Two dimensional nonseparable adaptive directional lifting structure of discrete wavelet transform
ABSTRACT In this paper, we propose a two dimensional (2D) nonseparable adaptive directional lifting (ADL) structure for discrete wavelet transform (DWT) and its image coding application. Although a 2D nonseparable lifting structure of 9/7 DWT has been proposed by interchanging some lifting, we generalize a polyphase representation of 2D nonseparable lifting structure of DWT. Furthermore, by introducing the adaptive directional filteringingto the generalized structure, the 2D nonseparable ADL structure is realized and applied into image coding. Our proposed method is simpler than the 1D ADL, and can select the different transforming direction with 1D ADL. Through the simulations, the proposed method is shown to be efficient for the lossy and lossless image coding performance. key words: two dimensional nonseparable lifting structure, discrete wavelet transform, adaptive directional filtering, lossy and lossless image coding

Dataset: ICIP13 NonSep3D cmr
 SourceAvailable from: Masahiro Iwahashi
Conference Paper: Non Separable 3D Lifting Structure Compatible with Separable Quadruple Lifting DWT
[Show abstract] [Hide abstract]
ABSTRACT: This report reduces the total number of lifting steps in a 3D quadruple lifting DWT (discrete wavelet transform). In the JPEG 2000 international standard, the 9/7 quadruple lifting DWT has been widely utilized for image data compression. It has been also applied to volumetric medical image data analysis. However, it has long delay from input to output due to cascading four (quadruple) lifting steps per dimension. We reduce the total number of lifting steps introducing 3D direct memory accessing under the constraint that it has backward compatibility with the conventional DWT in JPEG 2000. As a result, the total number of lifting steps is reduced from 12 to 8 (67 %) without significant degradation of data compression performance.AsiaPacific Signal and Information Processing Association (APSIPA) Annual Summit and Conference 2013, Kaohsiung, Taiwan; 11/2013  SourceAvailable from: Masahiro Iwahashi[Show abstract] [Hide abstract]
ABSTRACT: This report reduces the total number of lifting steps in a threedimensional (3D) double lifting discrete wavelet transform (DWT), which has been widely applied for analyzing volumetric medical images. The lifting steps are necessary components in a DWT. Since calculation in a lifting step must wait for a result of former step, cascading many lifting steps brings about increase of delay from input to output. We decrease the total number of lifting steps introducing 3D memory accessing for the implementation of low delay 3D DWT. We also maintain compatibility with the conventional 5/3 DWT defined by JPEG 2000 international standard for utilization of its software and hardware resources. Finally, the total number of lifting steps and rounding operations were reduced to 67 % and 33 %, respectively. It was observed that total amount of errors due to rounding operations in the lifting steps was also reduced.IEEE International Conference on Image Processing (ICIP); 09/2013
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1920
IEICE TRANS. FUNDAMENTALS, VOL.E94–A, NO.10 OCTOBER 2011
PAPER
Two Dimensional Nonseparable Adaptive Directional Lifting
Structure of Discrete Wavelet Transform
Taichi YOSHIDA†a), Student Member, Taizo SUZUKI††, Seisuke KYOCHI†, and Masaaki IKEHARA†, Members
SUMMARY
separable adaptive directional lifting (ADL) structure for discrete wavelet
transform (DWT) and its image coding application. Although a 2D non
separable lifting structure of 9/7 DWT has been proposed by interchanging
some lifting, we generalize a polyphase representation of 2D nonseparable
lifting structure of DWT. Furthermore, by introducing the adaptive direc
tional filteringingto the generalized structure, the 2D nonseparable ADL
structure is realized and applied into image coding. Our proposed method
is simpler than the 1D ADL, and can select the different transforming di
rection with 1D ADL. Through the simulations, the proposed method is
shown to be efficient for the lossy and lossless image coding performance.
key words:
two dimensional nonseparable lifting structure, discrete
wavelet transform, adaptive directional filtering, lossy and lossless image
coding
In this paper, we propose a two dimensional (2D) non
1.Introduction
The discrete wavelet transform (DWT) has been a funda
mental tool for image and video processing for the last few
decades. It is applied to image coding standard, JPEG 2000
[1], and high quality digital cinema [2]. Despite its success,
DWT has a serious disadvantage. Since they can only trans
form images along the vertical and horizontal directions by
separable implementation, they fail to provide sparse repre
sentation of transforming images which consist of various
angles of directionallyoriented textures except vertical and
horizontal directions. Therefore, if DWT is applied to im
ages which contain a rich directional high frequency com
ponent, such as edges and contours, the coding efficiency is
severely degraded.
To avoid this degradation, an adaptive directional filter
ing based on a lifting structure has been proposed [3]–[9].
The adaptive directional filtering can flexibly switch the fil
tering direction according to the direction of edges and con
tours, and compression efficiency can be improved. Espe
cially, an one dimensional (1D) adaptive directional lifting
(ADL) based wavelet transform [5] achieves high compres
sion efficiency for the lossy image coding. However, since
1D ADL has to be separately applied twice for two dimen
sional (2D) signals, it is redundant and produces a distor
tion of desired filtering directions due to the downsampling
Manuscript received January 13, 2011.
Manuscript revised May 17, 2011.
†The authors are with the Department of Electronics and
Electrical Engineering, Keio University, Yokohamashi, 2238522
Japan.
††The author is with the College of Engineering, Nihon Univer
sity, Koriyamashi, 9638642 Japan.
a)Email: yoshida@tkhm.elec.keio.ac.jp
DOI: 10.1587/transfun.E94.A.1920
between the vertical and horizontal filtering. In this paper,
hence, we propose the 2D direct and adaptive directional fil
tering basedon DWT, called a 2D nonseparable ADL struc
ture of DWT.
The proposed structure is realized by introducing an
adaptive directional filtering framework into a 2D direct lift
ing structure based on DWT. The lifting structure of DWT
has been proposed for the integertointeger transform [10].
For 2D signals such as images, a 2D separable lifting struc
ture of DWT is realized by applying the 1D lifting structure
to images twice, vertically and horizontally. Iwahashi et al.
have proposed the 2D direct lifting structure based on 9/7
DWT, by interchanging and merging some lifting in the 2D
separable lifting structure of 9/7 DWT [11],[12]. The new
structure is called a 2D nonseparable lifting structure of
9/7 DWT. The nonseparable structure requires less round
ing operators than the separable one, and is suitable for the
lossless image coding application. In this paper, we general
ize the polyphase representation of 2D nonseparable lifting
structure based on 2channel 1D filter banks (FBs) such as
DWT. It contains the class of the 2D nonseparable lifting
structure of 9/7 DWT [12]. Our proposed structure can de
sign various 2D nonseparable lifting structures.
Furthermore, to improve the efficiency of image cod
ing, we develop the generalized polyphase representation
for the adaptive directional filtering. Changing the sam
pling matrix of the proposed structure, we can change the
direction of the transform. The proposed adaptive direc
tional filtering is realized by changing the sampling matrix
by subregions of images, according to feature directions of
the subregions. With advantages of the 2D nonseparable
structure and the adaptive directional filtering, the proposed
structure improves the performance of the lossytolossless
image coding application. The proposed method is simpler
than the 1D one and can select the filtering directions which
are different from 1D ADL. Finally, lossy and lossless im
age coding results of the proposed structure are shown to
validate the advantage of the proposed structure.
Section 2 summarizes FB and show a polyphase repre
sentation of the 2D separable lifting structure and 1D ADL
structure of DWT. We propose the generalized polyphase
representation of the 2D nonseparable lifting structure of
DWT in Sect.3, and its adaptive directional filtering in
Sect.4. Section5showssome image codingresultsandcon
clusions are presented in Sect.6.
Notations: Vectors are denoted by boldfaced lower
case characters, whereas matrices are denoted by boldfaced
Copyright c ? 2011 The Institute of Electronics, Information and Communication Engineers
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YOSHIDA et al.: 2D NONSEPARABLE ADL STRUCTURE OF DWT
1921
uppercase characters. ATand A−1denote the transpose and
the inverse of the matrix A, respectively. I and 0 are identity
and the null matrices, respectively. M is a 2×2 nonsingular
integer matrix. An absolute determinant of the factor M is
described as M = det(M). ↓ M and ↑ M also represent the
down and upsamplers of M, respectively. Using vectors
z = [zx,zy]Tand k = [kx,ky]T, a multiplication of vectors
is defined as zk= zkx
matrix as
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
xzky
y. diag(·) denotes the block diagonal
diag(A0,A1,··· ,An) =
A0
0
...
0
0
···
···
...
···
0
0
...
A1
...
0An
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.
2. Review
2.1Polyphase Representation of FB
The 1D maximally decimated FB with Mchannel can be
also represented by polyphase matrices.
are the typeI polyphase matrix of the analysis bank and
the typeII polyphase matrix of the synthesis bank [13].
These polyphase matrices are related to Hk(z) and Fk(z)
(k = 0,1,··· , M − 1) which denote analysis filters and syn
thesis filters as
E(z) and R(z)
H(z) = [H0(z),H1(z),··· ,HM−1(z)]T
= E(zM)dM(z),
F(z) = [F0(z),F1(z),··· ,FM−1(z)]T
= dT
M(z−1)R(zM),
(1)
where dM(z) = [1,z−1,z−2,··· ,z−(M−1)]T. If the polyphase
matrices satisfy the condition as E(z)R(z) = z−nI, the result
is said to be the perfect reconstruction, where n is an order
of the FB. Hence, the synthesis matrix R(z) is designed as
z−nE−1(z−1). For simplicity, the synthesis bank is omitted in
this paper.
In a similar way to 1D FB, the 2D maximally deci
mated FB with M is represented by the polyphase matrix
as
H(z) = E(zM)dM(z),
(2)
where M is referred to as a sampling matrix, z = [zx,zy],
zxand zyare the vertical and horizontal delay elements, re
spectively,
dM(z) = [z−k0,z−k1,··· ,z−kM−1]T,
zM= [zm0,0
01
zm1,0
,zm0,1
0
zm1,1
1
]T,
kiis an integer vector of a set Ξ(M), called as a delay vector,
k0is restricted to be [0,0]T, Ξ(M) is a set of integer vectors,
which is defined as
Ξ(M) = {Mx  x ∈ [0,1)},
x ∈ [0,1) denotes a set of 2 × 1 real vectors x whose the
ith component satisfies 0 ≤ xi< 1 and mi,jdenotes the (i, j)
element of M.
2.22D Separable Lifting Structure of DWT
1D DWT is classified into 2channel FB. Its lifting structure
can be expressed in the polyphase representation as follows
[10],[13].
?HL(z)
ˆE(z) =
0 1/s
i=N−1
This structure with N = 2 is shown in Fig.1, where R de
notes a rounding operator. This figure shows that 1D signals
are divided into even samples and odd samples, sample is
transformed with the other sample, and even and odd sam
ples are conclusively output as lowpass and highpass coef
ficients, respectively. In general, downarrows from even
signals to odd signals are called a prediction step, and up
arrows are called an update step.
The pair of scaling factors can be realized as a lifting
structure [10]. The scaling factors are factorized as
?s
Figure 2 show the lifting realization of scaling factors,
where s0 = s − 1, s1 = −1/s and s2 = s − s2. The lift
ing structure with rounding operators can transform signals
from integer to integer. Note that scaling parts can also be
constructed by lifting steps.
According to the separable implementation, the 2D
separable lifting structure of DWT can be given by
HH(z)
?
=ˆE(z2)d2(z)
?s
0
?
0 ?
??1 Ui(z)
01
??
10
Pi(z) 1
??
(3)
0
0 1/s
?
=
?1 s − s2
01
??
10
−1/s 1
??1 s − 1
01
??1 0
1 1
?
.
(4)
Fig.1
9/7 DWT.
Fig.2
Lifting realization of scaling.
Page 3
1922
IEICE TRANS. FUNDAMENTALS, VOL.E94–A, NO.10 OCTOBER 2011
Fig.3
2D separable lifting structure of 9/7 DWT.
H(z) =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
HLL(z)
HHL(z)
HLH(z)
HHH(z)
HL(zy)
HH(zy)
0
0
?ˆE(z2
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0
HL(zx)HL(zy)
HL(zx)HH(zy)
HH(zx)HL(zy)
HH(zx)HH(zy)
0
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
HL(zy)
HH(zy)
0
ˆE(z2
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
×
?HL(zx)
HH(zx)
?
=
y)
0
y)
??d2(zy)
0
0d2(zy)
?
ˆE(z2
x)d2(zx),
(5)
for the four subband filters, HLL(z), HHL(z), HLH(z) and
HHH(z). To define the polyphase matrix of 2D separable
structure, we introduce the relationships as
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
tionships (6), (5) is rewritten as
?ˆE(z2
×
0 1/s I
i=N−1
×ˆd2(zy)d2(zx).
ˆd2(zy)
?s
?1 Ui(z2
?
0
01/s
x)
1
?
?
?
=
?sI
?I Ui(z2
?
0
0
1/sI
x)I
I
?
ˆd2(zy),
ˆd2(zy)
0
=
0
?
?
ˆd2(zy),
ˆd2(zy)
10
Pi(z2
x) 1
=
I0
Pi(z2
x)I I
ˆd2(zy),
(6)
whereˆd2(zy) = diag(d2(zy),d2(zy)). From the above rela
H(z) =
y)
0
0
ˆE(z2
?
y)
0 ?
?
?sI0
??I Ui(z2
x)I
0I
??
I0
Pi(z2
x)I I
??
(7)
Consequently, the polyphase representation of the above 2D
separable structure is described as
H(z) = E(zM)dM(z),
?ˆEy
dM(z) = [1,z−1
E(z) =
0
0
ˆEy
y,z−1
??sI
x,z−1
0
0 1/s I
xz−1
?
0 ?
i=N−1
??I Ux
iI
I0
??
I
iI I
0
Px
??
,
y]T,
(8)
whereˆEj, Pj
Ui(zj). Since the 2D FB is realized by the separable im
plementation of 1D FB, the decimation matrix M should be
iand Uj
i(j : x or y) denoteˆE(zj), Pi(zj) and
Fig.4
Lifting direction of 1D ADL.
diag(2,2). For example, Fig.3 shows the 2D separable lift
ing structure of 9/7 DWT. This figure shows that images are
divided into four sets and transformed while maintaining in
tegers.
2.3 1D ADL Structure of DWT
Based on the lifting structure of 1D DWT, 1D ADL is re
alized by permitting the other directions of the transform
except horizontal and vertical [5].
Figure 4 illustrates a 2D signal x(n), where n =
[nx,ny]T, and nxand nyare defined as vertical and horizon
tal indices in 2D signals. In the lifting structure of 1D DWT,
the input signals are divided into even samples xeand odd
samples xoas follows.
?xe(n) = x([nx,2ny]T)
In the prediction step, odd samples xo(n) are predicted from
the neighboring even samples xe(n) as
xo(n) = x([nx,2ny+ 1]T)
(9)
ˆ xo(n) = xo(n) + pe(n),
(10)
where pe(n) is expressed as
pe(n) = pi
?
xe
??nx− tanθd
ny
??
+ xe
??nx+ tanθd
ny+ 1
???
,
piis a coefficient of filter and θdis arbitrary direction of
filtering. Figure 4 shows θd= {π/2,π/4,−π/4}. Next, in the
update step, even samples xe(n) are updated from the new
neighboring odd samples ˆ xo(n) as
ˆ xe(n) = xe(n) + uo(n),
(11)
where uo(n) is expressed as
uo(n) = ui
?
ˆ xo
??nx− tanθd
ny− 1
??
+ ˆ xo
??nx+ tanθd
ny
???
,
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YOSHIDA et al.: 2D NONSEPARABLE ADL STRUCTURE OF DWT
1923
uiis a coefficient of filters. This set of lifting operations is
repeated and conclusive samples are produced as lowpass
and highpass coefficients.
3.Polyphase Representation of 2D Nonseparable Lift
ing Structure of DWT
3.1Generalized Structure
In this paper, we introduce the generalized polyphase repre
sentation of the 2D nonseparable lifting structure of DWT.
Here, some matrices are defined as follows.
??s
Uj
i
0101
??1 0
By using these matrices, (8) is rewritten as
S = diag
0
0 1/s
??1 Uj
?
,
?s
?1 Uj
?1 0
0
0 1/s
??
,
ˆS =
?sI
?I Uj
?I
0
0 1/sI
?
,
,
i= diag
?
?
,
i
??
??
,
ˆUj
i=
iI
I
0
0
?
?
Pj
i= diag
Pj
i1
,
Pj
i1
,
ˆPj
i=
Pj
iI I
.
(12)
E(z) = S
0 ?
i=N−1
{Uy
iPy
i}ˆS
0 ?
i=N−1
{ˆUx
iˆPx
i}.
(13)
Without a loss of generalities, the following relation
ships are satisfied.
⎧⎪⎪⎪⎨⎪⎪⎪⎩
Uy
pˆUx
iPy
qˆPx
iˆS =ˆSUy
q=ˆUx
iPy
qˆPx
i,
Uy
pPy
qUy
pPy
p,
(14)
where p,q = 0,1,...,N − 1. These relationships mean that
some matrices can be moved, and (13) is rewritten as
E(z) = SˆS
0 ?
i=N−1
{Uy
iPy
iˆUx
iˆPx
i}.
(15)
Some matrices are merged with each other and factorized
into the nonseparable structure. If Gp,q(z) is defined a prod
uct of matrices as
Gp,q(z) = Uy
pPy
pˆUx
qˆPx
q,
(16)
where p and q are arbitrary integers, substituting (12) in
(16), Gp,q(z) is consequently described as
Gp,q(z) =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
Gy
Py
Gy
Py
pGx
pGx
pPx
pPx
q
Uy
pGx
Gx
q
Uy
Px
q
Gy
Py
pUx
pUx
Gy
p
Py
p
q
Uy
pUx
Ux
q
Uy
p
1
q
qq
q
pPx
q
qq
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
,
(17)
where Gj
Gp,q(z) can be factorized into a lifting structure as
k= 1 + Pj
kUj
k(j : x or y, k : p or q). Consequently,
Gp,q(z) =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
1 Uy
0
0
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
p Ux
1
0
0
q−Uy
0
1
0
00 0
10 0
01 0
qPx
pUx
0
0
1
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
q
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
1 0 0
Py
Px
0 0 0
0
p 1 0 Ux
q0 1 Uy
q
p
1
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
×
1
0
0
pPx
Py
qPy
p 1
.
(18)
This nonseparating factorization achieves to reduce round
ing operators through merging some matrices. Substituting
(18) in (15), the generalized polyphase representation of 2D
nonseparable lifting structure of DWT is defined as
E(z) =˜S
0 ?
i=N−1
Gi,i(z),
(19)
where˜S = SˆS = diag(s2,1,1,1/s2).
In this section, we propose the generalized polyphase
representation of 2D nonseparable lifting structure of DWT
by using the relationships (14) and the nonseparating fac
torization (18). We show some examples in the next section.
3.25/3 and 9/7 DWT
The generalized polyphase representation is proposed in the
previous section. In this section, we present its examples
based on 5/3 and 9/7 DWT. These DWTs are well known to
be applied to the JPEG 2000.
From (19), the polyphase matrix of 2D nonseparable
lifting structure of 5/3 DWT is described as
E(z) = G0,0(z).
(20)
On the other hand, in the case of 9/7 DWT, the polyphase
matrix is factorized from (19), as
E(z) =˜SG1,1(z)G0,0(z).
(21)
However, the other polyphase matrix can be designed from
the relationships (14) and the nonseparating factorization
(18). From (13) with N = 2, the other polyphase matrix is
designed as
E(z) = SUy
= SˆS × Uy
=˜SUy
1Py
1Uy
1Py
1G0,1(z)ˆUx
0Py
1× Uy
0×ˆSˆUx
1ˆPx
0ˆUx
0.
1ˆUx
1ˆPx
0ˆPx
1×ˆUx
0
0Py
0ˆPx
0ˆPx
0
1Py
(22)
This structure has been proposed in [14], and is shown in
Fig.5. Compared with Fig.3, it is shown to require less
rounding operators. Therefore, it is proven that the non
separable structure has better compatibilities to the conven
tional 9/7 DWTs in the JPEG 2000 than the separable struc
ture [14]. In that sense, we design the polyphase matrix of
2D nonseparable lifting structure of 9/7 DWT as (22).
4.2D Nonseparable ADL Structure of DWT
4.1Realization
Inthis section, we proposethe 2DnonseparableADLstruc
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IEICE TRANS. FUNDAMENTALS, VOL.E94–A, NO.10 OCTOBER 2011
Fig.5
2D nonseparable lifting structure of 9/7 DWT.
Fig.6
Lifting direction of 2D ADL.
ture of DWT based on the 2D nonseparable lifting structure
of DWT shown in Sect.3. The ADL structure can transform
a subregion of images along its direction. Therefore, the
structure efficiently reduces the entropy of transformed co
efficients and achieves better image coding performance.
By changing the sampling matrix M depending on sub
regions of images, the ADL structure is realized. Figure 6
shows one lifting operation, where these sampling matrices
are defined as
?2
These dots are pixels, and black dots are transformed with
neighbor white dots along arrows. This figure shows that
neighbor pixels used for the lifting operation are changed
by changing the sampling matrix in the subregion. In other
words, the direction of the transform can be changed by
changing the sampling matrix.
The arbitrary sampling matrix whose absolute determi
nant is 4 can be selected. The following sampling matrices
Md(d = 0,1,...,6) are used in this paper.
M0=
0
20
?
, M1=
?22
20
?
.
(23)
Md=
??2 0
0 2
?
,
?2 2
0 2
?
,
?2 −2
02
?
,
?2 0
2 2
?
,
?20
−2 2
?
,
?2 4
0 2
?
,
?2 −4
02
??
With changing the sampling matrix, the delay vectors have
Table 1
Delay vectors.
d
k0
k1
k2
k3
0
1
2
3
4
5
6
[0,0]T
[0,0]T
[0,0]T
[0,0]T
[0,0]T
[0,0]T
[0,0]T
[0,1]T
[1,1]T
[−1,1]T
[0,1]T
[0,1]T
[2,1]T
[−2,1]T
[1,0]T
[1,0]T
[1,0]T
[1,1]T
[1,−1]T
[1,0]T
[1,0]T
[1,1]T
[2,1]T
[0,1]T
[1,2]T
[1,0]T
[3,1]T
[−1,1]T
to be changed. For maintaining the linear phase property of
filters, we define the delay vectors to be symmetrical with
respect to a point which is a center of Md. Table 1 shows
the delay vectors corresponding to Mdand the directions of
the transform are represented in Fig.7. It shows that the
proposed structure can transform along various directions
besides vertical and horizontal shown in Fig.7(a). Since 2D
signals are transformed by E(z) after downsampling with
Mdand some delays, in this proposed structure, a sampling
matrix can share the same transformed system produced by
the polyphase matrix E(z) as the others.
4.2 Optimal Direction Decision
For the image coding application, the optimal directions are
decided according to the energy of highpass subband. Espe
cially, we focus on the energy of HH subband in this paper.
For all the sampling matrix Md, the HH subband compo
nents at a index n are described as hd(n). Practically, the
directional information is assigned to not the pixels but the
blocks divided by the quad tree decomposition in order to
reduce the side information. For this purpose, RD opti
mization is processed as in the following. The full quad tree
T is constructed by applying the quad tree decomposition to
the image until reaching the predefined block size. B, D(B)
and R(B) are defined as an arbitrary subtree, a distortion and
a rate of bits. The most suitable subtree B∗with optimal
direction is provided by minimizing the cost function J(B)
expressed by
B∗= min
B
J(B) = min
B(D(B) + λR(B)),
(24)
Page 6
YOSHIDA et al.: 2D NONSEPARABLE ADL STRUCTURE OF DWT
1925
Fig.7
Sampling matrices.
Table 2
Reduced numbers of computational costs.
Direction decision
5/39/7
(1/2)DL
(3/2)DL
(1/2)DT
(5/4)DL
DL
2DL
Image Transformation
5/3
−2L
(−1/2)L
L
9/7
L
Add.
Mult.
Round.
(7/4)L
(17/4)L
where
D(B) =
?
?
τ∈B
?
rC(τ) +
n∈Nτ
hτ,d(n)2,
?
R(B) =
τ∈B
v∈B
rT(v),
τ denotes a node of B, Nτis a support region in images at
a node τ, hτ,d(n) is hd(n) at a node τ, rC(τ) and rT(v) are
defined as the rate of the entropy encoding highpass coef
ficients in τ and coding the side information in node in v,
respectively.
4.3 Comparison of Computational Costs
In Table 2, we show the differences of the computational
costs between the 1D ADL structure and the proposed 2D
ADL of 5/3 and 9/7 DWTs by subtracting 2D ADL from
1D ADL. “Add.”, “Mult.”, “Round.”, “5/3” and “9/7” mean
addition, multiplication, rounding, 5/3 DWT and 9/7 DWT,
respectively. These numbers depend on the size L of input
images and their lifting steps. Additionally, the numbers
D of direction candidates for lifting should be taken into
account for the lifting direction decision.
Table 2 shows that the proposed structure is simpler
than the 1D ADL in terms of the computational costs. For
the adaptive directional filtering, we need at least three di
rections which are vertical or horizontal, diagonal from top
left to bottom right, and diagonal from top right to bottom
left. Hence, by substituting D = 3 and adding the num
bers of the directional decision and image transformation,
total numbers of the reduced addition and multiplication are
(−1/2)L and L, in the case of the 5/3 DWT. The multiplica
tion has more loads in the computation and implementation
than the addition. Therefore, the both proposed structures
are considered to be simpler than the 1D ADL structures.
5.Simulation
We apply the proposed ADL structure of 5/3 and 9/7 DWTs
into the lossy and lossless image coding application and
compare the coding efficiency with the conventional struc
ture of 5/3 and 9/7 DWTs.
For the comparisons, we use the 2D separable lifting
structure of DWT in (8), the 2D nonseparable lifting struc
ture of DWT in (20), and the 1D ADL structure of DWT
shown in Sect.2.3. In this section, for simple notations,
these conventional structures and the proposed structure are
denoted as 1D DWT, 2D DWT, 1D ADL and 2D ADL, re
spectively. The 1D and 2D DWTs are applied to images
with 6level octave decomposition. In cases of the 1D and
2D ADLs, the 1D and 2D DWTs are applied to images at the
second to sixth levels after the 1D and 2D ADL structures
of DWT are applied at only the first level, respectively†.
For reducing the side information and computational
costs, the candidate directions are restricted three and seven
in the 5/3 and 9/7 DWTs, respectively. There exists a trade
off between better compression rate and less computational
costs. These restrictions are experimentally decided for the
valid comparison.
The embedded zerotree wavelet based on intraband
partitioning (EZWIP) [15] is used as the encoder. Since
transformed coefficients are integer and encoded images are
lossless bit streams, lossy bit streams are produced by dis
carding the backward bits of the bit streams. The side infor
mation is encoded by the arithmetic coding algorithm [16].
At the lossy image coding application, the peak signalto
noise ratio (PSNR) is used as an objective function measur
ing an image quality of reconstructed images. The PSNR is
formulated as
PSNR = 10log10
2552
MSE,
(25)
where MSE means the mean square error.
†The 1D ADL is applied at an only vertical direction in the
first level for fairy comparison. In that case, the number of side
information bits is equal to case of the 2D ADL.
Page 7
1926
IEICE TRANS. FUNDAMENTALS, VOL.E94–A, NO.10 OCTOBER 2011
Table 3
Image coding results of 5/3 DWT [bpp].
1DDWT2DDWT
6.786.78
4.874.86
4.504.49
1DADL
6.73
4.84
4.49
2DADL
6.12
4.78
4.48
Zone plate
Barbara
Lena
5.15/3 DWT for Lossless
The 2D nonseparable ADL structure of 5/3 DWT is applied
into the lossless image coding. The 2D ADL is realized by
imposing the directional selectivity on the 2D nonseparable
lifting structure of 5/3 DWT described in (20), as shown in
Sect.4. For reducing the side information, the selected di
rections areexperientiallyrestrictedtobe threeinthis paper.
θdin Sect.2.3 is selected as {π/2,π/4,−π/4} in the 1D ADL
structure of DWT, and Md(d = 0,1,2) is selected in the
2D ADL. They are not correctly same directions due to the
downsampling, but considered to be a reasonable compari
son.
Table 3 indicates results of the lossless image coding
application. These values are entropies of compressed im
ages described in bitperpixel (bpp), and the entropy of 1D
ADL and 2D ADL includes the encoded side information.
In the case of Zone plate and Barbara, the 2D ADL is effi
cientforimages havingrichdirectionalcomponents because
of the small number of rounding operators and the adaptive
directional filtering. On the other hand, the proposed struc
ture is slightly effective for images whose energy is concen
trated on the low frequency region such as Lena.
5.29/7 DWT for LossytoLossless
In a similar way to the 5/3 DWT, the 2D ADL structure of
9/7DWTis appliedintothe lossyandlosslessimage coding.
The 2D ADL is based on (22), with Md(d = 0,1,··· ,6),
and tanθd= {0,1,−1,2,−2,0.5,−0.5} in the 1D ADL.
Table 4 indicates results of the lossy and lossless image
coding application. In the lossy, these numbers are PSNR of
reconstructed images in dB, and rates are described in bpp.
The proposed structure improves image coding application
in the lossy and lossless for the same reason of 5/3 DWT.
Compared with the 1D and 2D DWTs, the 2D ADL shows
more efficient image coding results because of fewer round
ing operations and adaptive directional filtering. However,
the adaptive directional filtering is considered to be slightly
effective for some images such as Lena due to the same rea
son as the 5/3 DWT, as in the discussion in Sect.5.1. On the
other hand, compared with 1D ADL, some coding results
of the 2D ADL show worse performance because the 1D
and 2D ADLs are not in a same class of the adaptive direc
tional filtering. However, it is anadvantage that the structure
of the 2D ADL is the simpler than one of the 1D ADL. In
this methods, the image coding performance depends on the
decided directions for subregions. In a future work, there
exists a better optimal directional decision for the 2D ADL.
Table 4
Image coding results of 9/7 DWT.
Lossless [bpp]
1DDWT
6.11
4.84
4.54
2DDWT
6.08
4.81
4.51
1DADL
4.93
4.81
4.54
2DADL
5.33
4.75
4.51
Zone plate
Barbara
Lena
Lossy [dB]
Rate
0.25
0.5
1
0.25
0.5
1
0.25
0.5
1
1DDWT
12.01
15.19
19.98
27.24
30.46
34.85
32.52
35.52
38.39
2DDWT
12.02
15.19
19.98
27.23
30.45
34.84
32.50
35.48
38.34
1DADL
13.32
17.91
24.56
27.43
30.82
35.14
32.54
35.60
38.44
2DADL
13.48
17.50
22.94
27.60
30.94
35.28
32.54
35.52
38.38
Zone plate
Barbara
Lena
Fig.8
Original images and reconstructed images.
In Fig.8, the original and reconstructed images are rep
resented, where the compression rate is 0.25[bpp]. Com
pared with the conventional structure which does not have
an adaptive directional selectivity, the proposed structure
can represent various directions besides vertical and hori
zontal. Especially, stripe textures in Fig.8(f) are represented
clearly.
Page 8
YOSHIDA et al.: 2D NONSEPARABLE ADL STRUCTURE OF DWT
1927
6.Conclusion
In this paper, we generalize the polyphase representation of
the 2D nonseparable lifting structure of DWT and propose
ADL of the structure. With maintaining the advantage of the
2D nonseparable and the adaptive directional filtering, the
proposed structure achieves efficient image coding results.
Acknowledgments
This work was supported by Global COE Program “High
Level Global Cooperation for LeadingEdge Platform on
Access Spaces (C12)”.
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Taichi Yoshida
degrees in electrical engineering from Keio Uni
versity, Yokohama, Japan, in 2006 and 2008,
respectively. He is currently a Ph.D. candidate
at Keio University, Yokohama, Japan, under the
supervision of Prof. Masaaki Ikehara. His re
search interests are in the field of filter bank de
sign and its image coding application.
received the B.E. and M.E.
Taizo Suzuki
and Ph.D. degrees in electrical engineering from
Keio University, Yokohama, Japan, in 2004,
2006 and 2010, respectively. He joined Toppan
Printing Co., Ltd., Tokyo, Japan, from 2006 to
2008. From 2008 to 2011, he was a Research
Assistant (RA) of Global Center of Excellence
(GCOE) at Keio University. Also, he was a Re
search Fellow of the Japan Society for the Pro
motion of Science (JSPS) from 2010 to 2011.
Moreover, he was a Visiting Scholar at the Uni
received the B.E., M.E.
versity of California, San Diego (Video Processing Group supervised by
Prof. T. Q. Nguyen) from 2010 to 2011. He is currently an Assistant Pro
fessor at College of Engineering, Nihon University. Hiscurrent research in
terests are Mchannel filter bank design and its application for image/video
signal processing.
Seisuke Kyochi
mathematics from Rikkyo University, Toshima,
Japan in 2005 and the M.E. and Ph.D. degrees
from Keio University, Yokohama, Japan in 2007
and 2010, respectively. He is currently with the
Video Coding Group of NTT Cyber Space Lab
oratories. His research interests in the areas of
wavelets, filterbanksandtheirapplicationtoim
age and video coding.
received the B.S. degree in
Masaaki Ikehara
and Dr.Eng. degrees in electrical engineering
from Keio University, Yokohama, Japan, in
1984, 1986, and 1989, respectively. He was Ap
pointed Lecturer at Nagasaki University, Naga
saki, Japan, from 1989 to 1992. In 1992, he
joined the Faculty of Engineering, Keio Univer
sity. From 1996 to 1998, he was a visiting re
searcher at the University of Wisconsin, Madi
son, and Boston University, Boston, MA. He is
currently a Full Professor with the Department
received the B.E., M.E.
of Electronics and Electrical Engineering, Keio University. His research
interests are in the areas of multirate signal processing, wavelet image cod
ing, and filter design problems.