A Two-Dimensional Non-Separable Implementationof Dyadic-Valued Cosine-Sine Modulated Filter
Banks for Low Computational Complexity
Seisuke Kyochi∗ , Taizo Suzuki† and Yuichi Tanaka‡∗ The University of Kitakyushu, Japan, Email: s-kyochi@kitakyu-u.ac.jp† University of Tsukuba, Japan, Email: taizo@cs.tsukuba.ac.jp‡ Tokyo University of Agriculture and Technology, Japan, Email: ytnk@cc.tuat.ac.jp

Abstract—This paper aims to design a new two-dimensional(2D) non-separable implementation of dyadic-valued cosine-sinemodulated filter banks (2D D-CSMFBs) for low computational
complexity. CSMFBs satisfy rich directional selectivity (DS) andshift-invariance (SI), and they can be easily designed by themodulation of a prototype filter. In addition, our previous work
introduced a one-dimensional (1D) D-CSMFBs (1D D-CSMFBs).
By restricting real-valued filter coefficients to rational-valued
ones, they can save computational cost while keeping DS and SI.
The proposed 2D implementation in this paper can further reducethe computational cost by unifying the conventional 2D separable
structure into a 2D non-separable one directly. Furthermore,experimental results of image non-linear approximation showthat the proposed 2D D-CSMFBs are comparable or even better
than the 1D D-CSMFBs.

I. I NTRODUCTION
For decades, various extensions of discrete wavelet transform, which satisfy “rich directional selectivity1 ” (DS), havebeen widely studied and presented, such as Contourlet [1],Curvelet [2], Shearlet [3], and so on. Among them, dual-treecomplex wavelet transforms (DTCWTs) have been paid muchattention due to their low redundancy and low computational
complexity [4]. While directional selectivity usually increases
redundancy or computational complexity, DTCWTs can restrict their redundancy ratio to 2 and save computational complexity by two-dimensional (2D) separable implementationof parallel DWTs, i.e., cascading the filtering along verticaland horizontal directions. Furthermore, it also satisfies “shiftinvariance2 ” (SI). Thanks to DS and SI, it can be successfully
applied to various kinds of practical image processing, suchas image denoising, image analysis, image compression.
As well as the DTCWT, another DS and SI transforms,
cosine-sine modulated filter banks (CSMFBs), have beenshown recently [5]. DTCWT is difficult to design M -channelcases with good filter performances, e.g., stopband attenuation or coding gain, due to the fractional delay requirement
imposed on parallel DWTs, which degrades design degree of
freedom. On the other hand, because CSMFBs require NO
1 Directional selectivity means sparse representation capability for directionally oriented components of images, e.g., lines and edges. For that, basis
functions should be oriented along various kinds of directions [4].
2 Shift-invariance means the stability of wavelet coefficients under the shiftof input signals [4].

fractional delay requirement (just the modulation of prototype
filter), M -channel CSMFBs can be better performance thanM -channel DTCWTs and effectively work in application.
In this paper, we address on finding a simplified structureof CSMFBs for low computational complexity. In [6], onedimensional dyadic-valued CSMFBs (1D D-CSMFBs) basedon lifting structure have been introduced. Despite of realrational conversion, 1D D-CSMFBs can still keep DS and SI
properties and show better performance compared with 1D
rational-valued DTCWTs (D-DTCWT) [7]. However, since
the previous structure does NOT consider removing redundant
lifting steps, there is possibility to exist much simpler structureof 1D D-CSMFBs.
This paper presents a new 2D non-separable implementationof D-CSMFBs (2D D-CSMFBs) for lower computational
complexity. While the conventional 2D transform of 1D DCSMFBs is based on separable implementation, the proposed
implementation consists of a 2D non-separable structure via
Kronecker product. Due to this 2D unification, redundant lifting steps can be removed and lower computational complexity
can be achieved. Moreover, in spite of the simplification,
experimental results of image non-linear approximation (NLA)
show that 2D D-CSMFBs has the comparable or even better
performance than 2D √separable structure.
Notations:j:=−1. H(z) is defined as H(z) :=
∑
−n
h(n)z
.
The
(M,
L) FB means the M -channel filternbank with the filter length of L. The (M × M, L × L) 2D
FB means the M × M -channel filter bank with the filter
size of L × L. I, J, and diag(a0 , . . . , aN −1 ) are the identity,
the reversal identity, and the diagonal matrices, respectively.
Γ = diag(1, −1, . . . , 1, −1).
II. R EVIEW
A. Dual-Tree Complex Wavelet Transform
DTCWTs are constructed by two maximally decimatedperfect reconstruction (PR) FBs, as illustrated in Fig. 1. In(1)(2)
this case, the pair {Hk (z), Hk (z)} (0 ≤ k ≤ M − 1) isdesigned to satisfy the following equations [4]:Hk (ejω ) = e−jθk (ω) Hk (ejω ),)(1(M − 1)ω − pω,θ0 (ω) = d +
2
(2)

(1)

(1)

C. Dyadic-Valued Implementation of CSMFB and DTCWT
In general, the rotation matrix can be factorized into lifting
steps as follows [10]:
[
] [
][
][
]
cos θ − sin θ
1 p 1 0 1 p=,(6)sin θ cos θ
0 1 u 1 0 1

Fig. 1. M -channel DTCWTs and CSMFBs

where d ∈ Z denotesthedelay
the [primal FB and the{ between})
2πdual FB, ∀p ∈ 0, . . . , M
−
1
,
∀ω
∈ 2π
2
M p, M (p + 1) ,
and
{
(
)
π
− d + 12 ω ω ∈ (0, 2π)2θk (ω) =(2)0
ω = 0,
where 1 ≤ k ≤ M − 1. Practically, M -channel DTCWTs aredesigned by the dual-tree complex wavelet packet (DTCWP)[8] which cascades 2-channel DTCWTs and 2-channel PRFBs.
B. Cosine-Sine Modulated Filter Banks
As well as DTCWTs, a CSMFB also consists of two maximally decimated filter banks as in Fig. 1. Its filter coefficients(1)(2)
of {Hk (z), Hk (z)} are expressed as [5]:
{
(
)
)
(
(1)
π
n − N 2−1 + θkhk (n) := 2p(n) cos (k + 21 ) M
(
(
)
)
(3)
(2)
π
hk (n) := 2p(n) sin (k + 21 ) M
n − N 2−1 + θkwhere k = 0, . . . , M − 1, θk = (−1)k π4 and p(n) is(2)
the prototype filter. In the paraunitary CSMFB, hk (n) =(1)(1)(−1)k fk (n) = (−1)k hk (N − 1 − n). Thus, if the primal
FB satisfies the PR property, the dual FB system also satisfies
it. Our previous work clarified the relationship of cosine andsine modulation derived both DS and SI [5].
The polyphase matrices E(1) (z) and E(2) (z) corresponding
to the CMFB and the SMFB can be decomposed as lattice
structures. They can be expressed by [9]:
K−1
)
∏ (
(1)
(1)
Λ(z 2 )Dk
E(1) (z) =CIV WΛ(z)D0
[

] k=1
[ −1
]
Sk J
z I 0
, Λ(z) =
,
JCk J
0
I
K−1
)
∏ (
(2)
(2)
E(2) (z) =SIV WΛ(z)D0
Λ(z 2 )Dk
(1)
Dk =

[

−Ck
JSk

(4)

] [
]
[
]
Sk J
I 0
(1) −I 0=
Dk.
−JCk J
0 I
0 −I
(5)
(
)
where Ck = diag cos θk0 , . . . , cos θ[k M/2−1] , Sk =()
0 I
diag sin θk0 , . . . , sin θk M/2−1 , W =, and CIV
I 0
and SIV denote the type-IV DCT and DST, respectively. The
structure is depicted in Fig. 2 (a).
(2)

Dk =

θ−1
where p = cossin θ , u = sin θ are called lifting coefficients. The lifting factorization enables us to design rationalvalued transformation simply by rounding lifting coefficientsrp = round(p × 2N )/2N , ru = round(u × 2N )/2N , while
preserving PR condition. By converting each rational-valued
multiplication with rounding operation to additions and bitshifters, for example,rp =P= α` 2` + α`−1 2`−1 + · · · ,2N(7)where P, ` ∈ Z, and αk = 1 or 0 (k ≤ `), a multiplierless(1)
CSMFB can be achieved. As shown in Fig. 2 (b), each Dk
can be reduced as
[
][
][
][
]
−I 0 I Pk JI
0 I Pk J
(1)
Dk =
, (8)
0 I 0
I
JUk I 0I()
where( Pk = diag) pk0 , . . . , pkM/2−1 and Uk =diag uk0 , . . . , ukM/2−1 . Since CIV is constructed by somerotation and butterfly matrices [11], it can also be converted
to a lifting-based factorization. Similarly, according to the(1)(2)
relationship of Dk and Dk , and SIV = ΓCIV J, both(2)
Dk and SIV can be expressed by a lifting factorization.
Interestingly, the lifting coefficients for E(1) (z) can be reusedfor E(2) (z).
On the other hand, M -channel D-DTCWT is designed basedon the lifting factorization of all 2-channel DTCWTs and 2channel FBs used in wavelet packets [7].
III. T WO - DIMENSIONAL R ATIONAL - VALUED C OSINE -S INE
M ODULATED F ILTER BANKS BASED ON THE L IFTING
FACTORIZATION
A. The Proposed StructureThis section derives a non-separable structure of 2D DCSMFB from separable 2D implementation. In general, a 1D
FB is applied to images in separable way. Specifically, each(1)
building block, e.g., Dk in a lattice structure (4), is applied
to a local block X in a image as:
(1)

(1)

Y = Dk X(Dk )T ,

(9)

k=1

Ck
JSk

where Y is the output block. The separable (lifting) implementation for (9) is illustrated in Fig. 3.
Here, by letting x be the 1D vector by rearranging the 2D
block X, the operation (9) can be reformulated as
(1)

(1)

y = Dk ⊗ Dk x,(10)where ⊗ denotes Kronecker product. Since multiplication of(1)
Dk is essentially based on 2 × 2 matrices as shown in Fig.

(b)

(a)

Fig. 2. The structure of the 4 channel CMFBs. (a) rotation-matrix-based structure, (b) lifting-based structure

1
2
3
4

5 9 136 10 14
7 11 15
8 12 16

2

2

3

14

14

3

15

15

1
2
3
4

1
2
3
4

5 9 136 10 14
7 11 15
8 12 16

Fig. 3. An example of the separable lifting implementation for (11).

2(a), (10) can be reduced to several submatrix multiplications
expressed as follows:
]
]
[
[
(k)
(k)
Rn
Rm
(11)
yi =
(k) xi(k) P0Rn
Rm= R(k)⊗ R(k)
n xi (0 ≤ m, n ≤ M/2 − 1),
] [
[ m
]
(k,m)
(k,m)
− cos θkm sin θkm
R00
R01
(k)
,
Rm =
(k,m) =(k,m)
sin θkmcos θkm
R11
R10
where P0 denotes the permutation matrix corresponding to
Fig. 3 and xi (0 ≤ i ≤ M/2 − 1) is the part of x. For example
in Fig. 3, xi = [X2 , X3 , X14 , X15 ]T , where Xk denotes the
sample at the index of k in X. The Kronecker product of(k)(k)
Rm ⊗ Rn can be factorized as the following equation:
(k)
(k)
R(k)
m ⊗ Rn = HΓ0 Rm,n Γ1 H
cos φm,n0
sin φm,n0cos ψm,n0(k)Rm,n = sin φm,n0− cos φm,n0
sin ψm,n
0

(12)


0
sin ψm,n ,0− cos ψm,nφm,n = θm − θn , ψm,n = θm + θn ,
Γ0 = diag(1, −1, −1, 1), Γ1 = diag(1, 1, −1, −1),


1 0 0
1
1 0 1 10.
H= √ 

2 0 1 −1 0 
1 0 0 −1
This factorization is analogous to the one for hierarchicallapped biorthogonal transform [(HLBT) employed in ]JPEGcos φm,nsin φm,n
XR [12]. Since Φm,n :=
and
sin
φ
−
cos φm,nm,n[]cos ψm,nsin ψm,nΨm,n :=
can be factorized into 3
sin ψm,n − cos ψm,nlifting steps as in (6). Moreover, by moving some scaling

5 9 136 10 14
7 11 15
8 12 16

2

2

3

14

14

3

15

15

1
2
3
4

5 9 136 10 14
7 11 15
8 12 16

Fig. 4. An example of the direct 2D lifting implementation for (12). pφ
, uφ
,
k
k
ψ
ψ
pk , and uk denote the lifting coefficients obtained from Φm,m and Ψm,m .

√√factors diag( 2, √12 , 2, √12 ) of the right H to the left H
in (12) or vice versa, both H can be further decomposed into
lifting steps as shown in Fig. 4. Note
[ that] when indices m
1 0
and n are same, Φm,m becomes, the number of
0 −1
multiplications can be reduced.
In the proposed method, a different lifting factorization of
type-IV DCT is used compared with [11], which is expressed
as follows [14]:
[
]
[
]
CII
0
Tcos
Tsin J
CIV = SP1
T, T =,
0 CII ΓJTsin −JTcos J
(
)
)
(
[
]
[
]
(2k + 1)π(2k + 1)πTcos kk = cos, Tsin kk = sin
,
4M
4M


1


Rπ/4
[
]


1
1 1


.
..
S=
,
 , Rπ/4 = √
−1 1


2


Rπ/4
1
CII is the type-II DCT and P1 is a permutation matrix,
respectively. S and T can be decomposed into 2D lifting steps
with the same way.
The 2D lifting implementation of the SMFB E(2) (z) canbe realized in the same way of the CMFB.
B. Computational Complexity
This section discusses about computational complexity of
1D and 2D D-CSMFBs in terms of the number of lifting
steps per block. As shown in Fig. 3, in order to process
an M ×(M ) block, the separable implementation requires
2
3×4× M= 3M 2 lifting steps. On the other hand, a2( )2
direct 2D implementation consists of 6 × M− 3 × M/2 =
2
3
2
(M
−
M
)
lifting
steps.
The
total
numberofadditions and2

TABLE I
T HE NUMBER OF ADDITIONS AND BIT- SHIFTERS
Separable D-DTCWT (8 × 8-channel, 22 × 22-tap)
Word length
8 bit
7 bit
6 bit
5 bit
4 bit
additions
2464 2224 2096 1232
1232
bit-shifters
2464 2224 2096
1232
1232
Separable 1D D-CSMFB (8 × 8-channel, 16 × 16-tap)
Word length
8 bit
7 bit
6 bit
5 bit
4 bit
additions
2352 2096 1920 1584
1360
bit-shifters
2288 2032 1856
1520
1296
Non-separable 2D D-CSMFB (8 × 8-channel, 16 × 16-tap)
Word length
8 bit
7 bit
6 bit
5 bit
4 bit
additions
1477 1332 1242 1017930
bit-shifters
1461 1316 1226
1001
913

䢲

䢲

䢯䢷

䢯䢷

䢯䢳䢲

䢯䢳䢲

䢯䢳䢷

䢯䢴䢲

䢯䢴䢷

䢯䢳䢷

䢯䢴䢲

䢯䢴䢷

䢯䢵䢲

䢯䢵䢲

䢯䢵䢷

䢯䢵䢷

䢯䢶䢲

䢯䢶䢷

Based on the designed lifting coefficients of CSMFBs, the2D D-CSMFBs are constructed. Table I clearly shows that the
2D D-CSMFBs require fewer addition and bit-shifters thanthe D-DTCWT and the 1D D-CSMFB. In addition, the 2-D
wavelet basis functions obtained by the 4 × 4-channel 2D DCSMFB are shown in Fig. 6. Clearly, the basis functions areoriented under rounding operation.

䢷䣏䣣䣩䣰䣫䣶䣷䣦䣧䢢䣔䣧䣵䣲䣱䣰䣵䣧䢢䣝䣦䣄䣟䣏䣣䣩䣰䣫䣶䣷䣦䣧䢢䣔䣧䣵䣲䣱䣰䣵䣧䢢䣝䣦䣄䣟䢷

Fig. 6. 32 directional basis functions with size of 16 × 16 obtained by the
2D CSMFB whose lifting coefficents are rounded to 4 bit word length.

䢯䢶䢲

䢲

䢲䢰䢳

䢲䢰䢴

䢲䢰䢵

䢲䢰䢶
䢲䢰䢷
䢲䢰䢸
䣐䣱䣴䣯䣣䣮䣫䣼䣧䣦䢢䣈䣴䣧䣳䣷䣧䣰䣥䣻

䢲䢰䢹

䢲䢰䢺

䢲䢰䢻

䢳

䢯䢶䢷

䢲

䢲䢰䢳

䢲䢰䢴

䢲䢰䢵

䢲䢰䢶
䢲䢰䢷
䢲䢰䢸
䣐䣱䣴䣯䣣䣮䣫䣼䣧䣦䢢䣈䣴䣧䣳䣷䣧䣰䣥䣻

䢲䢰䢹

䢲䢰䢺

䢲䢰䢻

䢳

A. Non-Linear Approximation(b) Rational(a) Real

Fig. 5. Frequency responses of the designed (4, 8) CSMFB. (a): Real-valued
CSMFB, (b) Rational-valued CSMFB with 5 bit lifting coefficients.

bit-shifters of a designed CSMFB depends on the number of
additions and bit-shifters allocated to each lifting coefficients.
IV. S IMULATIONIn this section, practical computational complexity and
NLA performance of 2D D-CSMFBs are shown. In order
to construct 2D D-CSMFBs and 1D D-CSMFBs, M -channel
CSMFBs are firstly designed. As a comparison, M -channel
DTCWT is designed by using DTCWP [8] cascading a 2channel DTCWT [4] and 2-channel PUFBs. Both transforms
are optimized to minimize the stopband attenuation ∆ specified by
∆=

M
−1 ∫
∑
k=0

(1)

Ωs (k)|Hk (ejω )|2 dω,
(1)

(13)

where Ωs (k) is stoppband of the Hk (ejω ), (0 ≤ k ≤ M −1).
ππ
In this paper, we set as Ωs (k) = [0, M(k −2)]∪[ M(k +2), π].
ππ(k − 2) ≤ 0 or M
(k + 2) ≥ π, the corresponding interval
If M
is ignored. Then, each optimized transform is decomposed via
lifting factorization and performed rounding for constructing
1D and 2D D-CSMFBs and D-DTCWT. Fig. 5 depicts thefrequency responses of the designed (4, 8) CSMFB and DCSMFB (# of channel, # of filter length). Note that, our previous work [6] shows that the results of stopband attenuationof 1D CSMFBs are consistently superior to those of DTCWTs.
It is because the design degree of freedom of DTCWTs is quite
limited, due to the packet implementation. For example, thenumber of the filter length of the cascaded 2-channel PUFBs
for the (4, 16) and (8, 22) D-DTCWTs is 4.

Non-linear approximation (NLA) [13] is often used as acriteria of transform effectiveness for practical applications,e.g., image coding, denoising, and so on. In this section,
NLA is demonstrated in the following way. Let f , c1,n andc2,n be the input image and the transformed coefficients ofthe primal FB and the dual FB, e.g., the CMFB and the
SMFB. In addition, fM is defined as the reconstructed signal
obtained from the M -largest coefficients within {c1,n }, {c2,n }by performing the inverse transform. The accuracy of NLA
kf − fM k indicates how well the transform compacts theenergy of the input signal.
Here we evaluated the (8, 22) D-DTCWT, the (8, 16) 1D DCSMFB, and the (8 × 8, 16 × 16) 2D D-CSMFB using typicaltest images Lena and Barbara (512×512 pixels). For all thetransforms, the number of 2D decomposition level is set to 2.
Periodical extension is used for image boundary processing.
For reconstruction, 3% transformed coefficients of Lena and
6% transformed coefficients of Barbara are kept.
Table II shows the NLA PSNR results versus the numberof bit word length allocated to lifting coefficients in eachtransform. Although the 1D D-CSMFB shows higher PSNRs
in the case of high bit word length, it suffers from degradationof performance in low bit word length, and the proposed 2D
D-CSMFB can achieve comparable or even better performancecompared with D-DTCWT and 1D D-CSMFB in low bit word
length. Fig. 7 shows the 2D D-CSMFB provides the best visualquality even the case of lowest bit word length (5 bit). Since
the 2D D-CSMFB has less lifting steps (rounding produces
filter perfromance degradation), it is more stable in low bit
word length than the 1D D-CSMFB.
V. C ONCLUSION
In this paper we proposed the 2D implementation of DCSMFBs for low computational complexity. By unifying theconventional 1D D-CSMFB via Kronecker product, redundant

TABLE II
N ON - LINEAR A PPROXIMATION
Lena (3% coeffs): D-DTCWT (8 × 8-channel, 22 × 22-tap)
Word length
5 bit
6 bit
7 bit
8 bit
32 bitPSNR [dB]
29.20029.36129.36729.38429.381
Lena (3% coeffs): 1D D-CSMFB (8 × 8-channel, 16 × 16-tap)
Word length
5 bit
6 bit
7 bit
8 bit
32 bitPSNR [dB]27.66828.06430.91231.02331.035
Lena (3% coeffs): 2D D-CSMFB (8 × 8-channel, 16 × 16-tap)
Word length
5 bit
6 bit
7 bit
8 bit
32 bitPSNR [dB]
30.16830.33530.36330.47230.599
Barbara (6% coeffs): D-DTCWT (8 × 8-channel, 22 × 22-tap)
Word length
5 bit
6 bit
7 bit
8 bit
32 bitPSNR [dB]28.97229.084 29.0931
29.10329.113
Barbara (6% coeffs): 1D D-CSMFB (8 × 8-channel, 16 × 16-tap)
Word length
5 bit
6 bit
7 bit
8 bit
32 bitPSNR [dB]28.50228.69731.47131.54231.543
Barbara (6% coeffs): 2D D-CSMFB (8 × 8-channel, 16 × 16-tap)
Word length
5 bit
6 bit
7 bit
8 bit
32 bitPSNR [dB]30.82530.96330.97531.05031.142lifting steps can be removed from the entire transform. The
design examples in simulation require much fewer numberof additions and bit-shifters than the D-DTCWT and the 1D
D-CSMFB. Even if the 2D unification and rounding lifting
coefficients is introduced, simulation results show 2D DCSMFB keep good directional selectivity. Finally, in the NLA
simulation, it is clarified that the 2D D-CSMFB is robust
for low bit word length allocation, and thus provides better
numerical and visual reconstruction quality compared with DDTCWT and 1D D-CSMFB.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

ACKNOWLEDGMENT
This work was supported by JSPS KAKENHI Grant Number 24860055, 25820152.
R EFERENCES[1] M. N. Do and M. Vetterli, “The contourlet transform: An efficient
directional multiresolution image representation,” IEEE Trans. Image
Process., vol. 14, no. 12, pp. 2091–2106, Dec. 2005.
[2] E. J. Candès and D. L. Donoho, “New tight frames of curvelets
and optimal representations of objects with piecewise C 2 singularities,”
Comm. Pure Appl. Math., vol. 56, no. 2, pp. 219–266, Feb. 2004.
[3] S. Yi, D. Labate, G. R. Easley, and H. Krim, “A shearlet approach toedge analysis and detection,” IEEE Trans. Image Process., vol. 18, no.
5, pp. 929–941, May 2009.
[4] I. W. Selesnick, R. G. Baraniuk and N. C. Kingsbury, “The dual-treecomplex wavelet transform,” IEEE Signal Process. Magazine, vol. 22
pp. 123–151, Nov. 2005.
[5] S. Kyochi, T. Uto, M. Ikehara, “Dual-tree complex wavelet transformarising from cosine-sine modulated filter banks,” in Proc. ISCAS, Taipei,
Taiwan, pp.2189-2192, May 2009.
[6] S. Kyochi, T. Suzuki, Y. Tanaka, “A directional and shift-invariant
transform based on M-channel rational-valued cosine-sine modulated
filter banks,” in Proc. APSIPA ASC, Hollywood, CA, Dec. 2012.
[7] A. Abbas and Trac D. Tran, “Rational coefficient dual-tree complex
wavelet transform: design and implementation,” IEEE Trans. Signal
Process., vol. 56, no. 8, pp. 3523–3534, Aug. 2008.
[8] I. Bayram and I. W. Selesnick, “On the dual-tree complex wavelet packet
and M -band transform,” IEEE Trans. Signal Process., vol. 56, no. 6, pp.
2298–2310, June 2008.
[9] A. Viholainen, J. Alhava and M. Renfors, “Implementation of parallelcosine and sine modulated filter banks for equalized transmultiplexer
systems,” in Proc. ICASSP, vol. 6, pp. 3625–3628, Salt Lake City, UT,
June 2001.

Fig. 7. NLA results. (a) and (e): Original images of (zoomed) Lena andBarbara (b) and (f): D-DTCWT (8 × 22, Bit word length: 5 bit) (c) and (g):1D D-CSMFB (8 × 16, Bit word length: 5 bit) (d) and (h): 2D D-CSMFB(8 × 16, Bit word length: 5 bit).
[10] I. Daubechies and W. Sweldens, “Factoring wavelet transforms into
lifting steps,” J. Fourier Anal. Appl., vol. 4, no. 3, pp. 245-267, 1998.
[11] Z. Wang, “Fast algorithms for the discrete W transform and for thediscrete fourier transform,” IEEE Trans. Signal Process., vol. 40, no. 11,
pp. 803–816, June 1984.
[12] C. Tu, S. Srinivasan, G. J. Sullivan, S. Regunathan, and H. S. Malvar,“Low-complexity hierarchical lapped transform for lossy-to-lossless image coding in JPEG XR / HD Photo,” in Proc. of SPIE Applications ofDigital Image Processing XXXI, San Diego, CA, August 2008.
[13] S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rded. New York: Academic, 2008.
[14] Z. Wang, “On computing the discrete fourier and cosine transforms,”
IEEE Trans. Acoust. Speech Signal Process., Vol. 33, No. 4, pp. 13411344, 1985.
[15] S. Malvar, “Extended lapped transforms: properties, applications andfast algorithms,” IEEE Trans. Signal Process., vol. 40, no. 11, pp. 2703–

2714, June 1992.