IEICE TRANS. FUNDAMENTALS, VOL.E93–A, NO.8 AUGUST 2010

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PAPER

M-Channel Paraunitary Filter Banks Based on Direct Lifting
Structure of Building Block and Its Inverse Transform for
Lossless-to-Lossy Image Coding
Taizo SUZUKI†a) , Student Member and Masaaki IKEHARA† , Member

SUMMARY
This paper presents a paraunitary filter bank (PUFB)
based on a direct lifting structure of a building block and its inverse transform for lossless-to-lossy image coding. Although the conventional liftingbased filter banks (LBFBs), which are constructed by lifting structures withinteger coeﬃcients and rounding operations, suﬀer from degradation ofcoding performance due to much rounding error generated by cascading
lifting structures, our proposals can be applied to any PUFB without losing
many ones because building blocks can be applied to every lifting block as
it is. It is constructed with very simple structures and many rounding operations are eliminated. Additionally, the number of rounding operations is
reduced more by using two-dimensional block transform (2DBT) of separated transform to each building block. As result, even though the proposed
PUFBs require a little side information block (SIB), they show better coding performance in lossless-to-lossy image coding than the conventional
ones.
key words: paraunitary filter bank (PUFB), direct lifting, lossless-to-lossy
image coding, side information block (SIB)

1.

Introduction

With the rapid development of hardware such as PCs andmobile phones etc. and the continual expansion of broadband, lossless-to-lossy image coding, which is the unification of lossy and lossless image coding, is demanded to obtain higher quality and compression ratio. Because JPEG[1], which is the international standard in image coding, separately uses discrete cosine transform (DCT) [2] and diﬀerential pulse code modulation (DPCM) for lossy and lossless image coding, respectively, it can not achieve losslessto-lossy image coding. Besides it, JPEG2000 [3], whichis the next generation standard based on discrete wavelet
transform (DWT) [4], has achieved it by lifting structures[5] with integer coeﬃcients and rounding operations.
On the other hand, DCT and DWT in the international
standards and the next generation one are the special formsof linear-phase filter banks (LPFBs) [6]. LPFBs correspondto a subclass of nonlinear-phase cases. Therefore, nonlinearphase filter banks (NLPFBs) can be expected to have thepotential to improve the performance or the quality in many
applications. Paraunitary filter banks (PUFBs) [7] are therepresentative in NLPFBs and show good coding perforManuscript received October 28, 2009.
Manuscript revised March 25, 2010.
†
The authors are with the Department of Electronics andElectrical Engineering, Keio University, Yokohama-shi, 223-8522
Japan.
a) E-mail: suzuki@tkhm.elec.keio.ac.jp
DOI: 10.1587/transfun.E93.A.1457

mance in lossy image coding. Then, we have presented
lifting-based PUFBs (LBPUFBs) for lossless-to-lossy image coding and solved a problem of error generated by manyrounding operations in [8]. Moreover, many lifting-based
filter banks (LBFBs) have been researched [9]–[13]. But,
any LBFB suﬀer from degradation of coding performancedue to much rounding error generated by cascading many
lifting structures.
This paper presents a PUFB based on a direct lifting
structure of a building block and its inverse transform forlossless-to-lossy image coding. The direct lifting structure
is a family of block lifting structures [14] which achieves
reduction of the number of rounding operations. Although
our previous work [15] has reduced many rounding operations, a hard restriction must be imposed. Our proposals
can be applied to any PUFB without losing many advantages since many lifting structures do not need to be cascaded because building blocks can be applied to every lifting
block as it is. It is constructed with very simple structuresand many rounding operations are eliminated. Additionally,
the number of rounding operations is reduced more by using two-dimensional block transform (2DBT) of separated
transforms on each building block. As result, even though
the proposed PUFBs require a little side information block(SIB), they show better coding performance in lossless-tolossy image coding than the conventional ones.
Notations: I, MT , M† , M[M] and diag{.} are an identity
matrix, a transpose of matrix, a conjugate transpose of matrix, M×M matrix and a block diagonal matrix, respectively.
For simplicity, we omit matrix sizes when they are obvious.
2.

Preliminaries

2.1 Lifting Structure
The lifting structure [5] is researched widely for various applications. Since the structure achieves integer-to-integertransform, lossless-to-lossy image coding can be implemented. In Fig. 1(a), which shows basic lifting structures,the analysis input signals xi and x j , the analysis output andsynthesis input signals yi and y j , the synthesis output signals
zi and z j and lifting coeﬃcients T 0 and T 1 are represented by
y j = x j + round [T 0 xi ]yi = xi + round T 1 y j

c 2010 The Institute of Electronics, Information and Communication Engineers
Copyright IEICE TRANS. FUNDAMENTALS, VOL.E93–A, NO.8 AUGUST 2010

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Fig. 1 Lifting structures: (a) basic lifting structures, (b) block lifting
structures.

and



zi = yi − round T 1 y j = xi
z j = y j − round [T 0 zi ] = x j

in analysis and synthesis part, respectively, where round[.]is rounding of (.).
In this case, the lifting matrices and their inverse matrices are as follows:
 
−1 


1 −T 0
1 T0
1 T0
=
,
,
0 1
0 1
0 1


−1 


1 0
1 0
1 0
=
and
.
−T 1 1T1 1
T1 1
2.2 Block Lifting Structure
Conversely, the block lifting structure [14] is known as aneﬃcient one. The structure achieves higher compression ratio because it can merge many rounding operations. It isclear that the number of rounding operations is reduced fromN 2 to N when the size of the lifting coeﬃcient matrix T is
N × N. In Fig. 1(b), which shows block lifting structures,the analysis input signals vector xi and x j , the analysis output and synthesis input signals vector yi and y j , the synthesis
output signals vector zi and z j and lifting coeﬃcient matrices T0 and T1 are represented by
y j = x j + round[T0 xi ]yi = xi + round[T1 y j ]andzi = yi − round[T1 y j ] = xi
z j = y j − round[T0 yi ] = x j
in analysis and synthesis part, respectively.

Fig. 2 M-channel maximally decimated filter bank (FB): (a) basic
structure, (b) polyphase structure.

In this case, the block lifting matrices and their inverse
matrices are as follows:

 
−1 

I −T0
I T0
I T0
=
,
,
0 I
0 I
0 I


−1 


I 0
I 0
I 0
=
and
.
−T1 I
T1 I
T1 I
2.3 Paraunitary Filter Banks (PUFBs)
Filter banks (FBs) [6] have found many applications suchas in speech, audio, image and video compression, statistical signal processing, discrete multitone modulation andchannel equalization. Figure 2(a) shows an M-channel maximally decimated FB, where Hk (z) and Fk (z) are the k-th
(k = 0, 1, · · · , M − 1) analysis and synthesis filter, respectively. Also Fig. 2(b) shows a polyphase structure of FB.
The analysis and synthesis filters are represented by using
the polyphase matrices E(z) and R(z) as follows:T
H0 (z) H1 (z) · · · H M−1 (z) = E(z M )e(z)TTF0 (z) F1 (z) · · · F M−1 (z) = e(z)R(z M )
where



e(z) = 1 z−1 z−2 · · · z−(M−1) .

If E† (z−1 )E(z) = I and R(z) = E† (z−1 ), the FBs are calledparaunitary filter banks (PUFBs).
In this paper, we consider PUFBs where the number ofchannels is M (even) and all filter lengths are MK (K ∈ N).
They are eﬃciently designed and implemented by the lattice
structure as [7]
⎞
⎛ 1
⎟⎟⎟
⎜⎜⎜
⎜
Gk Λk (z)⎟⎟⎟⎠ G0
E(z) = ⎜⎜⎝k=K−1

SUZUKI and IKEHARA: PUFBS BASED ON DIRECT LIFTING FOR LOSSLESS-TO-LOSSY IMAGE CODING

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where

Wk =

 [M−γ ]kI0Λk (z) =0
z−1 I[γk ]and Gk is an M×M arbitrary orthogonal matrix. Although γk
is arbitrary integer 0 ≤ γk ≤ M−1 and Gk can be constructedby complex values, we set γk = M/2 and Gk is constructedby real values for simplicity. Thus Λk (z) is denoted as Λ(z).
3.

PUFB Based on a Direct Lifting Structure of a
Building Block and Its Inverse TransformIn this section, an eﬃcient lifting structure of PUFB for
lossless-to-lossy image coding is presented. The system
is implemented by consideration of the block parallel system [15] of each building block and its inverse transformin PUFB. It means that their original matrices are preserved
and directly used as each lifting block even after the system
was factorized into lifting structures.

Consequently, diag{G, GT } in (1) can be factorized into thecomplete block lifting structures such as


 



0 I I 0 I −GT I 0
G 0
.
(5)
=
G I
−I 0 G I 0 I
0 GT
Thus the block parallel system of each building blockand its inverse transform can be eﬃciently implemented byblock lifting structure as shown in the right part of Fig. 3.
Any building block in PUFBs can be directly applied tolossless-to-lossy image coding even if they are designed for
only lossy image coding. However the systems in (5) have aproblem that the transformed signals by inverse transformsof building block are not suitable for image compression. It
is a solution where a condition GT = G is imposed as in[15], but it may make filter performance degrade.
3.2 Realization of Eﬃcient PUFB Based on a Direct Lifting Structure by Side Information Block (SIB)

3.1 Direct Lifting Structure
We consider to process two individual signals by each building block G and its inverse transform G−1 = GT as shownin the left part of Fig. 3. The output signals are transformed
by
 
  
G 0 xi
yi
=
.
(1)
ti
0 GT siFirst, let us define Wk as



 
0 I G 0
0 GT
Wk =
.
=
I 0 0 GT
G 0






−I 0 I 0 I −GT I 0.
0 I G I 0 I
G I

(2)

And, the block lifting matrix is multiplied from the rightsides of (2) as


 
I 0
−I GT
.
(3)
=
Wk
G 0
−G I
Next, the other block lifting matrix is multiplied from theright sides of (3) as

 


−I 0
−I GT I GT=
G 0 0 IG I−I 0 I 0=.
(4)
0 I G IWk can be factorized into the lifting structures using (2)–(4)asFig. 3 A direct lifting structure of each building block and its inverse
transform in PUFB (white circles: rounding operations).

We can easily come up with a simple realization for losslessto-lossy image coding which is applied after an N × N image X was segmented to M × M block xk (0 ≤ k ≤ n − 1,
n = (N/M)2 ). The segmented block xk is sequentially transformed by building block in PUFB. In parallel, M × M block
sk have to be transformed by inverse transform of building
block in (5). However, the transformed signals by it are notsuitable for image compression as described in the previous
subsection. Then we prepare M × M side information block(SIB) s0 as null matrix and sk is iteratively transformed froms0 by
sk = GT sk−1

fork = 1, 2, · · · , n − 1

As result, the proposed realization for lossless-to-lossy image coding is presented by
⎡
⎤ ⎡
⎤⎡
⎤
⎢⎢⎢ y0 ⎥⎥⎥ ⎢⎢⎢ G⎥⎥⎥ ⎢⎢⎢ x0 ⎥⎥⎥⎢⎢⎢ y ⎥⎥⎥ ⎢⎢⎢⎥⎢⎥⎥⎥ ⎢⎢⎢ x1 ⎥⎥⎥⎥
G⎢⎢⎢ 1 ⎥⎥⎥ ⎢⎢⎢⎥⎥⎥ ⎢⎢⎢ . ⎥⎥⎥0..
⎢⎢⎢⎢ .. ⎥⎥⎥⎥ = ⎢⎢⎢⎢⎥⎥⎥ ⎢⎢⎢ . ⎥⎥⎥.
⎢⎢⎢ . ⎥⎥⎥ ⎢⎢⎢⎥⎥⎥ ⎢⎢⎢ . ⎥⎥⎥⎢⎢⎢y ⎥⎥⎥ ⎢⎢⎢⎥⎥⎥ ⎢⎢⎢x ⎥⎥⎥G⎢⎣ n−1 ⎥⎦ ⎢⎣⎥ ⎢ n−1 ⎥⎦T n ⎦⎣
sn−1
0
(G )
s0

(6)

and sn−1 is also encoded with all of yk s. The left part of
Fig. 4 shows the system in (6). Note that sn−1  0 due torounding error in each of lifting structures. Also it is clearthat the combination of G and GT can be factorized into lifting structures by (5) as shown in dashed line area in Fig. 4.
Finally the proposed lossless-to-lossy image coding issummarized as follows:
1) Lossless bit stream with SIB sn−1 can be obtained byusing (6) and a progressive encoder with PSNR scalability like SPIHT [16] and EZW-IP [17].
2) In lossless mode, the image is reconstructed from all

IEICE TRANS. FUNDAMENTALS, VOL.E93–A, NO.8 AUGUST 2010

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Fig. 4 Realization of a direct lifting structure of a building block and its inverse transform (whitecircles: rounding operations).
Table 1

The numbers of rounding operations.

filter
4×8
4 × 12
8 × 168 × 24Conv. PUFBs [8]
23
31
62
82

Prop. PUFBs
12
18
24
36


T
G (Gx)T = GxGT  G2D x.

Fig. 5 Two-dimensional block transform (2DBT) when n = 2: (a) standard transform, (b) separated transform to each building block.

lossless bit stream and sn−1 . The each block sk is successively inverse transformed by sk = Gsk+1 without
any loss.
3) In lossy mode, the image is reconstructed by interrupting the lossless bit stream without using sn−1 .
3.3 More Eﬃcient Improvement by Two-Dimensional
Block Transform (2DBT)When we apply a block transform matrix F to a twodimensional (2D) input signal x in column- and row-wise
separately, the 2D output signal y is expressed by
y = (F(Fx)T )T = FxFT .

(7)

We call it a 2D separable block transform. If the transform F
is factorized as F = Fn−1 · · · F1 F0 (n ∈ N), (7) is representedbyy = Fn−1 · · · F1 F0 x(Fn−1 · · · F1 F0 )T= Fn−1 · · · F1 F0 xFT0 FT1 · · · FTn−1 .
This equation means that 2D block transform (2DBT) by
Fk+1 is applied after 2DBT by Fk (0 ≤ k ≤ n − 2, k ∈ N).
Therefore standard transform and separated transform toeach building block when n = 2 are shown in Fig. 5(a) and(b), respectively. In this paper, we regard F is E(z). Thus
when each building block G in PUFB is applied to an M × M
input signal x in column- and row-wise separately, it is expressed byConsequently, (5) and (6) are represented by




 

G2D 0
I 0
0 I
I 0 I −GT2D
=
I
G2D I
0 GT2D−I 0 G2D I 0
and
⎤⎡
⎡
⎤ ⎡
⎤
⎥⎥⎥ ⎢⎢ x0 ⎥⎥⎢⎢⎢ y0 ⎥⎥⎥ ⎢⎢⎢ G2D⎢⎥⎥⎥ ⎢⎢ x ⎥⎥⎥⎢⎢⎢ y ⎥⎥⎥ ⎢⎢⎢G2D⎥ ⎢⎢ 1 ⎥⎥⎢⎢⎢ 1 ⎥⎥⎥ ⎢⎢⎢0 ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢ . ⎥⎥⎥⎥⎢⎢⎢ .. ⎥⎥⎥ ⎢⎢⎢.
.
⎥⎥⎥ ⎢⎢⎢ .. ⎥⎥⎥ .
.
⎢⎢⎢ . ⎥⎥⎥ = ⎢⎢⎢⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢⎢⎢y ⎥⎥⎥⎥ ⎢⎢⎢⎢
G2D⎥⎥⎥ ⎢⎢⎢xn−1 ⎥⎥⎥⎥⎢⎣ n−1 ⎥⎦ ⎢⎢⎣
⎦
⎦⎣
sn−1
s0
0
(GT2D )nObviously, delay matrix Λ(z) in PUFB is also denoted as
Λ2D (z)x  Λ(z)xΛ(z)T . These lifting matrices by 2DBTof separated transform to each building block are more efficient improvement for lossless-to-lossy image coding because the number of rounding operations in one system isreduced from 3M to 3M/2 and the rounding error is not generated so much. When filter sizes are 4 × 8, 4 × 12, 8 × 16
and 8 × 24, the number of rounding operations is shown in
Table 1.
4.

Results

In this paper, the proposed PUFBs are validated by lossless bit rate [bpp] in lossless image coding, and PSNR [dB]in lossy image coding. For lossless-to-lossy image coding,we used 4 × 8, 4 × 12, 8 × 16 and 8 × 24 PUFBs whichhave same filter properties as LBPUFB [8], and applied the
proposed direct lifting factorization to each building blockin them. Frequency responses of the designed PUFBs areshown in Fig. 6. As targets for comparison, LBPUFBs,
which use same filters as our PUFBs and diﬀerent lifting
factorization, and 5/3- and 9/7-tap DWTs [4] were chosen.

SUZUKI and IKEHARA: PUFBS BASED ON DIRECT LIFTING FOR LOSSLESS-TO-LOSSY IMAGE CODING

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Fig. 6 Frequency responses of the proposed PUFBs and the conventional PUFBs: (a) 4 × 8, (b) 4 × 12,(c) 8 × 16, (d) 8 × 24.
Table 2Image
512×512BaboonBarbaraBoat
ElaineFinger1
Finger2
Goldhill
Grass
Lena
PepperAvg.

5/3-tap
DWT [4]
6.17
4.87
5.10
5.11
5.84
5.60
5.01
6.06
4.49
4.85
5.31

4×8
6.24
4.88
5.14
5.17
5.82
5.63
5.14
6.11
4.66
4.95
5.37

Comparison of lossless image coding.
Lossless bit rate [bpp]Conv. PUFBs [8]4 × 12
8 × 168 × 24
6.21
6.25
6.23
4.79
4.88
4.82
5.10
5.16
5.13
5.12
5.12
5.06
5.74
5.70
5.68
5.50
5.48
5.43
5.11
5.15
5.14
6.07
6.07
6.05
4.61
4.72
4.68
4.90
5.00
4.95
5.32
5.35
5.32

To evaluate transform performance equally, a very common
wavelet-based coder EZW-IP [17] was adopted for all. Also,
for image boundary, the periodic extension was used in our
proposals and LBPUFBs, and the symmetric extension wasused in 5/3- and 9/7-tap DWTs. Moreover, we used ten
512 × 512 gray scale images such as “Barbara” and “Lena”
etc.

4×8
6.23
4.83
5.11
5.16
5.81
5.60
5.11
6.10
4.62
4.92
5.35

Prop. PUFBs
4 × 12
8 × 16
6.20
6.24
4.74
4.77
5.07
5.10
5.11
5.08
5.73
5.67
5.48
5.42
5.09
5.09
6.07
6.06
4.58
4.63
4.88
4.95
5.30
5.30

8 × 24
6.22
4.70
5.07
5.01
5.64
5.36
5.08
6.04
4.59
4.89
5.26

4.1 Application to Lossless Image Coding
First, the designed PUFBs are applied to lossless image coding. SIB sn−1 is just encoded with no compression because
it has only very few bit data size:

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Table 3
Image
512×512BaboonBarbaraBoat

ElaineFinger1

Finger2

Goldhill

Grass

Lena

Pepper

Bit rate
[bpp]
0.25
0.50
1.00
0.25
0.50
1.00
0.25
0.50
1.00
0.25
0.50
1.00
0.25
0.50
1.00
0.25
0.50
1.00
0.25
0.50
1.00
0.25
0.50
1.00
0.25
0.50
1.00
0.25
0.50
1.00

9/7-tap
DWT [4]
22.82
24.60
27.61
27.25
30.49
34.88
28.46
31.39
34.51
31.51
32.94
34.61
23.51
25.99
29.07
24.18
27.53
30.90
29.67
31.53
34.51
24.36
26.09
28.68
32.53
35.52
38.65
31.94
34.48
36.09

4×8
22.61
24.53
27.58
27.11
30.67
35.17
28.11
31.18
34.58
30.97
32.26
34.51
22.87
25.70
29.37
22.52
25.83
29.76
28.79
31.35
34.59
24.18
26.14
28.89
31.38
34.93
38.69
30.58
33.40
35.94

Comparison of lossy image coding.
PSNR [dB]
Conv. PUFBs [8]4 × 12
8 × 16
22.73
22.69
24.70
24.66
27.74
27.64
27.62
27.97
31.23
31.54
35.68
35.70
28.33
28.32
31.36
31.25
34.76
34.56
31.29
31.08
32.96
32.60
34.80
34.86
23.20
23.66
26.12
26.52
29.80
30.11
23.33
23.82
26.61
27.03
30.61
31.65
29.00
29.09
31.52
31.59
34.74
34.67
24.35
24.48
26.30
26.41
29.08
29.15
32.09
31.90
35.39
35.07
38.88
38.37
31.64
30.81
34.33
33.20
36.19
35.73

8 × 24
22.77
24.73
27.71
28.38
31.98
36.07
28.41
31.38
34.65
31.34
33.29
35.24
23.79
26.59
30.21
24.55
27.62
32.40
29.14
31.67
34.69
24.57
26.46
29.22
32.23
35.39
38.54
31.67
34.17
35.91

4×8
22.61
24.54
27.60
27.12
30.70
35.28
28.11
31.20
34.68
30.99
32.37
34.61
22.87
25.70
29.40
22.53
25.84
29.80
28.80
31.39
34.67
24.19
26.15
28.93
31.42
35.00
38.89
30.60
33.52
36.06

Prop. PUFBs
4 × 12
8 × 16
22.74
22.69
24.71
24.68
27.76
27.68
27.64
27.99
31.25
31.62
35.79
35.96
28.33
28.35
31.38
31.32
34.85
34.74
31.32
31.12
33.00
32.88
34.89
35.12
23.19
23.66
26.10
26.53
29.80
30.20
23.33
23.82
26.61
27.07
30.64
31.89
29.02
29.12
31.56
31.67
34.82
34.87
24.35
24.49
26.30
26.43
29.10
29.22
32.13
31.95
35.49
35.22
39.10
38.92
31.68
30.89
34.38
33.44
36.33
35.97

8 × 24
22.78
24.73
27.75
28.41
32.13
36.44
28.44
31.47
34.88
31.40
33.44
35.54
23.79
26.71
30.46
24.58
27.88
32.66
29.16
31.73
34.90
24.59
26.52
29.32
32.31
35.59
39.09
31.73
34.29
36.19

Fig. 7 Enlarged images of “Barbara” in lossy image coding (bit rate: 0.25 [bpp]): (a) 9/7-tap DWT[4], (b) conventional 8×24 PUFB [8], (c) proposed 8×24 PUFB.

SIB bit rate [bpp] =log2 (|sn−1 |max ) + 1M 2 [bit]Total number of pixels [pixel]

where . is ceiling of (.). For example, the size is only
512 [bits] (0.01953125 [bpp]) in 512×512 image “Barbara”
transformed by the proposed 8 × 24 PUFB. The comparisonofLossless bit rate [bpp] =Total number of bits [bit]+ SIB bit rate [bpp]Total number of pixels [pixel]

in lossless image coding is shown in Table 2. The proposed PUFBs showed better coding performance than all of
the conventional methods in especially 8 × 24 PUFB. Even
though the proposed PUFBs and the conventional PUFBs
have same transfer function, all proposed PUFBs with lessrounding operations show better coding performance thanthe conventional PUFBs. We consider that this is mainly
due to the reduction of rounding operations as shown in Table 1.

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4.2 Application to Lossy Image Coding

References

If lossy compressed data is required, it can be achieved byinterrupting the obtained lossless bit stream without SIB
sn−1 . In this paper, sn−1 is replaced by 0 for simplicity. The
comparison of2552
PSNR [dB] = 10 log10
MSE[1] W. Pennebaker and J. Mitchell, JPEG, Still Image Data Compression
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[2] K.R. Rao and P. Yip, Discrete Cosine Transform Algorithms, Academic Press, 1990.
[3] A. Skodras, C. Christopoulis, and T. Ebrahimi, “The JPEG2000 still
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[4] G. Strang and T. Nguyen, Wavelets and Filter Banks, WellesleyCambridge Press, 1996.
[5] W. Sweldens, “The lifting scheme: A custom-design construction ofbiorthogonal wavelets,” Appl. Comput. Harmon. Anal., vol.3, no.2,
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[6] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Englewood
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[7] X. Gao, T.Q. Nguyen, and G. Strang, “On factorization of Mchannel paraunitary filterbanks,” IEEE Trans. Signal Process.,
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[8] T. Suzuki, Y. Tanaka, and M. Ikehara, “Lifting-based paraunitary filterbanks for lossy/lossless image coding,” Proc. EUSIPCO’06, Florence, Italy, Sept. 2006.
[9] K. Komatsu and K. Sezaki, “Design of lossless block transforms andfilter banks for image coding,” IEICE Trans. Fundamentals, vol.E82A, no.8, pp.1656–1664, Aug. 1999.
[10] M. Okuda, S.K. Mitra, M. Ikehara, and S. Takahashi, “Filter bankswith nonlinear lifting steps for lossless image compression,” IEICE
Trans. Fundamentals, vol.E84-A, no.3, pp.797–801, March 2001.
[11] T.D. Tran, “M-channel linear phase perfect reconstruction filter bankwith rational coeﬃcients,” IEEE Trans. Circuits Syst. I, vol.49, no.7,
pp.914–927, July 2002.
[12] Y.J. Chen and K.S. Amaratunga, “M-channel lifting factorizationof perfect reconstruction filter banks and reversible M-band wavelet
transforms,” IEEE Trans. Circuits Syst. II, vol.50, no.12, pp.963–
976, Dec. 2003.
[13] Y.J. Chen, S. Oraintara, and K.S. Amaratunga, “Dyadic-based factorizations for regular paraunitary filterbanks and M-band orthogonal wavelets with structural vanishing moments,” IEEE Trans. Signal Process., vol.53, no.1, pp.193–207, Jan. 2005.
[14] S. Iwamura, Y. Tanaka, and M. Ikehara, “An eﬃcient lifting structure of biorthogonal filter banks for lossless image coding,” Proc.
ICIP’07, San Antonio, TX, Sept. 2007.
[15] T. Suzuki and M. Ikehara, “Lifting factorization based on block parallel system of M-channel perfect reconstruction filter banks,” Proc.
EUSIPCO’09, Glasgow, Scotland, Aug. 2009.
[16] A. Said and W.A. Pearlman, “A new, fast, and eﬃcient image codec
based on set partitioning in hierarchical trees,” IEEE Trans. Circuits
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[17] Z. Liu and L.J. Karam, “An eﬃcient embedded zerotree wavelet image codec based on intraband partitioning,” Proc. ICIP’00, Vancouver, British Columbia, Canada, Sept. 2000.

where MSE is the mean squared error in lossy image coding are shown in Table 3. Furthermore, Fig. 7 shows theenlarged images of “Barbara” which are lossy compressed
images by the proposed PUFBs and the conventional methods when bit rate is 0.25 [bpp]. In Fig. 7, it is obvious thatthe high frequency region of the reconstructed image by our
PUFB is well-approximated because M-channel PUFB has
more excellent frequency resolution in high frequency than
DWTs. Moreover, although JPEG2000 is implemented by
two diﬀerent filter in lossy and lossless mode, our proposals
can show better coding performance with only one filter.
5.

Conclusion

In this paper, we presented a paraunitary filter bank (PUFB)
based on a direct lifting structure of a building block and its
inverse transform for lossless-to-lossy image coding. Our
proposals are obtained by the following processes:
1) The system diag{G, G, · · · , G, {GT }n }, where G is abuilding block in PUFB, is prepared.
2) Side information block (SIB) s0 to be iteratively transformed by inverse transform GT is prepared.
3) Each combination of G and GT in 1), which isdiag{G, GT }, is factorized into direct lifting structures.
4) Additionally, the number of rounding operations is
reduced by using two-dimensional block transform(2DBT) of separated transforms on each building
block.
5) In lossless mode, the image is reconstructed from all
encoded lossless bit stream and SIB.
6) In lossy mode, the image is reconstructed by interrupting the lossless bit stream without SIB.
As result, our proposals show better coding performance
than the conventional methods for lossless-to-lossy image
coding. In future work, side information and computational
cost should be deleted and reduced, respectively.
Acknowledgment
This work is supported in part by a Grant-in-Aid for the
Global Center of Excellence for High-Level Global Cooperation for Leading-Edge Platform on Access Spaces from the
Ministry of Education, Culture, Sport, Science, and Technology in Japan.

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Taizo Suzuki
received the B.E. and M.E.
degrees in electrical engineering from Keio University, Yokohama, Japan, in 2004 and 2006, respectively. He joined Toppan Printing Co., Ltd.,Tokyo, Japan, from 2006 to 2008. He is currently working toward a Ph.D. degree and belongs a Global Center of Excellence (G-COE)
research assistant (RA) at Keio University. His
research interests are in the field of filter bankdesign and its application for multimedia signal
processing.

Masaaki Ikehara
received the B.E., M.E.,
and Dr.Eng. degrees in electrical engineeringfrom Keio University, Yokohama, Japan, in
1984, 1986, and 1989, respectively. He was Appointed Lecturer at Nagasaki University, Nagasaki, Japan, from 1989 to 1992. In 1992, hejoined the Faculty of Engineering, Keio University. From 1996 to 1998, he was a visiting researcher at the University of Wisconsin, Madison, and Boston University, Boston, MA. He iscurrently a Full Professor with the Department
of Electronics and Electrical Engineering, Keio University. His researchinterests are in the areas of multirate signal processing, wavelet image coding, and filter design problems.