1

Redefined Block-Lifting-based Filter Banks with
Efficient Reversible Nonexpansive Convolution
Taizo Suzuki, Senior Member, IEEE, Naoki Tanaka, Hiroyuki Kudo, Member, IEEE

Abstract—This study redefines a block-lifting structure of M channel (M ∈ N, M ≥ 2) filter banks and proposes an efficient
reversible nonexpansive convolution at the boundaries for lossyto-lossless image coding. The previous studies left two problems.
One is that the conventional lifting-based FBs are restricted tohaving equal analysis/synthesis filter lengths. We derive blocklifting-based FBs (BLFBs) with not only equal analysis/synthesis
filter lengths but also the longer synthesis filter lengths than thoseof the analysis banks. The other problem is that the conventional
lifting-based FBs without the linear-phase (LP) property, such as
BLFBs, cannot implement the conventional smooth nonexpansive
convolution at the boundaries because of the rounding error ineach lifting. We solve the boundary problem by using an efficient
reversible nonexpansive convolution derived from a nonexpansive
convolution for nonlinear-phase FBs with paraunitariness. We
show that the redefined BLFBs (ReBLFBs) with the efficient
reversible nonexpansive convolution perform well at lossy-tolossless image coding.
Index Terms—Block-lifting structure, filter bank, filter length,
lossy-to-lossless image coding, nonexpansive convolution.

I. I NTRODUCTIONMAGE compression (coding), which is one of the mostuseful preprocessing technologies of communications, isclassified into lossy and lossless. Lossy image coding dramatically compacts the data size of natural images taken by digital
cameras and smartphones at the expense of image quality,
whereas lossless image coding losslessly compacts the data
size of medical, satellite, and art images at the expense of
the compression rate; i.e., image quality and compression rate
are in a trade-off. To cover all images in a cross-sectoral
manner, lossy-to-lossless image coding, which has scalability
from lossless to lossy data, has been incorporated in image
coding standards such as JPEG 2000 [1] and JPEG XR [2].1
The lifting structure [5], that maps integer input signals
to integer output signals by performing a rounding operation
before the addition in each lifting step, contributes to achieving
the lossy-to-lossless image coding. The JPEG 2000 uses 5/3and 9/7-tap discrete wavelet transforms (DWTs), and theirvarious related works have been presented [6–9]. However,since DWTs, which are considered as two-channel filter banks

I

T. Suzuki and H. Kudo are with the Faculty of Engineering, Information and
Systems, University of Tsukuba, Tsukuba, Ibaraki, 305-8573 Japan (e-mail:{taizo, kudo}@cs.tsukuba.ac.jp).
N. Tanaka is with the Department of Computer Science,
University of Tsukuba, Tsukuba, Ibaraki, 305-8573 Japan (e-mail:tanaka@imagelab.cs.tsukuba.ac.jp).
1 This study focuses on “lossy-to-lossless image coding.” Although there are
the newer image coding methods with machine learning and neural networks
as in [3], [4], which may outperform the traditional transform coding as inthis study, they are specialized for only lossy image coding and cannot beapplied to lossless image coding.

(FBs), may be limited to design the better transforms becauseof a few free parameters, many M -channel (M ∈ N, M ≥ 2,
M is even) lifting-based FBs have been presented in theliterature [10–19]. In [11], Tran et al. introduced time-domain
lapped transforms (TDLTs), which can be directly applied to
the JPEG framework. In [14], Tu et al. introduced a 4 × 8
lifting-based hierarchical lapped transform (HLT) for the JPEG
XR by extending TDLTs. However, since FBs with linearphase (LP) property such as TDLT and HLT may be also
limited to design the better transforms, several nonlinear-phase
FBs have been studied for more efficient image coding. In[17], Iwamura et al. presented nonlinear-phase FBs based ona block-lifting structure that reduces the rounding error by
merging many rounding operations, called block-lifting-based
FBs (BLFBs), because the rounding error reduces the coding
efficiency. In [18], we generalized the BLFBs; i.e., we allowed
the McMillan degree rk to be 1 ≤ rk < M and allowed
both even and odd block size. In [19], we presented their 2-D
nonseparable structures.2
On the other hand, it is known that the analysis banks
basis functions (or impulse response) should be short toavoid ringing artifacts around regions with high-frequency
components, whereas the synthesis banks should be long andtheir coefficients should decay to zero smoothly at both ends
to avoid blocking artifacts around regions with low-frequency
components [23]. Nevertheless, the conventional lifting-based
FBs are restricted to having equal analysis/synthesis filter
lengths. Moreover, although our previous work [24] allows
the traditional smooth nonexpansive convolution, a symmetricextension [25], at the boundaries in lifting-based FBs with a
LP property, it does not allow any conventional nonexpansive convolution to be used in M -channel nonlinear-phase
cases [24–27] because of the rounding error. In particular,
the M -channel nonlinear-phase cases except for M = 2 musthave a common periodic extension, whose discontinuity at theboundaries yields the annoying artifacts.
This study redefines a block-lifting structure of M -channel
FBs for lossy-to-lossless image coding. The redefined BLFBs(ReBLFBs) can derive not only equal analysis/synthesis filter lengths but also the longer synthesis filter lengths than
the analysis ones. Also, we produce an efficient reversible
nonexpansive convolution unrestricted by the LP property.
2 This study focuses on “transforms for general natural images,” i.e., they
are basic technologies that can be customized if necessary and widelyused. Although there are some newer studies for lossy-to-lossless image
coding [20–22], they have proposed more efficient uses of the traditionaltransforms such as DWTs and the methods for specified images such as
medical and hyperspectral image.

2

A preliminary work of the nonexpansive convolution will
be presented in [28], where we discussed only FBs withparaunitariness and equal analysis/synthesis filter lengths. This
paper describes the unrestricted case for the characteristics. We
show that the ReBLFBs with the nonexpansive convolution
perform well at lossy-to-lossless image coding.
The remaining part of this paper is organized as follows.
Section II reviews the BLFBs. Section III redefines the BLFBs
for a structure embodying the conventional block-lifting structure and proposes an efficient reversible boundary processing.
Section IV presents the filter design and the experimental
results. Section V concludes this paper.
Notations: I, J, O, det(·), superscript ⊤, and subscript [m](m ∈ N) denote an identity matrix, reversal matrix, zeromatrix, determinant of a matrix, transpose of a vector/matrix,
and m × m matrix, respectively. The subscript [m] is omitted
if its size is clear.

addition, when V k = U k , the FBs are paraunitary filterbanks (PUFBs) and the others are biorthogonal FBs (BOFBs).
This study eases the restriction on the filter lengths andparaunitariness, i.e., W k ̸= I[rk ] , and produces FBs withlonger synthesis filter lengths than those of the analysis banks.

II. R EVIEW AND D EFINITION−1
Ek (z) = L−1
k Uk Λk (z)Uk Lk ,B. Block-Lifting-based Filter Banks
The lifting structure [5] is a cascade of elementary matrices,which are identity matrices with one single nonzero offdiagonal element. It preserves reversibility if the rounding
operation is performed before the addition in each lifting step.
The block-lifting structure [17–19], which is a special class oflifting structure, achieves good lossy-to-lossless image coding
because it reduces the rounding error, which adversely affects
coding efficiency, by merging many rounding operations.
Through the restriction W k = I[rk ] , the block-lifting-based
building block Ek (z) is expressed as

A. Filter Banks
The polyphase matrix of causal M -channel maximally decimated (K − 1)th order (M × M K, K ∈ N) FBs is factorized
as follows [29]:E(z) = EK−1 (z)EK−2 (z) · · · E1 (z)X0 ,[]I[qk ]
O
Lk = −⊤ ⊤v v
I[rk ][ k1 k0]⊤
I[qk ] uk0 vk1
Uk =.
O
I[rk ](1)

where the initial block X0 is an M × M nonsingular matrix,
and Ek (z) (1 ≤ k ≤ K − 1) can be factorized into−1
U kV⊤
Ek (z) = I[M ] − U k V ⊤
k,
k +z

where

k

for some integer 1 ≤ rk < M , where × indicates possibly
nonzero elements. Let U k and V k be
[
]
uk0
Uk =
(4)
uk1
[ ]
v
V k = k0 ,(5)
vk1
where uk0 and vk0 are qk × rk (qk = M − rk ) matrices and
uk1 and vk1 are rk ×rk square matrices, respectively, for later
discussion. The synthesis polyphase matrix R(z) is given by−1−1
R(z) = X−1
0 E1 (z) · · · EK−1 (z),which is anticausal as a result of (3) and satisfies R(z)E(z) =I for perfect reconstruction [30]. Since the rank of W k in (3)is rk , the McMillan degree of Ek (z), as in (2), is rk and thematrix Ek (z) is a degree-rk building block. If rk = M
2 (MM
is even), ⌈ M
⌉
(M
and
k
are
odd),
or
⌊
⌋
(M
is
odd
and k
2
2
is even) [31], the Ek (z) is an order-1 building block. K must
be odd when M is odd.
Almost all studies impose the restriction W k = I[rk ] thatguarantees equal filter lengths in analysis/synthesis banks. In

(7)
(8)

The inverse building block E−1
k (z) is obtained as follows:
−1
−1
−1
E−1
k (z) = Lk Uk Λk (z)Uk Lk .

(2)

where the M × rk parameter matrices U k and V k satisfy
1 × ··· ×.
0 1 . . . .. ⊤
Vk Uk =  .
≜ Wk
(3)

.
.
.
.
.
.
.
. ×
0 · · · 0 1 [r ]

(6)

(9)

Note that the initial block X0 is restricted to | det(X0 )| = d(d ∈ N) for the purpose of making a lifting factorization. It canbe factorized into non block-lifting structures, e.g., the singlerow elementary reversible matrices (SERMs) presented in [10].
In addition, lifting-based FBs without the LP property, such
as BLFBs, cannot directly use the conventional smooth nonexpansive convolution [24–27] at the boundaries because of
the rounding error. A periodic extension, which is commonlyused for boundary processing in the M -channel nonlinearphase cases except for M = 2, causes annoying boundary
artifacts in low-bit compression because of the discontinuity.
III. R EDEFINED B LOCK -L IFTING - BASED F ILTER BANKSWITH R EVERSIBLE N ONEXPANSIVE C ONVOLUTION
A. Redefined Block-Lifting Structure for Filter Banks
Theorem: By not setting the restriction W k = I[rk ] thatis used in [18], we can redefine a building block Ek (z) as amore generalized block-lifting structure as follows (Fig. 1):−1
Ek (z) = L−1
k Wk (z)Uk Λk (z)Uk Lk ,

where
[

I[qk ]Wk (z) =
O

O
fk
W

] [I

[qk ]

O

(10)

)
( O−1
f − I[r ]I[r ] + z W
k

k

]

k

(11)
f k = v−⊤ W k v⊤ .
W
k1
k1

(12)

3

×M
signals, respectively): (left) analysis bank,Fig. 1. Building block of ReBLFB (areas enclosed by dashed lines mean the new part, and arrows mean M22(right) synthesis bank.

Wk (z) is the new part of the block-lifting structure. The
f k in the first term diag(I[q ] , W
f k ) in (11) can be easilyWkfactorized into a variety of lifting structures becausef k ) = det(v−⊤ ) · det(W k ) · det(v⊤ ) = 1,
det(W
k1
k1

(13)

where
det(W k ) = 1

and

−⊤
⊤
det(vk1) · det(vk1) = 1,(14)

which is a sufficient condition for the purpose of making
a lifting factorization. The second term diag(I[qk ] , I[rk ] +−1
f − I[r ] )) in (11) can already be considered a lifting
z(W
k

k

−1

f −I[r ] ). The inverse
structure with a lifting coefficient z(Wkk−1building block Ek (z) is obtained as follows:
−1
−1
−1
−1
E−1
k (z) = Lk Uk Λk (z)Uk Wk (z)Lk ,

where
W−1
k (z) =[

I[qk ]O)( O−1
f − I[r ]I[r ] − z W
k

k

k

][

I[qk ]O

By using the upper block-lifting matrix Uk expressed as (8),its inverse U−1
k , and Λk (z), (18) can be further factorized as−1e
Lk Ek (z)L−1
k = Wk Uk Λk (z)Uk ≜ Ek (z).

(19)

Consequently, Ek (z) can be factorized into the lifting matrices
as in (10) because−1−1e
Ek (z) = L−1
k Ek (z)Lk = Lk Wk (z)Uk Λk (z)Uk Lk .
(20)

(15)
]
O
−1 .
f
W
k

(16)
When W k = I[rk ] , i.e., Wk (z) = I[M ] , (10) and (15)are completely equivalent to the conventional building block
Ek (z) and the inverse E−1
k (z) in [18].
Proof: In accordance with [18], Ek (z) in (2)-(5) isrewritten as
Ek (z)()[⊤
I[qk ](+ z −1 −1 uk0 vk0)=−1⊤
z − 1 uk1 vk0

The lower block-lifting matrix Lk expressed as (7) and its
inverse L−1
k are respectively multiplied from the left and rightsides of Ek (z) as follows:
( −1
)
]
[
⊤
I[qk ]
z − 1 uk0 vk1()(18)
Lk Ek (z)L−1=kfk .
O I[rk ] + z −1 − 1 W

( −1
)
]
⊤
z ( − 1 uk0
vk1
)
⊤ .
I[rk ] + z −1 − 1 uk1 vk1(17)

B. Reversible Nonexpansive Convolution for Lifting-based Filter BanksThis subsection uses the following FB:
E(z) = XK−1 ΛK−1 (z) · · · X1 Λ1 (z)X0 ,(21)

where Xk is a nonsingular matrix, M is even, and rk = M
2 for
simplicity. The top images of Figs. 2 and 3 show the upperboundary processing of the FBs when K = 2 and 3. They
extra signals have to be extended atmean that L = (K−1)M2each boundary. To obtain smooth signals, the extra signals arecommonly extended by using an L × L extension matrix:
E [L] = J[L] ;(22)i.e., it is a symmetric extension [25]. We derive an efficient reversible nonexpansive convolution for lifting-based FBs froma nonexpansive convolution for PUFBs [26] as follows:4Fig. 2. Upper boundary processing of M × 2M FBs (dashed lines and arrows indicate the upper boundary and M
×M
signals, respectively): (top) simple22
signal extension; (right) nonexpansive convolution.

Fig. 3. Upper boundary processing of M × 3M FBs (dashed lines and arrows indicate the upper boundary and M
×M
signals, respectively): (top) simple22
signal extension; (right) nonexpansive convolution.

1) Remove the paraunitariness restriction.
2) Use an arbitrary extension matrix E [L] , not a symmetricextension matrix J[L] .
3) Restrict the determinant of the resulting boundary matrix
Fk to | det(Fk )| = d (d ∈ N) for the purpose of making
a lifting factorization [10].
Here, we have a nonexpansive problem because the signals
transmitted to the synthesis bank should only be the output
yn (n ∈ N). This subsection presents a reconstruction method
at the synthesis bank without any extra signal; i.e., the input
signal xn is reconstructed from only the output signal yn . To
facilitate the discussion, let an M × M matrix M be
[
M=

[M]α
[M]γ

]
[M]β
,
[M]δ

(23)

M
where each submatrix is M
2 × 2 in size. Hereafter, we consider
only the upper boundary processing. The signals at the lower
boundary processing can be reconstructed in the same way asin the upper boundary processing.

1) Case of K = 2: From the top of Fig. 2, the middle
signal a0 is expressed as[][ ][] E[ M ]Ox02
a0 = [X0 ]α [X0 ]β= F0 x0 , (24)O
I [ M ] x02

where
F0 = [X0 ]α E [ M ] + [X0 ]β2s.t.

| det(F0 )| = d.

(25)

The problem in the synthesis banks is to reconstruct the input
signal x0 from the middle signal a0 , which is transmitted tothe synthesis bank. The input signal x0 is represented asx0 = F−1
0 a0 .

(26)

The nonexpansive convolution of the FBs in case of K = 2
is shown at the bottom of Fig. 2.
2) Case of K = 3: From the top of Fig. 3, the middle
signals b−1 , b0 , and b1 are expressed as
[ ]
[
]
x
b−1 = [X0 ]α [X0 ]β E [M ] 0
(27)
x1
[ ]
[ ]
b1
x
= X0 0 .
(28)
b0
x1

5

By substituting
[

x0
x1

]

[ ]
b1
= X−1
,
0
b0

(29)

which is obtained from (28), into (27), the middle signal b−1
is represented as
[ ]
[
]
−1 b1
= F b0 + Gb1 ,b−1 = [X0 ]α [X0 ]β E [M ] X0
b0
(30)
where

(
[
]
[
] )[
]
F = [X0 ]α E [M ] α + [X0 ]β E [M ] γ X−1
0
β
(
[
]
[
] ) [ −1 ]+ [X0 ]α E [M ] β + [X0 ]β E [M ] δ X0 δ
(
[
]
[
] )[
]
G = [X0 ]α E [M ] α + [X0 ]β E [M ] γ X−1
0
α
(
[
]
[
] ) [ −1 ]+ [X0 ]α E [M ] β + [X0 ]β E [M ] δ X0 γ .

(31)

(32)

Substituting (30) into (33) yields
(34)

where
F1 = [X1 ]α F + [X1 ]βs.t.

| det(F1 )| = d

G1 = [X1 ]α G.

(35)
(36)

Unlike the determinant of F1 , that of G1 is not restricted toa natural number because the processing including G1 canalready be considered a lifting structure. The problem in thesynthesis banks is to reconstruct the middle signal b0 from the
middle signals a0 and b1 , which is transmitted to the synthesisbank. The middle signal b0 is represented asb0 = F−1
1 (a0 − G1 b1 ) .

z −M

1

z=1

0

where [1, 1, · · · , 1]⊤ are DC components, c is a nonzero
constant, and any building block Ek (z) in the ReBLFB E(z)is skipped as follows:()−1
Ek (z)|z=1 = L−1
k Wk (z)Uk Λk (z)Uk Lk z=1−1= L−1
k I[M ] Uk I[M ] Uk Lk = I[M ] .

Also, the middle signal a0 is expressed as
[
]
] b−1
[
a0 = [X1 ]α [X1 ]β= [X1 ]α b−1 + [X1 ]β b0 .
b0(33)

a0 = F1 b0 + G1 b1 ,the ReBLFBs can be structurally imposed in the same way asin [27], because [29]   11c
 z −1 
1 0   = E0  .  =  .  ,(38)
E(z)  ..  . 
 ..   .. (37)

The nonexpansive convolution in the case of K = 3 is shownat the bottom of Fig. 3. For any K, the signals can bereconstructed as the solution of a simultaneous matrix equation
with (K − 1) unknowns.
The new nonexpansive convolution can be used by all
lifting-based FBs (not only ReBLFBs) having a lattice structure. In addition, the new part Wk (z) of the ReBLFBs withlonger synthesis filters than the analysis ones is skipped at theboundaries for simplicity.

C. Regularity for Redefined Block-Lifting-based Filter BanksFB theory indicates that regularity, especially one degree of
regularity (one-regularity), is an important property to prevent
the DC leakage for image compression [29]. One-regularity in(39)

Moreover, we must consider regularity in the proposedboundary processing. If the resulting ReBLFBs achieve structural one-regularity, we can also achieve it at the boundaries
by preserving the DC components in the input signals; i.e.,
the extension matrix E [L] should be structurally restricted to   1
1
1 1   (40)
E [L]  .  =  .  .
 ..   .. 
1

1E.g., the extension matrix E [L] can be set as


∑L−2
e0,0
···
e0,L−2
1 − j=0 e0,j
 .

..
..
..
 ..

.
.
.

,
E [L] = ∑L−2eL−2,0 · · · eL−2,L−2 1 − j=0 eL−2,j ∑L−2
eL−1,0 · · · eL−1,L−2 1 − j=0 eL−1,j(41)
where ei,j (i = 0, 1, · · · , L − 1, j = 0, 1, · · · , L − 2) is anarbitrary parameter.
D. Specific Filter Design
We specifically designed 4 × 8, 4 × 8/4 × 12, and 4 × 12
ReBLFBs, which had rk = 2 and structural one-regularity. The
structures of 4 × 8 and 4 × 12 ReBLFBs are completely sameas the conventional structures in [18]. We used the following
cost function ϕ, which is a weighted linear combination of thecoding gain [32], the determinant control of matrices, and thesmoothness of the extension matrix E [L] :ϕ = −w0 Ccg + w1 Cdet + w2 Csmooth ,

(42)

where

∑M −1 2
1σ
Ccg = 10 log10 ( M i=0 )i 1
(43)
∏M −1 2 2 N
i=0 σi fi
(
( )
)2 (
( )
)2
Cdet = det F↑k − d + det F↓k − d
(44)
{
)
L−1
∑ L−1
∑ ([ ↑ ][] 2
Csmooth =E [L] − S [L] ij
i=0 j=0

ij

) }
([
]
[
] 2
↓
, (45)+ E [L] − S [L] ij
ij

6

TABLE IC ODING GAINS [ D B].

Ccg [dB]

HLT
4×8
[14]
8.45

4×8
same as [18]8.64

ReBLFBs
4 × 8/4 × 12
new
8.66

TABLE IIL OSSLESS IMAGE CODING RESULTS (LBR [ BPP ]; () IS IN A PERIODIC
EXTENSION ).
4 × 12same as [18]8.79

and fminunc.m in the Optimization ToolBox of
MATLAB. σi2 is the variance of the ith subband, and fi2 isthe L2 norm of the ith synthesis filter. When the transformhas paraunitariness, the synthesis scaling fi2 drops out of thecoding gain equation. The input signal is assumed to be an
AR(1) model with autocorrelation coefficient ρ = 0.95 incommon use. The super scripts ↑ and ↓ respectively mean
the matrices at the upper and lower boundary processings ofthe FBs. [·]ij means the (i, j)th element of a matrix. Since it isdifficult to make | det(Fk )| = d exactly by using only the costfunction Cdet , which brings | det(Fk )| as close as possible tod, we used Fdet
k , which is forcibly controlled,
(
) M2
d
=
Fdet
Fk
(46)
k
|det (Fk )|instead of Fk , where we set d = 1 for simplicity. The
cost function Cdet should be imposed because the boundary
processing cannot be implemented smoothly when | det(Fk )|is substantially different from d, i.e., Fdet
is substantiallykdifferent from Fk . For example, | det(F↑k )| in the resulting
4 × 8/4 × 12 ReBLFB was 0.92 ≈ 1. In addition, we set thesmoothness condition as a symmetric extension S [L] = J[L] .
The weights wk (k = 0, 1, 2) are heuristically determinedbecause the resulting filter coefficients may lead to a local
minimum, and this is due to depense of the solution on theinitial values and the weights.
IV. E XPERIMENTAL R ESULTS
A. Coding Gain and Frequency/Impulse Responses
Table I shows the coding gains Ccg of the HLT [14] in JPEG
XR and the resulting ReBLFBs. We can see that the ReBLFBs
have higher coding gains than that of the HLT restricted by
the LP property and the ReBLFBs with longer filter lengths
have higher coding gains than those with shorter filter lengths.
Fig. 4 shows frequency and impulse responses of theresulting ReBLFBs. Those of 4×8/4×12 ReBLFB resembles
those of 4×8 ReBLFB in appearance because the difference instructure is trivial; i.e., W 1 = I[ M ] or W 1 ̸= I[ M ] . However,22
it is clear that unlike the others, the 4 × 8/4 × 12 ReBLFB has
longer synthesis filter lengths than the analysis filter lengths.
B. Lossy-to-Lossless Image Coding
The resulting ReBLFBs were implemented with a rounding
operation at each lifting step and compared in terms of thelossless bitrate (LBR) [bpp]:
Total number of bits [bit]
LBR [bpp] =Total number of pixels [pixel]

Test
ImagesBike
Building
Cafe
Car
Falls
FlowerGirl
House
SakuraWoman
r0ebb0bb4t
r0ed019a4t
r0f24ca4dt
r0fb0a690t
r02d732f8t
r03b0c944t
r08f2d4c6t
r069e346et
r0773471dt
r01170470t
mLBR
SD

HLT
4×8
[14]
5.26
4.17
6.09
4.10
4.18
4.44
4.37
4.94
4.48
4.86
4.64
3.27
4.65
4.35
3.83
4.70
3.92
3.70
5.11
3.96
4.06
0.71

4×8
same as [18]5.18 (5.19)
4.17 (4.18)
6.04 (6.05)
4.10 (4.11)
4.17 (4.19)
4.40 (4.41)
4.39 (4.40)
4.93 (4.94)
4.41 (4.42)
4.83 (4.85)
4.58 (4.58)
3.33 (3.34)
4.58 (4.59)
4.30 (4.30)
3.85 (3.85)
4.61 (4.61)
3.94 (3.95)
3.73 (3.73)
5.02 (5.02)
3.96 (3.97)
4.01 (4.02)
0.70 (0.70)

ReBLFBs
4 × 8/4 × 12
new
5.16 (5.17)
4.12 (4.13)
6.03 (6.04)
4.07 (4.08)
4.12 (4.14)
4.39 (4.40)
4.34 (4.35)
4.87 (4.88)
4.39 (4.40)
4.81 (4.83)
4.55 (4.56)
3.24 (3.24)
4.53 (4.53)
4.26 (4.27)
3.78 (3.79)
4.58 (4.58)
3.90 (3.91)
3.68 (3.69)
5.00 (5.00)
3.86 (3.87)
3.96 (3.97)
0.71 (0.71)

4 × 12same as [18]5.16 (5.17)
4.14 (4.15)
6.02 (6.03)
4.11 (4.13)
4.14 (4.15)
4.42 (4.42)
4.36 (4.37)
4.88 (4.90)
4.42 (4.43)
4.83 (4.84)
4.56 (4.56)
3.27 (3.28)
4.52 (4.53)
4.28 (4.28)
3.80 (3.80)
4.59 (4.59)
3.92 (3.92)
3.71 (3.72)
4.99 (4.99)
3.87 (3.88)
3.99 (3.99)
0.70 (0.70)

in lossless image coding and in terms of the peak signal-tonoise ratio (PSNR) [dB]:(()2 )2BIT − 1
PSNR [dB] = 10 log10,
MSE
where BIT and MSE are the number of bits of the image andthe mean squared error, and multi-scale structural similarity(MS-SSIM) [35], which is an image quality assessment basedon human visual characteristics unlike PSNR, in lossy image
coding. To evaluate the transform performance fairly, we employed three-level decompositions on all filters. The image set
included 1024 × 1024 and 2048 × 2048 clipped 8-bit standard
grayscale images in [33], [34] (see Fig. 5). Also, in addition to
the results of images in Fig. 5, the mean LBR (mLBR), mean
PSNR (mPSNR), mean MS-SSIM (mMS-SSIM), and standard
deviation (SD) of the other 100 images in [34] are shown. A
quadtree-based embedded image coder [36], which is moresuited to block transforms than are the popular zerotree-based
coders [37], [38], was used to encode the transformed images.
Tables II-IV and Fig. 6 show the results of lossless andlossy image coding. Although the HLT, 4 × 8 ReBLFB, and4 × 12 ReBLFB sometimes performed the best, the 4 × 8/4 ×
12 ReBLFB basically outperformed the others. In addition,
the 4 × 8/4 × 12 ReBLFB avoided ringing artifacts aroundregions with high-frequency components as well as blocking
artifacts around regions with low-frequency components andthe proposed nonexpansive convolution helped to inhibit theboundary artifacts.
V. C ONCLUSION
This study redefined a block-lifting structure of M -channel
FBs and proposed an efficient reversible nonexpansive convolution for lossy-to-lossless image coding. We solved two

7Fig. 4. Frequency and impulse responses of the ReBLFBs: (top) analysis bank, (bottom) synthesis bank, (left) 4 × 8, (middle) 4 × 8/4 × 12, and (right)
4 × 12.

problems, i.e., the restriction of equal analysis/synthesis filter lengths and lack of an efficient reversible nonexpansive
convolution, left by our previous work. We showed that the
ReBLFBs with the nonexpansive convolution performed wellat lossy-to-lossless image coding.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers
for providing many constructive suggestions that significantly
improve the presentation of this paper. This work was supported by JSPS Grant-in-Aid for Young Scientists (B), Grant
Number 16K18100.
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8Fig. 5. Test images excerpted from data set: (top) Bike, Building, Cafe, Car, Falls, Flower, Girl, House, Sakura, and Woman in [33], (middle and bottom)
r0ebb0bb4t, r0ed019a4t, r0f24ca4dt, r0fb0a690t, r02d732f8t, r03b0c944t, r08f2d4c6t, r069e346et, r0773471dt, and r01170470t in [34].

Fig. 6. Comparison of results in a particular area of the image Cafe (top and left sides are boundaries; bitrate: 0.25 [bpp]): (top-left) original image, (otherleft-to-right) 4 × 8 HLT, 4 × 8 ReBLFB, 4 × 8/4 × 12 ReBLFB, and 4 × 12 ReBLFB, (top) a periodic extension, and (bottom) the proposed nonexpansive
convolution.

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9

TABLE IIIL OSSY IMAGE CODING RESULTS (PSNR [ D B]; () IS IN A PERIODIC
EXTENSION ).

TABLE IV
L OSSY IMAGE CODING RESULTS (MS-SSIM; () IS IN A PERIODIC
EXTENSION ).

Bitrate
[bpp]

HLT
4×8
[14]

Bitrate
[bpp]

HLT
4×8
[14]

0.25
0.50
1.00

24.52
28.16
33.22

24.35 (24.37)
28.20 (28.21)
33.57 (33.56)

0.25
0.50
1.00

0.892
0.944
0.977

0.25
0.50
1.00

20.16
23.02
26.97

20.16 (20.19)
22.93 (22.94)
27.09 (27.07)

0.25
0.50
1.00

0.869
0.923
0.962

0.25
0.50
1.00

31.30
34.66
38.85

31.03 (30.79)
34.42 (34.26)
39.29 (39.22)

0.25
0.50
1.00

0.945
0.977
0.991

0.25
0.50
1.00

33.75
36.30
39.23

34.17 (33.97)
36.59 (36.48)
39.23 (39.17)

0.25
0.50
1.00

0.961
0.979
0.990

0.25
0.50
1.00

28.05
31.75
37.00

29.12 (29.12)
32.89 (32.90)
37.22 (37.18)

0.25
0.50
1.00

0.954
0.980
0.994

0.25
0.50
1.00

28.54
31.63
35.38

28.91 (28.86)
32.33 (32.29)
36.28 (36.26)

0.25
0.50
1.00

0.937
0.969
0.987

0.25
0.50
1.00

27.36
30.63
34.71

27.63 (27.57)
31.14 (31.11)
35.52 (35.49)

0.25
0.50
1.00

0.932
0.969
0.987

0.25
0.50
1.00

33.05
37.07
41.22

33.51 (33.43)
37.72 (37.67)
41.49 (41.48)

0.25
0.50
1.00

0.981
0.990
0.994

0.25
0.50
1.00

34.07
37.40
40.95

34.49 (34.45)
37.90 (37.87)
41.26 (41.24)

0.25
0.50
1.00

0.968
0.985
0.993

0.25
0.50
1.00

27.24
30.10
33.46

27.27 (27.25)
30.17 (30.16)
33.82 (33.83)

0.25
0.50
1.00

0.890
0.953
0.981

0.25
0.50
1.00

32.95
35.93
39.24

33.02 (32.95)
36.05 (36.00)
39.44 (39.41)

0.25
0.50
1.00

0.958
0.980
0.991

0.25
0.50
1.00

5.69
5.16
4.42

5.66 (5.65)
5.08 (5.08)
4.14 (4.14)

0.25
0.50
1.00

0.028
0.014
0.006

ReBLFBs
4×8
4 × 8/4 × 12
same as [18]newBike
24.26 (24.19)
24.48 (24.44)
28.16 (28.10)
28.35 (28.28)
33.50 (33.49)
33.73 (33.72)
Cafe
20.32 (20.31)
20.45 (20.45)
23.20 (23.18)
23.39 (23.39)
27.19 (27.15)
27.41 (27.38)
Falls
31.34 (30.91)
31.39 (30.98)
34.93 (34.56)
35.12 (34.70)
39.03 (38.97)
39.42 (39.35)
Girl
34.11 (33.94)
34.27 (34.12)
36.51 (36.41)
36.74 (36.63)
39.07 (39.02)
39.46 (39.40)
Sakura28.84 (28.79)
29.03 (28.98)
32.64 (32.62)
32.89 (32.85)
37.25 (37.12)
37.55 (37.43)
r0ebb0bb4t
28.77 (28.72)
28.88 (28.82)
32.05 (32.04)
32.25 (32.24)
35.92 (35.92)
36.23 (36.23)
r0f24ca4dt
27.47 (27.42)
27.65 (27.60)
30.95 (30.93)
31.22 (31.18)
35.29 (35.25)
35.60 (35.59)
r02d732f8t
33.33 (33.32)
33.58 (33.53)
37.50 (37.53)
37.83 (37.85)
41.13 (41.14)
41.55 (41.56)
r08f2d4c6t
34.40 (34.33)
34.60 (34.53)
37.72 (37.71)
37.96 (37.94)
40.85 (40.84)
41.29 (41.29)
r0773471dt
27.17 (27.15)
27.27 (27.25)
29.99 (29.97)
30.12 (30.11)
33.53 (33.53)
33.85 (33.84)
mPSNR
32.92 (32.87)
33.10 (33.04)
35.89 (35.85)
36.16 (36.12)
39.09 (39.07)
39.62 (39.60)
SD
5.59 (5.60)
5.62 (5.63)
4.91 (4.93)
5.03 (5.05)
3.85 (3.86)
4.09 (4.10)

4 × 12same as [18]2008.
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coefficients,” IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3445–

ReBLFBs
4×8
4 × 8/4 × 12
same as [18]newBike0.910 (0.910)
0.914 (0.913)
0.953 (0.953)
0.955 (0.955)
0.981 (0.981)
0.982 (0.982)
Cafe
0.880 (0.879)
0.884 (0.884)
0.934 (0.934)
0.937 (0.936)
0.969 (0.969)
0.970 (0.970)
Falls
0.951 (0.945)
0.951 (0.947)
0.980 (0.978)
0.981 (0.978)0.992 (0.992)
0.993 (0.993)
Girl
0.966 (0.965)
0.968 (0.966)
0.982 (0.981)
0.983 (0.982)
0.990 (0.990)
0.991 (0.991)
Sakura0.965 (0.964)
0.966 (0.966)
0.985 (0.985)
0.986 (0.986)
0.994 (0.994)
0.995 (0.995)
r0ebb0bb4t
0.947 (0.946)
0.948 (0.947)
0.975 (0.975)
0.977 (0.976)
0.989 (0.989)
0.990 (0.990)
r0f24ca4dt
0.939 (0.938)
0.940 (0.939)
0.973 (0.973)
0.974 (0.974)
0.990 (0.990)
0.991 (0.991)
r02d732f8t
0.982 (0.982)
0.983 (0.982)
0.991 (0.991)
0.991 (0.991)
0.994 (0.994)
0.995 (0.995)
r08f2d4c6t
0.972 (0.972)
0.973 (0.973)
0.987 (0.987)
0.988 (0.988)
0.993 (0.993)
0.994 (0.994)
r0773471dt
0.899 (0.899)
0.902 (0.902)
0.953 (0.953)
0.955 (0.954)
0.983 (0.983)
0.984 (0.984)
mMS-SSIM
0.961 (0.960)
0.962 (0.962)
0.981 (0.981)
0.982 (0.982)
0.991 (0.991)
0.993 (0.992)
SD
0.026 (0.026)
0.025 (0.025)
0.013 (0.013)
0.012 (0.012)
0.005 (0.005)
0.005 (0.005)

4 × 12same as [18]0.901 (0.901)
0.951 (0.951)
0.980 (0.980)
0.862 (0.861)
0.924 (0.922)
0.966 (0.965)
0.944 (0.942)
0.975 (0.974)
0.992 (0.992)
0.963 (0.962)
0.980 (0.979)
0.990 (0.990)
0.964 (0.964)
0.985 (0.985)
0.994 (0.994)
0.938 (0.938)
0.974 (0.973)
0.989 (0.989)
0.930 (0.930)
0.969 (0.969)
0.990 (0.989)
0.981 (0.980)
0.991 (0.991)
0.995 (0.995)
0.969 (0.968)
0.986 (0.986)
0.994 (0.994)
0.884 (0.884)
0.945 (0.945)
0.981 (0.980)
0.956 (0.956)
0.979 (0.979)
0.991 (0.991)
0.030 (0.030)
0.015 (0.015)
0.006 (0.006)

3462, Dec. 1993.
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10

Taizo Suzuki (S’08-M’11-SM’17) received the
B.E., M.E., and Ph.D. degrees in electrical engineering from Keio University, Japan, in 2004, 2006
and 2010, respectively. From 2006 to 2008, he waswith Toppan Printing Co., Ltd., Japan. From 2008
to 2011, he was a Research Associate of the Global
Center of Excellence (G-COE) at Keio University,
Japan. From 2010 to 2011, he was a Research
Fellow of the Japan Society for the Promotion of
Science (JSPS) and a Visiting Scholar at the Video
Processing Group, the University of California, San
Diego, CA. From 2011 to 2012, he was an Assistant Professor in the
Department of Electrical and Electronic Engineering, College of Engineering,
Nihon University, Japan. Since 2012, he has been an Assistant Professor
in the Faculty of Engineering, Information and Systems, University ofTsukuba, Japan. His current research interests are image and video processing,source coding, and multidimensional transforms. Since 2017, he has been an
Associate Editor of IEICE Trans. Fundamentals.

Naoki Tanaka received the B.E. and M.E. degrees
from the University of Tsukuba, Japan, in 2015 and2017, respectively. In 2017, he joined Sony LSIDesign Inc., Japan. His research interests are signal/image processing and integrated circuit design.

Hiroyuki Kudo (M’88) received the B.Sc. degreefrom the Department of Electrical Communications,
Tohoku University, Japan, in 1985, and the Ph.D.
degree from the Graduate School of Engineering,Tohoku University, in 1990. In 1992, he joined the
University of Tsukuba, Japan, as an Assistant Professor. He is currently a Professor with the Divisionof Information Engineering, Faculty of Engineering,
Information and Systems, University of Tsukuba,
Japan. He is involved in tomographic image reconstruction for X-ray CT, PET, SPECT, and electron
tomography. His research areas include medical imaging, image processing,and inverse problems. He is a member of the Japanese Society of Medical
Imaging Technology and IEICE, Japan. He received the best paper award
several times from various domestic and international academic societies. In
particular, his papers on interior tomography published in 2008 were selected
as High Lights of the two journals, Physics in Medicine and Biology andInverse Problems. Since 2011, he has been an Editor-in-Chief of the Journalof Medical Imaging Technology at MIT.