A Directional and Shift-Invariant Transform Basedon M -channel Rational-Valued Cosine-Sine
Modulated Filter Banks
Seisuke Kyochi∗, Taizo Suzuki† and Yuichi Tanaka‡∗ The University of Kitakyushu, Japan, Email: s-kyochi@kitakyu-u.ac.jp† Nihon University, Japan, Email: taizo@ee.ce.nihon-u.ac.jp‡ Tokyo University of Agriculture and Technology, Japan, Email: ytnk@cc.tuat.ac.jp

Abstract—This paper proposes a directional and shift-invariant
transform based on M -channel rational-valued cosine-sine modulated filter banks (R-CSMFBs) for the practical implementationon hardware devices. M -channel CSMFBs can be easily designedby the modulation of a prototype filter and achieve a goodstopband attenuation. In addition, in our previous work, thedirectionality and the shift-invariance of CSMFBs have beentheoretically clarified. Thus, they can be an alternative choice ofthe dual-tree complex wavelet transform (DTCWT) which is oneof the most popular directional and shift-invariant transforms. In
this paper, it is shown that the proposed lifting-based structureof the R-CSMFB can also achieve rich directional selectivity andthe shift-invariance even if the lifting coefficients are rounded
to rational values. Finally, the R-CSMFB can provide better
stopband attenuation and image denoising performance than thatof the conventional M -channel rational-valued DTCWT in thesimulation.

I. I NTRODUCTION
The dual-tree complex wavelet transform (DTCWT), conventionally, have been introduced to achieve “rich directionalselectivity” and “shift-invariance” which discrete wavelet
transforms (DWTs) cannot satisfy [1]. It can achieve better
performance than DWTs in various kinds of image processing,such as image denoising [1].
In this paper, we address on the design of the directionaland shift-invariant complex WT (CWT) satisfying the numberof channel M > 2 and the rational value constraint simultaneously. Conventionally, an efficient design method for 2-channel
rational-valued DTCWT (R-DTCWT) has been proposed forthe hardware implementation (add/bit-shift operations) [2].
However, it has not been proposed yet for the M -channel
ones. Although M -channel R-DTCWT can be realized byusing the dual-tree complex wavelet packet (DTCWP) [3],a cascade of the 2-channel R-DTCWT suffers from poor
stopband attenuation owing to the small number of the design
parameters, especially in the case of the short filter length
constraints on the entire system.
In our previous work, the directional selectivity andthe shift-invariance of cosine-sine modulated filter banks(CSMFB) [4] have been theoretically clarified [5]. Thus, it canbe considered that CSMFBs are in the same class of the directional and shift-invariant CWTs where the DTCWT belongs
to. In this paper, the rational-valued CSMFB (R-CSMFB)
is proposed by introducing the lifting factorization and theFig. 1. M -channel DTCWTs and CSMFBs

rounding operation into the CSMFB. Unlike the DTCWP, RCSFMB can be designed directly as a M -channel system, andthus it can obtain better stopbband attenuation even if lifting
coefficients are rounded. In the simulation, the directionality
and the shift-invariance of the R-CSMFB are demonstrated.
Moreover, the R-CSMFB is applied to image denoising as apractical application and shown its better performance thanthat of the R-DTCWT.
Notations: I and J are the identity√ and the reversal iden−1. α is the conjugate
tity matrices, repectively. j :=∗(z)andH̃(z),H(ω)
are definedof α ∈ C. H(z),H
−n−nas H(z) :=, H ∗ (z) :=, andn h(n)zn h(n)z∗ −1H̃(z) := H (z ). The M × L FB means the M channel
filter bank with the filter length of L.
II. R EVIEW
A. Dual-Tree Complex Wavelet Transform and Dual-TreeComplex Wavelet Packet
The DTCWT is constructed by two maximally decimatedperfect reconstruction FBs, as illustrated in Fig. 1. In this case,(1)(2)
the pair {Hk (z), Hk (z)} (0 ≤ k ≤ M − 1) is designed tosatisfy the following equations [1]:
(2)

(1)

Hk (ejω ) =e−jθk (ω) Hk (ejω ),
1θ0 (ω) = d +(M − 1)ω − pω,2(1)

where d ∈ Z denotesthedelay
the primal FB and the between


2π−1,
∀ω
∈ 2πdual FB, ∀p ∈ 0, . . . , M
2M p, M (p + 1) ,according to the relationship of cosine and sine modulationbetween two FBs [5].
III. R ATIONAL - VALUED C OSINE S INE M ODULATEDF ILTER BANKS BASED ON THE L IFTING FACTORIZATIONIn this section, the R-CSMFB is proposed. First, we consider a lattice strucutre corresponding to the CMFB. It isknown as the extended lapped transform [6] expressed by:(a) Primal FB(1)

E(1) (z) = CIV WΛ(z)D0(1)
Dk =(b) Dual FB
Fig. 2. Example of 2r -band DTCWT (r = 2)

andθk (ω) =π
2 −0d + 12 ωω ∈ (0, 2π)
ω = 0,(2)where 1 ≤ k ≤ M − 1.
Conventionally, the dual-tree complex wavelet packet(DTCWP) which is the design method of 2r -channel DTCWT
has been proposed [3]. Specifically, it is obtained by cascad(1)(2)
ing a 2-channel DTCWT {Hk (z), Hk (z)}k=0,1 , and 2channel perfect reconstruction FBs (PRFBs) {F0 (z), F1 (z)}(1)(1)
and {H0f (z), H1f (z)}, as shown in Fig. 2. The 2-channel
DTCWT can be designed by the several methods, such as“q-shift solution” and “common factor solution” (spectralfactorization) in [1]. The problem of the DTCWP is that goodstopband attenuation cannot be obtained if the filter length of
the whole system is restricted to be short. It is because thedesign degree of freedom for filter optimization remains onlyin the cascaded 2-channel FBs with the limited filter length.
B. Cosine-sine modulated filter banks
A CSMFB also consists of two maximally decimated filter(1)(2)
banks as in Fig. 1. Its filter coefficients of {Hk (z), Hk (z)}are expressed as follows [4]:(1)πn − N 2−1 + θkhk (n) := 2p(n) cos (k + 12 ) M(2)1 πhk (n) := 2p(n) sin (k + 2 ) M n − N 2−1 + θk ,(1)

Λ(z 2 )Dkk=1 −1−CkSk Jz I 0,, Λ(z) =JSk J JCk J0I(4)where Ck = diag cos θk0 , . . . , cos θk M/2−1 , Sk =diag sin θk0 , . . . , sin θk M/2−1 and CIV denotes the typeIV DCT. The structure is depicted in Fig. 3 (a). Then, weconvert the lattice structure into lifting steps.
In general, the rotation matrix can be factorized into lifting
steps as follows: 
cos θ − sin θ1 p 1 0 1 p=,(5)sin θ cos θ0 1 u 1 0 1
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 coefficientsround(p × 2N )/2N , round(u × 2N )/2N , while preserving
the perfect reconstruction condition. In fact, the 2-channel RDTCWT is designed based on the lifting (or butterfly matrices)
factorization [2].
(1)
As shown in Fig. 3 (b), each Dk can be reduced as
−I 0 I Pk JI0 I Pk J
(1)
Dk =, (6)0 I 0I
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 [7] (described in Fig. 4), it canalso be converted to the lifting-based factorization.
On the other hand, the lifting coefficients of E(2) (z)can be determined immediately by using the relationship ofE(2) (z) = z −K ΓẼ(1) (z) = z −K Γ(E(1) (z))−1 . Specifically,
the lifting factorization of E(2) (z) is expressed as follows:1−1(1)
Dkz −2 Λ(z −2 )E(2) (z) =Γ
k=K−1(3)(−1)k π4(2)K−1 

where k = 0, . . . , M − 1, θk =and p(n) is theprototype filter. Let E(1) (z) and E (z) be the polyphasematrices of the primal and the dual FBs, respectively, then,from (3), it follows that E(2) (z) = ΓẼ(1) (z), where Γ isthe diagonal matrix Γ := diag((−1)k ) (0 ≤ k ≤ M − 1).
Therefore, if the primal FB satisfies the perfect reconstruction(PR) property Ẽ(1) (z)E(1) (z) = I, the dual FB E(2) (z)system also satisfies the PR property. In our previous work, thedirectionality and the shift-invariance of CSMFBs are derived−1(1)
× D0z −1 Λ(z −1 )WCIV(7)

According to the above expression, it can be verified that−1(2)(1)
Dk = Dk
I −Pk JI0 I −Pk J −I 0=.
0
I−JUk I 0I0 I(8)It indicates that the lifting coefficients of E(2) (z) are just thesign-altered versions of those of E(1) (z).

TABLE I
S TOPBAND ATTENUATION

Word length
Δ (L=16)
Δ (L=22)8 bit
2.23150.6356

Fig. 4. The rotation-matrix-based structure of the type-IV DCT. (Example:
4 × 4, θ0 = π/16, θ1 = 5π/16, θ2 = π/4)

Word length
Δ (L=8)Δ (L=16)8 bit
0.24770.3863

In [4], an another lattice structure of E(2) (z) based on typeIV DST has been presented. In that case, the lattice strucuture
can be expressed by

Word length
Δ (L=22)Δ (L=36)8 bit
135.0658
72.5471

Word length
Δ (L=16)
Δ (L=32)

8 bit
2.2984
1.3973(2)

Dk =K−1 

Λ(z2k=1Ck
JSk J(2))Dk 
Sk J−I 0(1) −I 0=
Dk.
−JCk J0 I0 I(9)From the above equations and the relationship SIV = ΓCIV J,the lifting coefficients for E(1) (z) can also be reused forE(2) (z).
We design M -channel R-CSMFBs (M > 2) based on
the lifting approach described in the previous section, andevaluate its directional selectivity andshift-invariance. For
a comparison, M -channel R-DTCWT (M > 2) is designed by the DTCWP [3] cascading the 2-channel spectralfactorization-based R-DTCWT [2]. The cascaded 2-channel(1)(1)
FBs {F0 (z), F1 (z)} and {H0f (z), H1f (z)} described in Sec.
II-A are implemented by the lifting steps of the rotation
matrices given in (5) and the delay matrices. Both M -channeltransforms are optimized to minimize the stopband attenuation
Δ specified byM−1(1)
Δ=|Hk (ejω )|2 dω,
(10)
k=0

Ωs (k)(1)

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].
ππIf M (k − 2) ≤ 0 or M (k + 2) ≥ π, the corresponding interval
is ignored.
A. Design Example
The design examples of the R-CSMFB and the R-DTCWT
are given in this section. First, the R-CSMFB and the RDTCWT are designed as a real-valued FB minimizing thestopband attenuation. Then, the real-valued lifting coefficientsare rounded to the rational coefficients from 8 to 4 bit wordlength. The results of the stopband attenuation Δ are listedin Table I. The numbers of the filter length of the cascaded
2-channel PUFBs for the 4 × 16 (# of channel × # of filterlength) and 8 × 22 R-DTCWTs and the 4 × 22 and 8 × 36 RDTCWTs are 4 and 6, respectively. Thus, the design degree offreedom for optimization is quite limited. On the other hand,since the CSMFBs with multiple channels can be designed

5 bit
3.46171.30354 bit
7.8892
3.46475 bit
0.30691.4547

4 bit
1.5346
1.77825 bit
143.3687
77.1526

4 bit
134.6823
75.1663

5 bit
31.5684
3.2597

4 bit
42.0679
11.3918

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IV. S IMULATION

8-channel R-DTCWT
7 bit
6 bit
135.0658
134.1306
71.391476.24798-channel R-CSMFB
7 bit
6 bit
3.065636.1532
1.51741.5949䣏䣣䣩䣰䣫䣶䣷䣦䣧䢢䣔䣧䣵䣲䣱䣰䣵䣧䢢䣝䣦䣄䣟(2)(z) =SIV WΛ(z)D0䣏䣣䣩䣰䣫䣶䣷䣦䣧䢢䣔䣧䣵䣲䣱䣰䣵䣧䢢䣝䣦䣄䣟E(2)4-channel R-DTCWT
7 bit
6 bit
2.28572.5866
0.53741.65004-channel R-CSMFB
7 bit
6 bit
0.22780.36640.84090.9212

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(b) Rational

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

directly, more design degree of freedom can be obtained, andthus it can achieve better stopband attenuation.
Fig. 5 depicts the frequency responses of the designed
4 × 8 CSMFB and R-CSMFB. It can be observed that thefrequency responses of R-CSMFB in Fig. 5(b) are robustly
approximating those of the original CSMFB given in Fig. 5(a)under rounding operation.
B. Directional Selectivity
The 2-D wavelet basis functions obtained by the 4-channel
R-CSMFB are shown in Fig. 6. Clearly, the basis functions
are oriented, which indicates the R-CSMFB achieves richdirectional selectivity under rounding operation.
C. Evaluation of Shift InvarianceIn this section, shift invariance of the R-CSMFB and theR-DTCWT (4 × 16, 4 bit word length) are evaluated. First, weapply the R-CSMFB and the R-DTCWT to the input impulse
and its shifted impulse independently. Then, compute the(1)correlation of the output subband signals. Specifically, let sk,r(2)and sk,r (0 ≤ k ≤ 3) be the k-th subband signals of the firstand the second system which are obtained from the r-sample(0 ≤ r ≤ 3) shifted original signal (the original one is denotedby r = 0). Then, we evaluate the shift invariant property
(1)
(2)
(1)
(2)
by the normalized correlation of sk,r + sk,r and sk,0 + sk,0denoted by Φ(k, r) (0 ≤ Φ(k, r) ≤ 1). Table
3 II shows theaveraged correlation values ΦA (k) = 14 r=0 Φ(k, r). As

Fig. 3. The structure of the 4 channel CMFBs. (Left: rotation-matrix-based structure, Right: lifting-based structure)

TABLE IIID ENOISING RESULTS
Lena R-DTCWT (2 level, filter length: 22)σσ = 15
σ = 20σ = 25σ = 30PSNR [dB]
30.4668
29.071828.0236
27.1358
Lena R-CSMFB (2 level, filter length: 16)σσ = 15
σ = 20σ = 25σ = 3031.0437
29.6159
28.4847
27.5970
PSNR [dB]
Barbara R-DTCWT (2 level, filter length: 22)σσ = 15
σ = 20σ = 25σ = 30PSNR [dB]
28.870327.368126.158125.2805
Barbara R-CSMFB (2 level, filter length: 16)σσ = 15
σ = 20σ = 25σ = 30PSNR [dB]29.8567
28.2664
26.9738
26.0463

Fig. 6. 32 directional basis functions with size of 16 × 16 obtained by the
4 × 16 CSMFB whose lifting coefficents are rounded to 4 bit word length.
TABLE IIC ORRELATION RESULTS
4-Channel R-DTCWT (filter length: 16)
Channel
k=0
k=1
k=2
k=3
ΦA (k)
0.95050.9252
0.94770.91244-Channel R-CSMFB (filter length: 16)
Channel
k=0
k=1
k=2
k=3
ΦA (k)
0.9962
0.99400.99480.9935shown in Table II, the R-CSMFB satisfies the better shift
invariant property than the R-DTCWT.

(a)

(b)

(c)

(d)

(e)

(f)

D. Image Denoising
In this simulation, image denoising is demonstrated toverify the performance of R-CSMFB. The gaussian random
noise with variance σ 2 (σ = 15, 20, 25, 30) is added to thetest images Lena and Barbara. The noisy images are firstdecomposed by 2-level 8-channel of the R-CSMFB and theR-DTCWT, then performed by hard-thresholding with 3σ.
The lifting coefficients of both transfomations are allocated
8 bit word length. For the numerical metric of the denoising
performance, we use PSNR. As shown in Table III and Fig.
7, R-CSMFB shows better visual quality than those of theR-DTCWT, thanks to the better stopband attenuation of theR-CSMFB.
V. C ONCLUSION
In this paper we proposed the directional and shift-invariant
transform based on M -channel R-CSMFB by the lifting factorization and the rounding operation. It is clarified that thelifting coefficients for the primal FB can be reused for thedual FB. The R-CSMFB provides not only better stopband
attenuation than the R-DTCWT, but also the properties of shift
invariance and rich directional selectivity under the rational
value constraint. Finally, the R-CSMFB is applied to image
denoising as one of the application, and shown its better
performance than that of the R-DTCWT.

Fig. 7. Denoising results. (a) and (d): Noisy images of Lena and Barbara(sigma = 30) (b) and (e): R-DTCWT (8 × 22) (c) and (f): R-CSMFB(8 × 16)

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