PSEUDO REVERSIBLE SYMMETRIC EXTENSION FOR
LIFTING-BASED NONLINEAR-PHASE PARAUNITARY FILTER BANKS
Taizo Suzuki1 , Naoki Tanaka2 , and Hiroyuki Kudo1
1: Faculty of Engineering, Information and Systems, University of Tsukuba, Japan
2: Department of Computer Science, University of Tsukuba, Japan
Email: taizo@cs.tsukuba.ac.jp, tanaka@imagelab.cs.tsukuba.ac.jp, and kudo@cs.tsukuba.ac.jp
ABSTRACTThis study presents a pseudo reversible symmetric extension (PRevSE) that solves the signal boundary problem of lifting-based
nonlinear-phase paraunitary filter banks (L-NLPPUFBs), whichhave high compression rates thanks to their not having a constraint
on the linear-phase property unlike the existing transforms used inimage coding standards. The conventional L-NLPPUFBs with aperiodic extension (PE) yield annoying artifacts at the signal boundaries. However, the P-RevSE is implemented smoothly at the signal
boundaries by using a nonexpansive convolution of a symmetricextension (SE) and determinant control for the lifting factorization.
Although the determinant control causes a pseudo SE, not a true SE,
the resulting L-NLPPUFB with P-RevSE outperforms not only theL-NLPPUFB with PE but also the current transform used in JPEG
XR.
Index Terms— Lifting structure, lossy-to-lossless image coding, nonlinear-phase paraunitary filter bank, signal boundary problem, symmetric extension.
1. INTRODUCTION
Image compression (coding) standards help to alleviate the burden
on servers and free up communication bandwidth. JPEG is the mostcommon image coding standard, but it uses the discrete cosine transform (DCT), which causes blocking artifacts in low bitrate compression. JPEG XR (eXtended Range) [1] is a more effective image coding standard that uses a lapped transform (LT) [2], which is a classof linear-phase filter banks (LPFBs) [3], and it solves the blocking
problem. Although LPFBs have to extend the signals at the signal
boundaries because of the overlapping processing, the output signals should not be larger than the input signals. A periodic extension (PE), which is one of the simplest boundary processing, causes
annoying artifacts due to discontinuities at the signal boundaries.
LPFBs solve the problem by using a symmetric extension (SE) [4],which extends the boundary signals smoothly and does not requiretransmission of extra signals for reconstruction.
Nonlinear-phase filter banks (NLPFBs) [5, 6] have high compression rates thanks to their not having a constraint on the linearphase property unlike LPFBs. NLPFBs also have the signal boundary problem, and some solutions for it have been presented [7–10].
On the other hand, lifting-based FBs (L-FBs) for lossy-to-losslessimage coding, whose image quality is scalable from lossless tohighly compressed lossy data, are not able to use the existingsmooth boundary solutions [7–10] because of the rounding error
This work was supported by JSPS Grant-in-Aid for Young Scientists (B),Grant Number 16K18100.

978-1-5090-2175-8/17/$31.00 ©2017 IEEE3265

in each lifting step. While our previous work presented a reversible
SE (RevSE) for lifting-based LPFBs (L-LPFBs) [11], the nonlinearphase case has not been treated yet.
This study presents a pseudo RevSE (P-RevSE) for liftingbased nonlinear-phase paraunitary filter banks (L-NLPPUFBs).
NLPPUFBs are NLPFBs with paraunitariness and are more practical than NLPFBs without paraunitariness. In addition, NLPPUFBs,
whose building blocks are orthonormal matrices, can be easily factorized into lifting structures for lossy-to-lossless image coding
because a minimum condition to realize a lifting factorization of amatrix is that the determinant is ±1 [12]. The P-RevSE is implemented smoothly at the signal boundaries by using a nonexpansive
convolution of an SE according to [9] and determinant control forthe lifting factorization. Although the determinant control causes apseudo SE, not a true SE, the resulting L-NLPPUFB with P-RevSE
outperforms not only L-NLPPUFB with PE but also the current LT
used in the JPEG XR standard.
Notations: Boldface small and capital letters represent vectors
and matrices, respectively. I[m] , J[m] , and ·T mean an m × m identity matrix, m × m reversal matrix, and transpose of a matrix, respectively. We omit the matrix sizes when they are obvious.
2. NONLINEAR-PHASE PARAUNITARY FILTER BANKS
Let M × M K NLPFB (M, K ∈ N, M is even, K ≥ 2) be an
NLPFB whose channel and filter length are M and M K, respectively. The polyphase matrix E(z) is expressed as [13]( 1)∏E(z) =Gi Λi (z) G0 ,
(1)
i=K−1

where Gk (k = 0, 1, · · · , K −1) is an M ×M arbitrary nonsingularmatrix and Λi (z) is a delay matrix with a delay element z as:[]
I0
Λi (z) = [M −γi ].
(2)0z −1 I[γi ]Although γi is an arbitrary integer 1 ≤ γi ≤ M − 1, we set
γi = M/2 for simplicity. Thus, Λi (z) is denoted as Λ(z). Since anNLPFB is not constrained by the linear-phase property, it achieves
a better frequency decomposition compared with the linear-phase
Tcase. When Gk is an arbitrary orthonormal matrix, G−1
k = Gk [5],an NLPFB is called an NLPPUFB.
3. SYMMETRIC EXTENSION FOR NLPPUFBS
Tanaka et al. proposed a nonexpansive convolution for NLPPUFBs [9]. The top images of Fig. 1 shows the upper boundary

ICIP 2017

Fig. 1. Upper boundary processing of the analysis banks of NLPPUFBs (dashed lines mean boundaries): (top) SE, (bottom) nonexpansive
convolution, (left) K = 2, (right) K = 3.

processing of the analysis banks of NLPPUFBs when K = 2 and3. They mean that (K − 1)M/2 extra signals have to be extended
at each boundary. To obtain smooth signals and for simplicity, theextra signals are commonly extended by using J[(K−1)M/2] ; i.e., theextension is an SE.
Here, we have a nonexpansive problem because the signalstransmitted to the synthesis bank should only be yn (n = 0, 1, 2, · · · ).
This section presents a reconstruction method at the synthesis bankwithout any extra signal; i.e., xn is reconstructed from only an ,which can be reconstructed from yn . Let Gk be][Ak Bk,(3)Gk =C k Dk
where each submatrix is M/2 × M/2. Hereafter, we consider only
the upper boundary processing. The signals at the lower boundary
processing can be reconstructed in the same way as in the upperboundary processing.
3.1. Case of K = 2
From the top left of Fig. 1, we obtain][[] Jx0= (A1 J + B1 ) x0 .
a0 = A1 B1x0|{z}GT2where G2 is an orthonormal matrix. We cannot easily calculate
b0 from the limited signals an unlike b1 . Substituting Eq. (7) intoEq. (6) yields[][ ][] AT2 CT2b1.
(8)
b−1 = B2 J A2 JTT
B 2 D2 b 0Also, we obtain[a0 = A1(4)B1][] b−1.
b0(9)Consequently, b0 can be calculated from Eqs. (8) and (9) as))−1((b0 = A1 B2 JCT2 + A2 JDT2 + B1|{z}≜V) (· a0 − A1 B2 JAT2 + A2 JBT2 b1  ,|{z} (10)≜W≜UThe problem is to reconstruct x0 from a0 , which is transmitted tothe synthesis bank. x0 is represented asx0 = U −1 a0 ,Also, x0 and x1 in Eq. (6) can be recalculated from b0 and b1 as[ ] [ T][ ]x0A2 CT2b1=,(7)
TTx1B 2 D2 b 0|{z}(5)where the boundary matrix U has to be nonsingular, i.e., det(U ) ̸=0. The nonexpansive convolution of the analysis banks in case of
K = 2 is shown at the bottom left of Fig. 1.
3.2. Case of K = 3
From the top right of Fig. 1, similar to the case of K = 2, we obtain[][ ]] Jx1[] x0[= B2 J A2 J.
(6)
b−1 = A2 B2
Jx0x1

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where the boundary matrix V has to be nonsingular, i.e., det(V) ̸=0. The nonexpansive convolution of the analysis banks in case of
K = 3 is shown at the bottom right of Fig. 1.
For any K, the signals can be reconstructed as the solution of asimultaneous matrix equation with (K − 1) unknowns.
4. PSEUDO SYMMETRIC EXTENSION FOR L-NLPPUFBS
This section presents a P-RevSE for L-NLPPUFBs that uses the nonexpansive convolution described in Sec. 3 and determinant control
for the lifting factorization. From the bottom of Fig. 1, if the structures are expressed as lifting structures, they achieve reversible transforms for lossless image coding. We can consider that the processing with W is already a lifting structure. Consequently, the residual matrices Gk , U , and V should be factorized into lifting structures. A minimum condition to realize a lifting factorization of aTable 1. Coding gain Ccg s of the resulting 4 × 12 L-NLPPUFBs.
BoundaryNot
Upper
Lower
Ccg
8.3168 8.2852 8.3173

Table 2. Lossless image coding results (LBR [bpp]).
Test
LT [2]
L-NLPPUFBs
Images
RevSE
PE
P-RevSE
Barbara
4.801
4.798
4.775
Boat5.124
5.103
5.093
Elaine
5.166
5.132
5.106
Lena
4.587
4.615
4.587
Pepper
4.954
4.907
4.897
Room4.344
4.452
4.427

matrix is that the determinant is ±1 [12]. Since Gk is an orthonormal matrix, i.e., det(Gk ) = ±1, the matrix can be factorized into
lifting structures. On the other hand, note that the determinants of
U and V might not satisfy this condition, i.e., det(U ) ̸= ±1 anddet(V) ̸= ±1. Thus, we control the matrices ine =U√

M/2

U|det (U )|ande =VV√,|det (V)|M/2(11)where det(U ) ̸= 0 and det(V) ̸= 0, and these conditions arecompletely equivalent to those as described in Sec. 3. Since thesee and Ve satisfy the condition for a lifting factorization,matrices Uee = ±1, U
e and Ve can be factorized intodet(U ) = ±1 and det(V)lifting structures. This means
thatthetop
half√of G1 at the signal√
boundaries is scaled by M/2 | det(U )| and M/2 | det(V)|,[1][1]
1f1 = s I 0 G1 = s A1 s B1 ,G(12)0 I
C1D1√√
where s = M/2 | det(U )| (K = 2) or M/2 | det(V)| (K = 3);
i.e., the NLPPUFBs losing the originalused at the√ properties are √signal boundaries. Therefore, if M/2 | det(U )| and M/2 | det(V)|are quite different from 1, the boundary processing cannot be implemented smoothly. Accordingly, we have to take this into consideration when designing the NLPPUFB.
5. EXPERIMENTAL RESULTS
5.1. Design of NLPPUFBs
We designed 4 × 12 (M = 4 and K = 3) NLPPUFB with the costfunction ϕ, which is a weighted linear combination of the coding
gain Ccg [14], the symmetric property of filters Csym [9], and thedeterminant control of matrices Cdet , as follows:
ϕ = −w0 Ccg + w1 Csym + w2 Cdet ,(13)Cdet = (|det (V u )| − 1)2 + (|det (V l )| − 1)2 .

(14)

whereV u and V l mean V at the upper and lower boundary processing of the NLPPUFBs. wk was experimentally-determined as{w0 , w1 , w2 } = {1, 0.01, 0.15}. Regularity, which is one of
the most important image coding properties to prevent the DC leakage, was considered structurally [12]. As described in the previous3267

Fig. 2. Frequency responses of the resulting 4 × 12 NLPPUFBs(black solid, pink dotted, and light blue chained lines mean the NLPPUFBs at non-signal boundaries, upper signal boundaries, and lower
signal boundaries, respectively): (top) analysis banks, (bottom) synthesis banks.
section,√ smooth boundary processing cannot be implemented if
M/2| det(V)| is quite different√from 1. However, the resulting
4 ×√12 NLPPUFB achieved M/2 | det(V u )| = 0.9339 ≈ 1 andM/2| det(V l )| = 0.9635 ≈ 1 thanks to the cost functions Csymand Cdet . Table 1 and Fig. 2 respectively show the coding gain Ccg sand the frequency responses of the resulting 4 × 12 NLPPUFBs at
non-signal boundaries, upper signal boundaries, and lower signal
boundaries. From the coding gains and frequency responses, weconsider that the differences that depend on the signal boundaries
are trivial.
5.2. Lossy-to-Lossless Image Coding
We evaluated the 4 × 8 LT (JPEG XR) with RevSE1 and the resulting 4 × 12 L-NLPPUFBs with PE and P-RevSE through lossy-tolossless image coding. A single-row elementary reversible matrixe As(SERM) [15] was used for the lifting factorization of Gk and V.
described in Sec. 4, the processing with W is considered to be a lifting structure without any lifting factorization. The resulting 4 × 12L-NLPPUFBs were implemented with a rounding operation at eachlifting step and compared in terms of the lossless bitrate (LBR) [bpp]in lossless image coding and peak signal-to-noise ratio (PSNR) [dB]1 Note that the RevSE in JPEG XR is not equivalent to [11] because it iscustomized for JPEG XR.

Fig. 3. Particular areas of images of Room, which was lossy compressed with 0.25 [bpp] (top and bottom boundaries of each image are
non-image boundaries): (top-to-bottom) original image, LT with RevSE, L-NLPPUFB with PE, and L-NLPPUFB with P-RevSE.

To evaluate transform performance fairly, we employed two-level
decompositions on all transforms. The image set included six 512 ×512 eight-bit standard grayscale images in [16]. A quadtree-based
embedded image coder EZW-IP [17] was used to encode the transformed images.
Table 2, Table 3, and Fig. 3 show the lossless and lossy image
coding results. The results show the advantage of the L-NLPPUFB
with P-RevSE over the L-NLPPUFB with PE. Especially, in Fig. 3,
the L-NLPPUFB with PE produces annoying artifacts, whereas noartifacts are apparent in the image from the L-NLPPUFB with PRevSE. In addition, the L-NLPPUFB with P-RevSE outperformed
the 4 × 8 LT with RevSE.

Table 3. Lossy image coding results (PSNR [dB]).
Test
Bitrate LT [2]
L-NLPPUFBs
Images
[bpp]
RevSE
PE
P-RevSE
0.25
26.569 27.436
27.578
Barbara0.50
30.334 31.097
31.2341.00
34.952 35.601
35.7280.25
27.261 28.129
28.21930.727 31.296
31.371
Boat
0.50
1.00
34.213 34.758
34.8050.25
30.825 31.178
31.381
Elaine0.50
32.502 32.876
33.01634.179 34.746
34.8941.00
0.25
31.603 31.965
32.25135.024 35.300
35.524
Lena
0.50
1.00
38.247 38.572
38.7080.25
31.026 31.078
31.586Pepper0.50
33.892 34.207
34.32835.428 36.200
36.2731.00
0.25
27.818 28.684
29.048Room
0.50
32.715 32.838
33.1161.00
38.288 37.927
38.407

6. CONCLUSION
This study presented a P-RevSE for L-NLPPUFBs. The conventional L-NLPPUFBs without any smooth boundary processing havethe annoying artifacts at signal boundaries. The L-NLPPUFB with
P-RevSE solved this problem by using a nonexpansive convolution of an SE and determinant control for the lifting factorization.
Although the determinant control caused a pseudo SE, not a true
SE, the L-NLPPUFB with P-RevSE outperformed not only the LNLPPUFB with PE but also the current transform used in JPEG
XR.

in lossy image coding:
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