstruction of wavenumber-limited functions in N-dimensional Euclidean spaces”, Information and Control, vol. 5, no. 4, pp. 279-323, Dec. 1962.
[2] R.M. Mersereau, “The processing of hexagonal sampled two-dimensional
signals”, Proc. IEEE, vol. 67, no. 6, pp. 930–949, Jun. 1979.
Sep. 1993, pp. 35–38.
[25] A. Cohen and J.-M. Schlenker, “Compactly supported bidimensional
wavelets bases with hexagonal symmetry”, Constr. Approx., vol. 9, no.
2/3, pp. 209–236, Jun. 1993.
[26] J.D. Allen, “Perfect reconstruction lter banks for the hexagonal grid”,
In Fifth International Conference on Information, Communications and
Signal Processing 2005, Dec. 2005, pp. 73–76.
[3] R.C. Staunton and N. Storey, “A comparison between square and hexag-
[27] C. Cabrelli, C. Heil, and U. Molter, “Accuracy of lattice translates of
onal sampling methods for pipeline image processing”, In Proc. of SPIE
several multidimensional renable functions”, J. Approx. Theory, vol. 95,
Vol. 1194, Optics, Illumination, and Image Sensing for Machine Vision
IV, 1990, pp. 142–151.
[4] J.D. Allen, “Coding transforms for the hexagon grid”, Ricoh Calif.
Research Ctr., Technical Report CRC-TR-9851, Aug. 1998.
[5] G. Tirunelveli, R. Gordon, and S. Pistorius, “Comparison of square-
no. 1, pp. 5–52, Oct. 1998.
[28] R.Q. Jia, “Convergence of vector subdivision schemes and construction
of biorthogonal multiple wavelets”, In Advances in Wavelets, SpringerVerlag, Singapore, 1999, pp. 199–227.
[29] W. Lawton, S.L. Lee, and Z.W. Shen, “Stability and orthonormality of
pixel and hexagonal-pixel resolution in image processing”, in Proceedings
multivariate renable functions”, SIAM J. Math. Anal., vol. 28, no. 4, pp.
of the 2002 IEEE Canadian Conference on Electrical and Computer
999–1014, Jul. 1997.
Engineering, vol. 2, May 2002, pp. 867–872.
[6] D. Van De Ville, T. Blu, M. Unser, W. Philips, I. Lemahieu, and R. Van de
Walle, “Hex-splines: a novel spline family for hexagonal lattices”, IEEE
Tran. Image Proc., vol. 13, no. 6, pp. 758–772, Jun. 2004.
[7] L. Middleton and J. Sivaswarmy, Hexagonal Image Processing: A Practical Approach, Springer, 2005.
[8] X.J. He and W.J. Jia, “Hexagonal Structure for Intelligent Vision”, in
Proceedings of the 2005 First International Conference on Information
and Communication Technologies, Aug. 2005, pp. 52–64.
[30] C.K. Chui and Q.T. Jiang, “Multivariate balanced vector-valued renable
functions”, in Mordern Development in Multivariate Approximation,
ISNM 145, Birhhäuser Verlag, Basel, 2003, pp. 71–102.
[31] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice
Hall, 1995.
[32] G. Strang and T. Nguyen, Wavelets and Filter Banks, WellesleyCambridge Press, 1996.
[33] R.Q. Jia, “Approximation properties of multivariate wavelets”, Math.
Comp., vol. 67, no. 222, pp. 647–665, Apr. 1998.
[9] X.Q. Zheng, G.X. Ritter, D.C. Wilson, and A. Vince, “Fast discrete
[34] Q.T. Jiang, “Orthogonal multiwavelets with optimum time–frequency
Fourier transform algorithms on regular hexagonal structures”, preprint,
resolution”, IEEE Trans. Signal Proc., vol. 46, no. 4, pp. 830–844, Apr.
University of Florida, Gainsville, FL, 2006.
1998.
[10] R.C. Staunton, “The design of hexagonal sampling structures for im-
[35] Q.T. Jiang, “On the design of multilter banks and orthonormal multi-
age digitization and their use with local operators”, Image and Vision
wavelet bases”, IEEE Trans. Signal Proc., vol. 46, no. 12, pp. 3292–3303,
Computing, vol. 7, no. 3, pp. 162–166, Aug. 1989.
Dec. 1998.
[11] L. Middleton and J. Sivaswarmy, “Edge detection in a hexagonal-image
[36] R.Q. Jia and S.R. Zhang, “Spectral properties of the transition operator
processing framework”, Image and Vision Computing, vol. 19, no. 14,
associated to a multivariate renement equation”, Linear Algebra Appl.,
pp. 1071-1081, Dec. 2001.
[12] A.F. Laine, S. Schuler, J. Fan, and W. Huda, “Mammographic feature
vol. 292, no. 1, pp. 155–178, May 1999.
[37] R.Q. Jia and Q.T. Jiang, “Spectral analysis of transition operators and its
enhancement by multiscale analysis”, IEEE Trans. Med Imaging, vol. 13,
applications to smoothness analysis of wavelets”, SIAM J. Matrix Anal.
no. 4, pp. 725–740, Dec. 1994.
Appl., vol. 24, no. 4, pp. 1071–1109, 2003.
√
[13] A.F. Laine and S. Schuler, “Hexagonal wavelet representations for rec-
[38] Q.T. Jiang and P. Oswald, “Triangular
3-subdivision
schemes: the
ognizing complex annotations”, in Proceedings of the IEEE Conference
regular case”, J. Comput. Appl. Math., vol. 156, no. 1, pp. 47–75, Jul.
on Computer Vision and Pattern Recognition, Seattle, WA, Jun. 1994, pp.
2003.
740-745.
[14] A.P. Fitz and R. Green, “Fingerprint classication using hexagonal fast
Fourier transform”, Pattern Recognition, vol. 29, no. 10, pp. 1587–1597,
1996.
[15] S. Periaswamy, “Detection of microcalcication in mammograms using
hexagonal wavelets”, M.S. thesis, Dept. of Computer Science, Univ. of
South Carolina, Columbia, SC, 1996.
[16] R.C. Staunton, “One-pass parallel hexagonal thinning algorithm”, IEE
Qingtang Jiang received his B.S. and M.S. degrees
Proceedings on Vision, Image and Signal Processing, vol. 148, no. 1, pp.
from Hangzhou University, Hangzhou, China, in
45–53, Feb. 2001.
1986 and 1989, respectively, and his Ph.D. degree
[17] A. Camps, J. Bara, I.C. Sanahuja, and F. Torres, “The processing
of hexagonally sampled signals with standard rectangular techniques:
application to 2-D large aperture synthesis interferometric radiometers”,
IEEE Trans. Geoscience and Remote Sensing, vol. 35, no. 1, pp. 183–190,
Jan. 1997.
[18] E. Anterrieu, P. Waldteufel, and A. Lannes, “Apodization functions for
2-D hexagonally sampled synthetic aperture imaging radiometers”, IEEE
Trans. Geoscience and Remote Sensing, vol. 40, no. 12, pp. 2531–2542,
Dec. 2002.
[19] D. White, A.J. Kimberling, and W.S. Overton, “Cartographic and
geometric components of a global sampling design for environmental
monitoring”, Cartography and Geographic Information Systems, vol. 19,
no. 1, pp. 5-22, 1992.
[20] K. Sahr, D. White, and A.J. Kimerling, “Geodesic discrete global grid
systems”, Cartography and Geographic Information Science, vol. 30, no.
2, pp. 121–134, Apr. 2003.
PLACE
PHOTO
HERE
from Peking University, Beijing, China, in 1992,
all in mathematics. He was with Peking University,
Beijing, China from 1992 to 1995. He was an NSTB
post-doctoral fellow and research fellow at the National University of Singapore from 1995 to 1997.
Before joined University of Missouri - St. Louis
in 2002, he held visting positions at Unversity of
Alberta, Canada, and West Virginia University, USA.
He is now a full professor in the Department of Math and Computer Science
at University of Missouri - St. Louis. Dr. Jiang's current research interests
include time–frequency analysis, wavelet theory and its applications, lter
bank design, signal classication, image processing and surface subdivision.