1

Reversible Symmetric Non-Expansive Convolution:
An Effective Image Boundary Processing for
M -Channel Lifting-based Linear-Phase Filter Banks
Taizo Suzuki, Member, IEEE and Masaaki Ikehara, Senior

Member, IEEE
Abstract—We present an effective image boundary processing for
M -channel (M ∈ N, M ≥ 2) lifting-based linear-phase filter banks(L-LPFBs) that are applied to unified lossy and lossless image compression (coding), i.e., lossy-to-lossless image coding. The reversible
symmetric extension (RevSE) we propose is achieved by manipulatingbuilding blocks on the image boundary and reawakening the symmetryof “each building block” that has been lost due to rounding error
on each lifting step. Moreover, complexity is reduced by extending
non-expansive convolution, called reversible symmetric non-expansive
convolution (RevSNEC), because the number of input signals does noteven temporarily increase. Our method not only achieves reversible
boundary processing, but also is comparable to irreversible SE (IrrSE)
in lossy image coding and outperformed periodic extension (PE) in lossyto-lossless image coding.
Index Terms—Lifting-based linear-phase filter bank (L-LPFB), lossyto-lossless image coding, reversible symmetric extension (RevSE), reversible symmetric non-expansive convolution (RevSNEC).

I. I NTRODUCTION
Filter banks (FBs) [1] have been contributing to signal processing
and communication tools for many years. They have often been
employed as transforms in image compression (coding) because they
have extensive frequency selectivity and high coding gain. FBs withlinear-phase (LP) properties, i.e., LPFBs [2–8], including lapped
transforms (LTs) [9–14] are particularly one of the most usefultransforms for image coding. LPFBs can be easily designed andthey simply overcome problems with image boundary distortion via
symmetric extension (SE) [2] that maintain continuity at the image
boundary. The output signals can be reconstructed without image
boundary distortion by using symmetry even if the extended signals
are not transmitted to the synthesis bank. Symmetry means that when
an input signal vector for a building block is the reflected vector ofanother input signal vector, their output signal vectors also have thesame relationship. Smith and Eddins [2] achieved such SE by using
the symmetry of “whole FB.”
Lifting structures with rounding operations have also been presented [15–17]. A transform only composed of lifting structures andinteger multipliers achieves an integer-to-integer transform that maps
integer input signals to integer output signals. Thus, many liftingbased FBs (L-FBs) [18–23] including lifting-based LTs [24–26] havebeen proposed, and the structures have led FBs to achieve lossyto-lossless image coding, which is unified lossy and lossless image
coding, such as JPEG 2000 [27] and JPEG XR [28]. Discrete wavelet
transforms (DWTs) [17] in JPEG 2000 have demonstrated excellent
coding, whereas L-FBs have greater potential for lossy-to-losslessimage coding due to their higher degrees of design freedom than
DWTs. Several reversible smooth extensions have in fact been applied
to 2-channel L-FBs including DWTs [21], [22], [29]. However, nosmooth boundary extensions can directly be applied to generalized
M -channel (M ∈ N, M ≥ 2) L-FBs. 4 × 8 hierarchical lapped
transform (HLT) [30] in JPEG XR is well-known as one of the mostT. Suzuki is with the Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Ibaraki, 305-8573 Japan (e-mail:taizo@cs.tsukuba.ac.jp).
M. Ikehara is with the Department of Electronics and Electrical Engineering, Keio University, Yokohama, Kanagawa, 223-8522 Japan (e-mail:ikehara@tkhm.elec.keio.ac.jp).

popular L-FBs. Although it uses a constrained case of the image
boundary solution we propose later, M ×2M L-LPFB case, it causes
a little bit boundary error because SE cannot be precisely achievedby ignoring the scaling coefficients. Periodic extension (PE), whichcauses image boundary distortion, is often reluctantly used for lossyto-lossless image coding based on M -channel L-FBs even if it has
LP properties because rounding error on each lifting step corrupt
symmetry [26].
We solve the image boundary problem in lossy-to-lossless image
coding based on M -channel lifting-based LPFBs (L-LPFBs) by
focusing on the symmetry of “each building block” unlike that of the“whole FB” [2]. Although the proposed reversible SE (RevSE) is notcompletely equivalent to SE, which is called irreversible SE (IrrSE)
to distinguish it from RevSE, it can obtain very similar smoothness
to IrrSE at the image boundary even if rounding operations areused. Moreover, complexity is reduced by extending non-expansive
convolution [13], called reversible symmetric non-expansive convolution (RevSNEC), because the number of input signals does noteven temporarily increase. The proposed RevSNEC can be applied
to M × M K (K ∈ N, K ̸= 0, and K must only be odd in odd
channel case) L-LPFBs which are extensions of M × 2M L-LPFBs
as HLT in JPEG XR. The RevSNEC not only achieves reversible
boundary processing, but also is comparable to IrrSE in lossy image
coding and outperformed PE in lossy-to-lossless image coding.
The remaining part of this paper is organized as follows: Sec. II
reviews the lattice structures of LPFBs and explains how we derive
their lifting structures. Sec. III presents IrrSE using symmetry of“each building block” newly in both even and odd channel cases.
This structures are extended to RevSE and RevSNEC by simple
matrix manipulations. Filter design examples, lossy-to-lossless image
coding simulations, and comparisons to the conventional methods arepresented in Sec. IV. Sec. V concludes the paper.
Boldface letters represent vectors or matrices. IN , JN , 0, ·T ,diag(· · · ), and det(·) respectively denote an N ×N (N ∈ N, N ̸= 0)
identity matrix, N ×N reversal identity matrix, null matrix, transposeof a matrix, block diagonal matrix, and determinant of a matrix.
II. L ATTICE AND L IFTING S TRUCTURES OF LPFB S
A. Linear-Phase Filter Banks (LPFBs)
When m ∈ N, m ̸= 0, M = 2m (M is even) and M = 2m + 1(M is odd), the type-II analysis polyphase matrix E(z) in M × M K
LPFBs are presented as in Tran [6]:E(z) = E0 G1 (z) · · · GK−2 (z)GK−1 (z)

(1)

where
E0 = Φ0 W, Gk (z) = Λ(z)WΦk W,(M is even)diag{Ue,k , Ve,k }Φk = diag{Uo,k , Vo,k }(M is odd, k is even)
diag{Uo,k , wk , Vo,k } (M is odd, k is odd),{diag{Im , z −1 Im }(M is even)Λ(z) =
diag{Im+1 , z −1 Im } (M is odd),
[
]


Im
Jm

1

√
(M is even)2Jm −Im



W=
Im
0
Jm

√




√1  02
0  (M is odd),


2


Jm
0
−Im
Ue,k , Ve,k , and Vo,k are m × m arbitrary nonsingular matrices,Uo,k is an (m + 1) × (m + 1) or m × m arbitrary nonsingular matrix

2

single-row elementary reversible matrices (SERMs) [18] have been
applied to PULPFBs designed in this paper because they have fewer
rounding operations than the others. Also, let W0 be W in the last
block E0 . W0 in PULPFBs is factorized by using the traditional
three steps lifting factorization of the Givens rotation matrix. On theother hand, BOLPFBs based on time-domain LTs (TDLTs) in [14]and lifting-based DCT in [31] are designed in this paper because
they not only are simple but also have high coding performance.
IrrSE cannot directly be applied to the L-LPFBs as explained in Sec.
III-B. We will explain our solution to the problem in the next section.

III. I MAGE B OUNDARY P ROCESSING FOR L-LPFB S

Fig. 1. Building blocks of M -channel L-LPFB in even channel case (whitecircles mean rounding operations, bold arrows mean m × 1 vector signals,
and narrow arrows mean scalar signals).

when k is even or odd, and wk is a nonzero scalar. Also, either
Ue,k or Ve,k and Vo,k are usually replaced by Im when k ≥ 1
to eliminate redundancy. If E(z) is invertible, synthesis polyphasematrix R(z) can be chosen as the inverse of E(z), i.e., the perfectreconstruction (PR) property R(z)E(z) = IM is satisfied. The
LPFBs are paraunitary LPFBs (PULPFBs) if all Ue,k , Ve,k , Uo,k ,and Vo,k are orthogonal matrices and wk is 1, and the others arebiorthogonal LPFBs (BOLPFBs). In PULPFBs, m must be m ≥ 2.
As described in Sec. III-A, IrrSE can simply be applied to the LPFBs
to improve coding performance.
B. Lifting-based Linear-Phase Filter Banks (L-LPFBs)
This subsection presents L-LPFBs based on the polyphase matrix
in Eq. (1). When all lifting steps are implemented with rounding
operations, L-LPFBs achieve integer-to-integer mapping, i.e., thelossless mode. The building block Gk (z) in Eq. (1) to achieve thisis represented as
Gk (z) = Λ(z)WΦk W = Λ(z)WL Φk WR ≜ Λ(z)Ξk
where

][
]
[

Im
0
Im − 12 Jm(M is even)0Im Jm Im



[
]]
[
WL =
0
1
[ Im ]

I
0
−
J
m
m



2

(M is odd)


 0


Im+1
0
Im+1

Jm

and

[

 Im




 0
WR = [

Im





 0

]
Im
0
Im
−Jm Im]]  I[m]
0 12 Jm [
 0
Im+1
−Jm
1
J
2 m

Let symmetry be that of “each building block” unlike that of the“whole FB” in Smith and Eddins [2]. PR is satisfied by using such
symmetry without receiving redundant signals in the synthesis bank.
First, IrrSE is investigated in both case of even and odd channels.
Furthermore, RevSE for the lossless mode is presented with Cases
I and II. Case I means building blocks that do not step over theimage boundary, and Case II means those that just step over theimage boundary. Moreover, the redundancy of RevSE is eliminatedby extending non-expansive convolution [13], called RevSNEC.

A. Irreversible Symmetric Extension (IrrSE)
Fig. 2 shows IrrSE in even channel case. Let Xk , JM Xk , Yk , andZk correspond to an M × 1 input signal vector for a building blockΞk , a reflected vector of Xk , an output signal vector of Ξk Xk , andan output signal vector of Ξk JM Xk as Yk = Ξk Xk and Zk =
Ξk JM Xk . Symmetry means
Yk = JM Zk .

(2)

We demonstrate this symmetry is satisfied in both Cases I and II.
1) Case I (Not Stepping Over Image Boundary): Xk is an input
signal vector for Ξk that does not step over the image boundary
in this subsection. Let Xk be Xk = [ATk , BkT ]T (M is even) or[ATk , ck , BkT ]T (M is odd) where Ak and Bk are m × 1 vectors andck is a scalar. Also, let Uo,k (k is even) be redefined as[
Uo,k =

uk
tk

sk
uk

]

where uk , sk , tk , and uk correspond to m×m, m×1, 1×m matrices,
and a scalar. The output signal vectors Yk and Zk are expressed by][(M is even)
0
Im+1





(M is odd).

The top half of Fig. 1 outlines the building block of L-LPFB ineven channel case. Let ˆ· be a matrix factorized into lifting structureswith rounding operations. Hence, Ξ̂k , Ûe,k , V̂e,k , Ûo,k , and V̂o,krespectively mean matrices factorizing Ξk , Ue,k , Ve,k , Uo,k , andVo,k into lifting structures with rounding operations. Although any
lifting factorization can be applied to them if each |det(Ue,k )|,|det(Ve,k )|, |det(Uo,k )|, |det(Vo,k )|, and |wk | is an integer value,

Yk = Ξk Xk
]
 [

U
e,k Uk + Jm Ve,k Vk1(M is even) 2 Jm Ue,k Uk − Ve,k Vk




√



uk Uk + 2ck sk + Jm Vo,k Vk


√




 21 
2tk Uk + 2uk ck√=Jm (uk Uk + 2ck sk ) − Vo,k Vk(M is odd, k is even)







Uo,k Uk + Jm Vo,k Vk





1 

w k ck (M is odd, k is odd)


2


Jm Uo,k Uk − Vo,k Vk

3Fig. 2. IrrSE by M × 3M LPFB in even channel case (dashed lines, bold arrows, narrow arrows, dark gray blocks, light gray blocks, and white blocks
correspond to image boundary, m × 1 vector signals, scalar signals, blocks that do not step over image boundary, blocks that just step over image boundary,and inner blocks).

and
Zk = Ξk JM Xk
]
 [

U
e,k Uk − Jm Ve,k Vk1(M is even) 2 Jm Ue,k Uk + Ve,k Vk




√



uk Uk + 2ck sk − Jm Vo,k Vk


√




 12 
2tk Uk + 2uk ck√=Jm (uk Uk + 2ck sk ) + Vo,k Vk(M is odd, k is even)







Uo,k Uk − Jm Vo,k Vk





1 

w k ck (M is odd, k is odd),


2


Jm Uo,k Uk + Vo,k Vk
where Uk = Ak + Jm Bk and Vk = Jm Ak − Bk , respectively.
Consequently, it is clear that symmetry is satisfied as Yk = JM Zk
in Eq. (2). Precisely, this means that when an input signal vector is
the reflected vector of another input signal vector, their output signal
vectors also have the same relationship.
2) Case II (Just Stepping Over Image Boundary): Xk is an input
signal vector for Ξk that just steps over the image boundary inthis subsection. Let Xk be Xk = [(Jm Bk )T , BkT ]T (M is even) or[(Jm Bk )T , ck , BkT ]T (M is odd) where Bk is an m × 1 vector andck is a scalar. Similar to Case I, it is clear that symmetry is satisfied
as Yk = JM Zk in Eq. (2) where][ Ue,k Jm Bk(M is even)Jm Ue,k Jm Bk 
Yk = Zk =Uo,k Jm Bk





wk ck

 (M is odd).
Jm Uo,k Jm Bk
B. Reversible Symmetric Extension (RevSE)
IrrSE in Sec. III-A is only for the lossy mode. We solve theproblem with symmetry lost due to rounding errors in the losslessmode in this subsection. It can be solved with a very simple
matrix manipulation where Ξ̂k for extended signals is replacedwith JM Ξ̂k JM except for the case where the process involves juststepping over the image boundary. Fig. 3 shows a realization of thesymmetry of L-LPFBs with rounding operations in even channel case.

1) Case I (Not Stepping Over Image Boundary): When Xk is an
input signal vector for Ξk that does not step over the image boundaryand rounding operations are considered, this is expressed as Yk′ ≜Ξ̂k Xk ̸= Yk and Zk′ ≜ Ξ̂k JM Xk ̸= Zk . Obviously, symmetry is lostas Zk′ ̸= JM Yk′ due to rounding error on each lifting step. Therefore,
we cannot use this extension as it is. We need to refocus on Ξk before
it is factorized into lifting structures. According to Eq. (2), Ξk canbe represented byΞk = JM Ξk JM .

(3)

However, when Ξ̂k is used instead of Ξk , this relationship is notpreserved completely as Ξ̂k ̸= JM Ξ̂k JM , where each building blockΞ̂k for extended signals is replaced by JM Ξ̂k JM . Although thistransform at the boundary is different from a normal transform withΞ̂k , this difference is trivial. By replacing Ξ̂k for extended signals,
the implementation in the case of reflected input signal vector JM Xk
is expressed as JM Ξ̂k JM · JM Xk = JM Ξ̂k Xk = JM Yk′ . Fig. 1
shows the symmetry in building blocks Ξ̂k and JΞ̂k J of L-LPFB ineven cannel case. As a result, it is clear that symmetry can be satisfied
by a simple matrix manipulation for extended signals as Eq. (3) even
if rounding operations are implemented.
2) Case II (Just Stepping Over Image Boundary): When Xk is an
input signal vector for Ξk that just steps over the image boundaryand rounding operations are considered, it is clear that symmetry as
Yk′ = JM Zk′ ,
where

]
[

 Ûe,k Jm BkJm Ûe,k Jm Bk ′′
Yk = Zk =Ûo,k Jm Bk




wk ck






Jm Ûo,k Jm Bk(4)(M is even)(M is odd),

is structurally satisfied even if the rounding operations are implemented in this case similar to those in Sec. III-A2. Ξ̂k for extended
signals can be replaced by JM Ξ̂k JM in this case because it is clearthat Ξ̂k = JM Ξ̂k JM unlike that in Sec. III-B1, where we did not
replace it for the sake of simplicity.

4

Fig. 3. RevSE by M × 3M L-LPFB in even channel case (dashed lines, bold arrows, narrow arrows, dark gray blocks, light gray blocks, and white blocks
correspond to image boundary, m × 1 vector signals, scalar signals, blocks that do not step over image boundary, blocks that just step over image boundary,and inner blocks).

C. Reversible Symmetric Non-Expansive Convolution (RevSNEC)
It is important for the input and output signals for Ξ̂k to always
achieve symmetry as explained in Sec. III-B. Therefore, only Ξ̂k thatjust stepping over the image boundary should be considered, and onlym or m + 1 output signals are used to the next block as follows:
1) When M is even, either Ûe,k Jm Bk or Jm Ûe,k Jm Bk in the
output signals in Eq. (4) are used.
2) When M is odd, either [(Ûo,k Jm Bk )T , (wk ck )T ]T or
Jm Ûo,k Jm Bk in the output signals in Eq. (4) are used.
Consequently, RevSE in the above subsection can be replaced by nonexpansive convolution [13], called RevSNEC, as seen in Fig. 4. This
RevSNEC is less complex because it does not need any temporarily
extensions to the input and output signals at the image boundary.
Also, since Ûe,k (k ̸= 0) usually adopts Im as discussed in Sec.
II-A, Jm Ûe,k and Ûe,k Jm are simply replaced by Jm .
IV. R ESULTS
A. Filter Optimization
We designed 8 × 16 and 8 × 24 PULPFBs, which have Ue,k = Im−1TT(k ̸= 0), U−1e,0 = Ue,0 and Ve,k = Ve,k , based on Sec. II-B
and 8 × 16 and 8 × 24 BOLPFBs based on TDLTs in [14] andlifting-based DCT in [31]. We optimized the design parameters byusing fminunc.m in the Optimization ToolBox of MATLAB
and only coding gain CCG as the cost function [1] for simple
design. Moreover, since less DC leakage is one of the most important
properties in FB theory for image compression, we parameterized theinitial blocks of LPFBs for one degree of regularity [32].
B. Application to Lossy-to-Lossless Image Coding
The resulting LPFBs were applied to lossy-to-lossless image coding. Integer-to-integer transforms can be obtained by using a rounding
operation at each lifting step. A wavelet-based coder (embedded
zerotree wavelet based on intraband partitioning: EZW-IP) [33]was used in the simulation to fairly evaluate the performance oftransforms. Also, RevSNEC, PE, and IrrSE were used for the image
boundary processing in the designed LPFBs. We compared the lossy

image coding results in Table I in the peak signal-to-noise ratio(PSNR): PSNR [dB] = 10 log10 (MAX2p /MSE), where MAXp and
MSE are the maximum possible pixel value of the image and themean squared error, respectively, at 0.25, 0.50, and 1.00 bit per pixel(bpp) for several test images: 512 × 512 8-bit standard grayscale
images, 512 × 768 8-bit Kodak grayscale images, and 2816 × 1600
16-bit clipped grayscale images in [34]. The bold numerals indicate
the best PSNRs. 9/7-tap DWT (9/7-DWT) and 4 × 8 HLT are
the transforms used in JPEG 2000 and JPEG XR lossy modes,
respectively. Table I shows that most LPFBs with the RevSNEC
achieves better lossy coding than the conventional methods. Fig. 5
illustrates the comparison of a particular area of the image Barbara.
It is obvious that the proposed RevSNEC is better than PE and theboundary processing for HLT in JPEG XR in the image boundary at
the right of the images in Fig. 5. Also, the LPFBs with the RevSNEC
achieved almost same performance compared with the IrrSE which
can achieve only lossy mode. Since the RevSNEC in 8 × 16 case iscompletely equivalent to the IrrSE in same case, Table II show theresults of IrrSE only in 8 × 24 case.
Since the resulting LPFBs are integer-to-integer transforms, we canalso obtain lossless reconstructed images at high bit rates. The performance of lossless coding at the lossless bit rate (LBR): LBR [bpp]= (Total number of bits [bit])/(Total number of pixels [pixel]) issummarized in Table II. The bold numerals mean the best LBRs.
5/3-tap DWT (5/3-DWT) and 4 × 8 HLT are the transforms used in
JPEG 2000 and JPEG XR lossless modes, respectively. Although the
LPFBs with the RevSNEC are often inferior as compared with DWTand HLT in Kodak images and 16-bit large images, they demonstrated
their effectiveness in images with high frequency components.
V. C ONCLUSION
This paper has presented reversible symmetric extension (RevSE)
for M -channel lifting-based linear-phase filter banks (L-LPFBs)
applied to lossy-to-lossless image coding. Since the proposed RevSE
has similar smoothness to an irreversible symmetric extension (IrrSE)
at the image boundary, it does not generate distortion at the image
boundary even if rounding operations are used. Moreover, complexity
is lessened by extending non-expansive convolution, called reversible

5Fig. 4. RevSNEC by M × 5M L-LPFB in even channel case (dashed lines, bold arrows, narrow arrows, dark gray blocks, light gray blocks, and whiteblocks correspond to image boundary, m × 1 vector signals, scalar signals, blocks that do not step over image boundary, blocks that just stepping over image
boundary, and inner blocks).

Fig. 5. Comparison of particular area of image Barbara reconstructed with
8 × 24 LPFBs when bit rate is 0.25 [bpp] (Left, top, and bottom boundaries
of each image are not image boundaries.): (top-left) 4 × 8 HLT, (top-right)
8 × 24 BOLPFBs with the PE, (bottom-left) 8 × 24 BOLPFBs with the IrrSE,
and (bottom-right) 8 × 24 BOLPFBs with the RevSNEC.

symmetric non-expansive convolution (RevSNEC). As a result, it
achieves better performance in lossy-to-lossless image coding thanperiodic extension (PE).
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers and
Dr. S. Kyochi for providing many constructive suggestions thatsignificantly improve the presentation of this paper. This work was
supported by JSPS Grant-in-Aid for Young Scientists (B) Grant
Number 25820152.
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6

TABLE IL OSSY IMAGE CODING RESULTS (PSNR[ D B]).
Test
images

9/7-DWT[17]

4 × 8 HLT
[30]

BarbaraElaineFinger
Kodim19
Kodim20
Kodim21
Arri
Face
Lake Locked

27.23
31.50
23.49
28.38
31.92
27.21
33.28
45.96
39.04

26.85
30.81
22.96
27.95
31.29
26.76
33.22
45.49
38.37

BarbaraElaineFinger
Kodim19
Kodim20
Kodim21
Arri
Face
Lake Locked

30.47
32.96
25.97
31.06
35.19
30.20
37.30
48.72
42.11

30.43
32.47
25.56
30.85
34.34
29.90
36.75
48.40
41.78

BarbaraElaineFinger
Kodim19
Kodim20
Kodim21
Arri
Face
Lake Locked

34.87
34.63
29.06
34.70
39.45
34.31
41.97
51.68
45.50

35.05
34.22
29.01
34.66
39.50
34.12
42.28
51.47
45.37

8 × 16 PULPFB
8 × 24 PULPFBRevSNEC (PE)
RevSNEC (PE, IrrSE)
bit rate: 0.25 [bpp]28.09 (27.97)
28.30 (28.15, 28.29)
31.27 (31.01)
31.35 (31.08, 31.35)
23.63 (23.60)
23.70 (23.68, 23.70)
28.78 (28.60)
28.87 (28.66, 28.87)
30.72 (30.57)
30.77 (30.63, 30.76)
27.24 (27.07)
27.28 (27.11, 27.28)
33.53 (33.42)
33.71 (33.59, 33.71)
46.27 (46.12)
46.42 (46.25, 46.42)
39.25 (39.10)
39.41 (39.24, 39.41)
bit rate: 0.50 [bpp]31.74 (31.58)
31.92 (31.73, 31.93)
33.12 (32.42)
33.17 (32.56, 33.17)
26.49 (26.44)
26.53 (26.48, 26.52)
31.40 (31.23)
31.45 (31.40, 31.46)
33.78 (33.69)
33.77 (33.68, 33.76)
30.28 (30.15)
30.27 (30.15, 30.27)
38.05 (37.93)
38.25 (38.13, 38.25)
49.04 (48.95)
49.11 (49.02, 49.11)
42.51 (42.41)
42.67 (42.55, 42.67)
bit rate:1.00 [bpp]35.95 (35.77)
35.91 (35.75, 35.92)
35.09 (34.87)
35.03 (34.82, 35.03)
30.09 (30.05)
30.13 (30.10, 30.12)
34.95 (34.80)
34.87 (34.70, 34.87)
38.23 (37.89)
38.08 (37.71, 38.07)
34.31 (34.15)
34.22 (34.07, 34.21)
42.75 (42.67)
43.32 (43.01, 43.32)
52.05 (51.98)
52.11 (52.04, 52.11)
46.02 (45.95)
46.13 (46.04, 46.13)

8 × 16 BOLPFBRevSNEC (PE)8 × 24 BOLPFBRevSNEC (PE, IrrSE)

28.42 (28.31)
31.65 (31.40)
23.72 (23.71)
28.96 (28.78)
30.85 (30.76)
27.49 (27.37)
34.09 (33.98)
46.59 (46.44)
39.71 (39.52)

28.50 (28.34, 28.50)
31.66 (31.39, 31.66)
23.72 (23.69, 23.72)
29.01 (28.80, 29.01)
30.89 (30.75, 30.89)
27.53 (27.42, 27.53)
34.02 (33.76, 34.02)
46.63 (46.49, 46.63)
39.76 (39.57, 39.76)

32.02 (31.85)
33.16 (32.72)
26.79 (26.75)
31.60 (31.45)
33.98 (33.91)
30.41 (30.27)
38.68 (38.58)
49.24 (49.15)
42.93 (42.82)

32.03 (31.84, 32.03)
32.93 (32.61, 32.93)
26.74 (26.70, 26.74)
31.63 (31.48, 31.63)
34.03 (33.93, 34.03)
30.41 (30.27, 30.41)
38.78 (38.68, 38.78)
49.28 (49.19, 49.28)
43.00 (42.88, 43.00)

36.28 (36.12)
35.19 (35.00)
30.58 (30.54)
35.20 (35.06)
38.50 (38.27)
34.46 (34.31)
43.40 (43.32)
52.17 (52.11)
46.32 (46.25)

36.20 (36.02, 36.20)
35.09 (34.89, 35.09)
30.55 (30.52, 30.55)
35.14 (34.99, 35.14)
38.29 (38.13, 38.29)
34.38 (34.24, 34.31)
43.55 (43.46, 43.55)
52.20 (52.15, 52.20)
46.37 (46.30, 46.37)

TABLE IIL OSSLESS IMAGE CODING RESULTS (LBR [ BPP ]).
Test
imagesBarbaraElaineFinger
Kodim19
Kodim20
Kodim21
Arri
Face
Lake Locked

5/3-DWT
[17]
4.87
5.11
5.84
4.90
3.85
4.96
11.40
10.37
11.35

4 × 8 HLT
[30]
4.81
5.17
5.71
4.97
4.18
5.06
11.28
10.33
11.28

8 × 16 PULPFBRevSNEC (PE)4.83 (4.86)
5.08 (5.12)
5.70 (5.71)
5.06 (5.10)
4.40 (4.42)
5.17 (5.20)
11.37 (11.39)
10.30 (10.31)
11.29 (11.30)

filter banks,” IEEE Trans. Signal Process., vol. 58, no. 4, pp. 2078–2087,
Apr. 2010.
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factorization of M -channel biorthogonal filter banks for lossy-to-losslessimage coding,” IEEE Trans. Image Process., vol. 21, no. 7, pp. 3220–
3228, July 2012.
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transforms and their applications to image coding,” IEEE Trans. Image
Process., vol. 11, no. 10, pp. 1152–1159, Oct. 2002.
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standard,” IEEE Signal Process. Mag., vol. 26, no. 6, pp. 195–199, 204,
Nov. 2009.

8 × 24 PULPFBRevSNEC (PE)4.85 (4.88)
5.08 (5.12)
5.69 (5.70)
5.10 (5.13)
4.48 (4.51)
5.19 (5.22)
11.35 (11.36)
10.28 (10.30)
11.27 (11.28)

8 × 16 BOLPFBRevSNEC (PE)4.81 (4.83)
5.07 (5.10)
5.72 (5.72)
5.04 (5.07)
4.24 (4.26)
5.15 (5.17)
11.35 (11.37)
10.34 (10.35)
11.30 (11.31)

8 × 24 BOLPFBRevSNEC (PE)4.83 (4.85)
5.09 (5.12)
5.74 (5.75)
5.06 (5.09)
4.26 (4.28)
5.16 (5.19)
11.34 (11.36)
10.33 (10.34)
11.30 (11.31)[29] M. D. Adams and R. K. Ward, “Symmetric-extension-compatible reversible integer-to-integer wavelet transforms,” IEEE Trans. Signal Process., vol. 51, no. 10, pp. 2624–2636, Oct. 2003.
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vol. E96-A, no. 4, pp. 762–768, Apr. 2013.
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