Proceedings of APSIPA Annual Summit and Conference 201516-19 December 2015

Design of Oversampled Cosine-Sine Modulated
Filter Banks for Directional Image Representation
Ryutaro Ogawa∗ , Seisuke Kyochi∗ , and Taizo Suzuki†∗ Dept. of Information and Media Engineering, The University of Kitakyushu, Fukuoka, Japan

E-mail: v4mca004@eng.kitakyu-u.ac.jp, s-kyochi@kitakyu-u.ac.jp† Faculty of Engineering, Information and Systems, University of Tsukuba, Ibaraki, Japan

E-mail: taizo@cs.tsukuba.ac.jp

Abstract—We propose an oversampled cosine-sine modulated
filter bank (OCSMFB) for richer overcompleteness contributing
sparse directional image representation. The CSMFB, proposed
as an alternative of the dual-tree complex wavelet transform(DTCWT), can be designed easily by modulating a prototype
filter and can work in image processing tasks better than the
DTCWT. However, the conventional CSMFB consists of two
critically-sampled FB, and its non-redundancy restricts design
degree of freedom and overcomplete image representation. This
paper relaxes the redundancy of the conventional CSMFB byoversampling each CMFB and SMFB. For design of the OCSMFB, we present the perfect reconstruction (PR) condition andthe parameterization, in order to satisfy the PR structurally.
Moreover, the parameterization for structural regularity, which
provides NO DC leakage, is discussed. The experimental resultsof image denoising show the OCSMFB can work better than the
CSMFB.

I. I NTRODUCTION
For decades, extended discrete wavelet transforms satisfying“rich directional selectivity1 ” (DS) have been widely studied,such as dual-tree complex wavelet transforms (DTCWTs)[1], [2], Contourlet [3], Curvelet [4], and so on. Amongthem, DTCWTs have been paid much attenuation due totheir low computational complexity [1], [2]. While directional selective transforms usually require high computational
complexity, such as two dimensional (2D) non-separable filtering, DTCWTs can save computational complexity by 2D
separable implementation of parallel filer banks (FBs), i.e.,cascading vertical and horizontal filtering. Furthermore, theyalso satisfy “shift-invariance” (SI). Thanks to DS and SI,
it can be successfully applied to various kinds of practicalimage processing, such as image denoising, image analysis,
and image compression [1].
As well as the DTCWT, another DS and SI transforms,
cosine-sine modulated filter banks (CSMFBs), have beenshown recently [5], [6], [7]. M -channel DTCWTs are difficult
to design with good filter performances, e.g., stopband attenuation or coding gain, in time-domain, due to the fractional delay
requirement imposed on parallel FBs, which degrades design
degree of freedom. On the other hand, because CSMFBs
require NO fractional delay requirement (just the modulation
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 [1].

978-988-14768-0-7©2015 APSIPA

of a prototype filter), M -channel CSMFBs can be better
performance than M -channel DTCWTs and effectively workin applications.
One problem on the conventional CSMFBs is the limitationof redundancy, i.e., both CMFB and SMFB are criticallysampled FBs and the resulting redundant ratio is 2. This poor
redundancy degrades the design degree of freedom. Since
the design degree of freedom of the CSMFB is only in a
prototype filter, filter performance cannot be improved undershort filter length. Moreover, rich redundancy is generally
desired to enhance the performance in many image processing
tasks, where redundant image representation is allowed.
This paper attempts to relax the limitation of critical sampling to oversampling and realizes oversampled CSMFBs(OCSMFBs). Note that the oversampled CMFB (OCMFB) has
already been proposed in [8], whereas the oversampled SMFB(OSMFB) has not yet. Thus the perfect reconstruction (PR)condition for the OSMFB is firstly discussed in this paper.
Moreover, the parameterization of the OCSMFB for the PR
is proposed. In addition, the parameterization for structural
regularity, which is important for NO DC leakage [9], is also
shown. The designed OCSMFB is applied to image denoising
as a practical application. Experimental results show that the
OCSMFB outperforms the CSMFB with respect to both of thereconstruction error and the visual quality.
The rest of this paper is organized as follows. Sec. II
briefly reviews about the CSMFB and the OCMFB. The PR
condition for OSMFB is discussed in Sec. III-A. Then, theparameterizations for PR and structural regularity are given inSec. III-B and Sec. III-C, respectively. Sec. IV evaluates the
OCSMFB in image denoising. Finally, this paper is concluded
in Sec. V.
∑−nNotations: H(z) is defined as H(z) :=n h(n)ze
and H(z):= H ∗ (z −1 ). The M × L FB means the M channel filter bank with the filter length of L. IN , JN , anddiag(a0 , . . . , aN −1 ) are the identity, the reversal identity, andthe diagonal matrices, respectively.
II. P RELIMINARIES
A. Cosine-Sine Modulated Filter Banks and Oversampled
Cosine Modulated Filter Banks
A CSMFB consists of two maximally decimated FBs as inFig. 1 (D = M ). Its filter coefficients of the analysis subband

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Proceedings of APSIPA Annual Summit and Conference 201516-19 December 2015
On the other hand, Bölcskei et al. have discussed the PR
condition for OCMFB (as D = 2ML in Fig. 1) [8]
4L−1
∑
`=0

e k+D` (z)Gk+D` (z) = 1 ,
G
2M

(3)

where k = 0, . . . , D
2 − 1.
III. OVERSAMPLED C OSINE -S INE M ODULATED F ILTER
BANKS
Fig. 1. M -channel CSMFB (D = M ). Upper FB: CMFB, Lower FB: SMFB.

7KH&0)%6HSDUDEOH'7UDQVIRUPThis section gives the PR condition of the OSMFB, which
has not been discussed yet, for constructing OCSMFB. Consider the same definition of the subband filters for SMFB (1),but the decimation factor is D = 2ML (1 ≤ L ≤ log2 M −1). In(2)(1)
the paraunitary OCSMFB, hk (n) = (−1)k hk (N − 1 − n).
From the time-domain relationship between the OCMFB andthe OSMFB, the relationship between the polyphase matrix of
the OCMFB Ec (z) and the OSMFB Es (z) can be expressedas:  

  





  

  

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6XEEDQG
6LJQDOV

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 

A. Perfect Reconstruction Condition for OSMFB

ES (z D ) = z −(N −(D−1)) ΓEC (z −D )JD ,where Γ = diag(1, −1, . . . , 1, −1). Assume that the OCMFB
e C (z)EC (z) = ID . By substituting
satisfies the PR condition E
S D
S D
e
(4) into E (z )E (z ), it can be reduced that

Fig. 2. Procedure of the directional image decomposition via M -channel
CSMFB.

The parameterization for the CMFB (and SMFB) waspresented in [10]. It is briefly reviewed in the following.
For simple discussion, the numbers of the channel and thedecimation factor are 4. Then, the PR condition on the 4channel and 4-decimated CMFB/SMFB can be expressed as:
e 2k (z)G2k (z) + G
e 2k+4 (z)G2k+4 (z) = 1 , (k = 0, 1). (6)G
8For given filter length N = 2mM , the parameterization for
the above equation can be given as [10]:
[
]
{
[
]} [
]
1
∏
G2k (z)1cos θk=Vk,`, (7)G2k+4 (z)z −1sin θk`=m−1(2)where k = 0, . . . , M
2 − 1.
In our previous work [5], the directionality of CSMFBs
is derived according to the relationship of cosine and sinemodulation between two FBs. For the directional image
representation, the original image is firstly decomposed by
two separable 2D transforms of the CMFB and the SMFB.
Then, the directional coefficients are obtained by addition andsubtraction of the CMFB and the SMFB coefficients. The
procedure is illustrated in Fig. 2.

978-988-14768-0-7©2015 APSIPA(5)

B. Parameterization for Perfect Reconstruction(1)

where k = 0, . . . , M − 1, N = 2mM − 1, θk = (−1)k π4 ,and p(n) is the prototype filter. In this paper, we assume theparaunitariness on FBs. Thus, the synthesis subband filters are
the time-reversed versions of the analysis filters, i.e., fkc (n) =hck (N − 1 − n) and fks (n) = hsk (N − 1 − n).
The PR condition is imposed on the prototype filter. Let
Gk (z) (k = 0, . . . , 2M − 1) be the polyphase∑componentsof the prototype
filter P (z), i.e., P (z) =Gk (z)z −k ,∑−qGk (z) =p(2M n + k)z . The PR property for CSMFB
can be expressed as:e k (z)Gk (z) + G
e M +k (z)GM +k (z) = 1 ,G
2Me S (z D )ES (z D ) = ID .
E

If the OCMFB satisfies the PR, the OSMFB also satisfies thePR.

filters {Hkc (z), Hks (z)} are expressed as follows [5]:

{
(
(
)
)
π
hck (n) := 2p(n) cos (k + 12 ) M
n − N 2−1 + θk
(
(
)
)
π
hsk (n) := 2p(n) sin (k + 12 ) M
n − N 2−1 + θk ,(4)

where Vk,` is a rotation matrix. The total number of the design
parameter is 2m.
On the other hand, according to (3), the PR condition for
the 4-channel and 2-decimated OCSMFB is reduced as:3∑
k=0

e 2k (z)G2k (z) = 1 .
G
8

(8)

Note that the parameterization of (8) can be realized byapplying (7) to {G0 (z), G4 (z)} and {G2 (z), G6 (z)}, independently, which is presented in [8]. However, this cannot fully

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Proceedings of APSIPA Annual Summit and Conference 201516-19 December 2015

exploit the design degree of freedom. This paper introduces awider class of the parameterization as follows: 
G0 (z)
a0
{
[
]}
1
∏
G2 (z)
a1 
I
2


  , (9)V0,k
G4 (z) =z −1 I2 a2 
k=m−1
G6 (z)
a3

where CIV indicates type-IV DCT. By substituting (12) into(11) as z = 1, it can be derived as:G0 (−1)
d0,3



2 
G2 (−1) = √c d0,1  .
√
(13)




G
(−1)
d
4
0,0
2M
M
G6 (−1)d0,2where V0,k is a 4 × 4∑
orthogonal matrix and {ai }3i=0 are
3real numbers satisfying i=0 a2i = 1. Generally, an M × M
orthogonal matrix V can be characterized by the product of−1
the M (M2 −1) givens rotation matrices as [11]: and {ai }M
i=0∑M −1 2
satisfying i=0 ai = 1 are parameterized by M − 1 anglesas:
a = cos θ 0 ( 0)k−1∏
. (10)
sin θi cos θk , (k = 1, . . . , M − 1)
 ak =

Here, if m = 1, three angles θ0 , θ1 , and θ2 for [a0 , a1 , a2 , a3 ]T
can be uniquely determined as:

(
)
d0,2

−1

θ2 = tan



d


)
( 0,0

1 d0,0θ1 = tan−1.
(14)cos θ2 d0,1


(
)


1 d0,1 θ0 = tan−1
cos θ1 d0,3i=0

Since M = 4 in this case, the number of the design parameteris 6(m − 1) + 3. Clearly, the proposed parameterization covers
the wider class of the OCSMFB and provides richer design parameters. The above discussion can be similarly applied to thegeneral M -channel D-decimated OCSMFB parameterization.
C. Parameterization for Structural Regularity
The regularity condition is important for many image
processing tasks [9]. Lack of regularity causes DC leakage
that sometimes degrades the energy compaction of natural
images, and thus efficient sparse representation cannot be
achieved. In [12], the parameterization of the CMFB for
structural regularity has been discussed. This section extends
it to the OCMFB/OSMFB. As well as the previous section, we
consider 4-channel and 2-decimated OCSMFB. First of all, thedefinition on the regularity is described as:




 
1
H0 (1)c H1 (1)  z −1 
0




 
= . .
(11)
 = E(z D ) 

..
..




 .. 
.
.
HM −1 (1)0z −(D−1) z=1
According to (1), the polyphase matrix of the OCMFB E(z D )is given as:
g0 (−z 4 ) z −2 g1 (−z 4 ) E(z 2 ) =Ca (12) z −4 g2 (−z 4 )  ,−64
z g3 (−z )√M (−1)m/2 CIV [ (I − J) − (I + J) ]
for m even,
Ca = √(m−1)/2

M
(−1)
C
[
(I
+
J)
(I − J) ]IV
for m odd
4

4

4

4

g0 (z) =diag [G0 (z), G1 (z)] , g1 (z) = diag [G2 (z), G3 (z)] ,g2 (z) =diag [G4 (z), G5 (z)] , g3 (z) = diag [G6 (z), G7 (z)] ,978-988-14768-0-7©2015 APSIPA

For general order m, (13) can be reduced as: 
1G0 (−1)∆d0 (m even)G2 (−1)  (−1)m/2=,
G4 (−1) 
1

d
(m
odd)

1
G6 (−1)(−1)(m−1)/2(15)=
where d0=
[d0,0 , d0,2 , d0,3 , d0,1 ]T , d1T
[d0,3 , d0,1 , d0,0 , d0,2 ] , ∆ = diag(1, 1, −1, −1). From(15), it is enough to impose the condition on only the matrix
V0,m−1 which can be expressed as:
V0,m−1 =I − 2γγ T
)
(∏
1
di −
k=m−2 V0,k a) , (i = 0, 1).
(∏
γ=
1
di −
k=m−2 V0,k a

(16)

2

The parameterization of structural regularity for general M channel D-decimated OCSMFBs can be given, similarly.
IV. E XPERIMENTAL R ESULTS
A. Design Examples
We designed the prototype filters of the 4-channel CSMFB
and 4-channel 2-decimated OCSMFB with the filter lengthof 16 (m = 2) and the regularity. The regularity for theconventional CSMFB was imposed by using the method
presented in [12]. The numbers of the design parameters are
2 and 3 for the CSMFB and the OCSMFB, respectively.
The parameters of the prototype filter P ∫(z) were optimizedπ
to minimize stopband attenuation: C = π |P (ω)|dω. The
2M
designed frequency responses are shown in Fig. 3. Clearly,
the frequency responses of the OCSMFB satisfy the regularity
condition and achieve better stopband attenuation.
B. Image Denoising
In this simulation, image denoising was demonstrated toverify the performance of OCSMFB. The Gaussian random
noise with variance σ 2 (σ = 30) is added to the test images
shown in Fig. 4. The 4×8, 4×16, 8×16, and 8×32 CSMFBs
and those of OCSMFBs with the decimation factor of 2 were
newly designed by minimizing the stopband attenuation. We

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APSIPA ASC 2015

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(a) Noisy(b) CSMFB(c) OCSMFB(d) LPCSMFB(e) CSMFB(f) OCSMFB(g) LPCSMFB(h) CSMFB(i) OCSMFB

1RUPDOL]HGb)UHTXHQF\(a) 4 × 16 CSMFB(b) 4 × 16 OCSMFB

Fig. 3. Examples of frequency responses.

(a)

(b)

(c)

(d)

Fig. 4. Original images: (a) Zone plate, (b) Lena, (c) Barbara, (d) Pepper.

also used the 4 × 10, 4 × 16, 8 × 22, and 8 × 36 DTCWTs
designed by [2], and the 5×8, 5×16, 9×16, and 9×32 linear
phase CSMFBs (LPCSMFBs) [7] for comparison. The noisy
images were firstly applied by 1-level decomposition of eachtransform, then performed by hard-thresholding with T h = 3σ[4]. We used PSNR for the numerical metric of the denoising
performance. As shown in Table I and Fig. 5, the OCSMFB
shows better numerical and visual quality than those of the
CSMFBs, the DTCWTs, and the LPCSMFBs, in most cases.

Fig. 5. (a): Noisy image (Barbara), (b), (c): Denoised images, and (d)-(f):Zoomed denoised images. LPCSMFB: 9 × 32, CSMFB/OCSMFB: 8 × 32.
TABLE IT HE NUMERICAL RESULTS OF IMAGE DENOISING (PSNR [ D B])

V. C ONCLUSION
In this paper we proposed the OCSMFB for better design
degree of freedom and richer overcompleteness. In order toconstruct the OCSMFB, we firstly showed the PR condition
for the OSMFB, which has not been discussed yet. Then, we
gave the design parameterization for the PR and the structural
regularity. Finally, in the image denoising simulation, it wasclarified that the proposed OCSMFB provides better numericaland visual reconstruction quality compared with the conventional CSMFBs and the DTCWTs.
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978-988-14768-0-7©2015 APSIPA

Test image
Noisy image
4 × 10 DTCWT [2]4 × 8 CSMFB [5]5 × 8 LPCSMFB [7]
4 × 8 OCSMFB
4 × 16 DTCWT [2]
4 × 16 CSMFB [5]5 × 16 LPCSMFB [7]4 × 16 OCSMFB

Zoneplate
18.59
20.02
22.64
21.21
24.46
21.04
23.90
23.32
25.01

Lena
18.59
27.76
28.28
27.18
28.96
27.90
28.39
28.30
28.92

Barbara
18.59
24.53
25.29
25.54
25.94
24.72
25.63
26.29
26.15

Pepper
18.59
27.65
28.07
27.31
28.81
27.79
27.90
28.26
28.67

8 × 22 DTCWT [2]
8 × 16 CSMFB [5]9 × 16 LPCSMFB [7]8 × 16 OCSMFB
8 × 36 DTCWT [2]
8 × 32 CSMFB [5]9 × 32 LPCSMFB [7]8 × 32 OCSMFB

20.92
25.81
22.68
27.85
21.98
25.35
25.06
28.01

27.43
28.41
27.12
29.32
27.77
28.46
27.87
29.27

25.41
26.76
25.99
27.75
25.81
26.87
26.75
27.80

27.63
28.32
27.13
29.10
27.70
28.05
27.68
29.12

[7] L. Liang and H. Liu, “Dual-tree cosine-modulated filter bank with linearphase individual filters: An alternative shift-invariant and directionalselective transform,” IEEE Trans. Image Process., vol. 22, no. 12, pp.
5168–5180, Dec. 2013.
[8] H. Bölcskei and F. Hlawatsch, “Oversampled cosine modulated filterbanks with perfect reconstruction,” IEEE Trans. Circuits and Syst. II,
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[9] G. Strang and T. Q. Nguyen, Wavelets and filter banks, WellesleyCambridge Press, 1997.
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[11] P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall,Englewood Cliffs, 1992.
[12] M. Ikehara T. Uto and K. Ohue, “A novel design of regular cosinemodulated filter banks for image coding,” IEICE Trans. on Fundamentals, vol. E92-A, no. 7, pp. 1633–1641, Jul. 2009.

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