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Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
1
Design of Regular Wavelets Using a Three-Step
Lifting Scheme
Ramin Eslami*, Member, IEEE, and Hayder Radha, Fellow, IEEE
EDICS: DSP-BANK Filter bank design and theory, DSP-WAVL Wavelet theory and applications
Abstract- We propose structural multidimensional multi-channel filter banks with desirable numbers of
vanishing moments for the analysis and synthesis banks. For a two-channel filter bank, we use a three-
step lifting scheme as opposed to the conventional two-step lifting method in order to provide more
symmetry between the analysis and synthesis filters. We show that the resulting filters have more
regularity, lower frame bounds ratio, and better frequency selectivity. We also extend our design to a
general multi-channel framework. Three steps of lifting scheme provides us a degree of freedom which
we can benefit from toward a more flexible design.
Index terms- Lifting scheme, multidimensional filter banks, vanishing moments, regular wavelet
transform, frame bounds.
I. INTRODUCTION
It is well known that iterated filter banks lead to a wavelet scheme under certain conditions.
Meanwhile, regularity of wavelets is a critical characteristic that at least ensures providing a d -
dimensional multiresolution scheme in 2 ( )dL � [18], [22], [28]. Therefore, designing filters for a wavelet
scheme where they fulfill perfect reconstruction (which is equivalent to biorthogonality) and regularity
has been an attractive, yet challenging aspect of wavelets analysis especially for the multidimensional
Manuscript received September 23, 2009.
R. Eslami was with the Department of Electrical and Computer Engineering, Michigan State University, East
Lansing, Michigan 48824. He is now with the Department of Biomedical Engineering, University of Rochester,
Rochester, NY 14627 (phone: 585-276-4378; fax: 585-276-2127; e-mail: [email protected]).
H. Radha is with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing,
Michigan 48824 (phone: 517-432-9958; fax: 517-353-1980; e-mail: [email protected]).
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
2
framework.
A necessary condition for regularity is having vanishing moments at aliasing frequencies of wavelet
highpass filters. However, this is not a sufficient condition for the regularity of multidimensional wavelets
[5], [15].
Owing to the lack of spectral factorization for multidimensional signals, designing filters for such
signals is not trivial. However, there are a few methods developed for this case. One of the approaches is
the cascade algorithm proposed in [15], [16]. Although perfect reconstruction is achieved through this
design method, imposing vanishing moments is not always possible. One to multidimensional approach
[25], an extension of the McClellan transformation, is another method which is widely used in
multidimensional filter design. This approach can also be performed in the polyphase domain where one
can take advantage of a separable filtering for nonseparable (e.g. quincunx) filter banks [17], [19].
Lifting scheme is another representation of wavelets where it provides a fast algorithm for computing
the wavelet transform [6], [23], [24]. Lifting can be interpreted as a representation of a filter bank in the
polyphase domain using ladder structure [1], [19]. Since perfect reconstruction is guaranteed for any kind
of selection of lifting steps, the filter design for this approach can be accomplished by only imposing the
desired characteristics such as vanishing moments into the filters.
Kovačević and Sweldens [14] proposed a structural design of multidimensional wavelets of any order
and any number of analysis and synthesis vanishing moments. For two-channel schemes they used a two-
step lifting technique. This approach could be considered as a generalization of the design method
proposed by Phoong et al. [19], where they proposed a ladder structure for one-dimensional (1-D) filer
banks and also they designed quincunx filter bank (QFB) using transformation.
Ansari et al. [1] proposed a two-channel filter bank using a triplet of halfband filters (equivalent to a
three-step lifting), where they could address the restrictions in double-halfband filter bank (equivalent to a
two-step lifting) [19]. While the magnitudes of the normalized analysis filter pair for the double-halfband
filter bank in [19] have to be 1 and 0.5 at / 2ω π= , they can be set to an equal value of 2 2 in the
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
3
triple-halfband design. Extensions of triple-halfband design have also been proposed [4], [13], [26].
In this work, we generalize Ansari’s method [1] to a multidimensional filter bank design with any
number of analysis and synthesis vanishing moments using a structural approach based on Kovačević’s
method [14]. In our design we only consider FIR filters. We demonstrate that the designed filters achieve
better regularity, lower frame bounds ratio, and better frequency selectivity when compared to the filters
designed in [14].
II. BACKGROUND
In this section we briefly outline some required background material in order to develop the proposed
filter design approach. Initially, we introduce notations that we use in this paper followed by a brief
background on two-channel filter banks. Then we explain about Neville filters and the two-step lifting
design proposed in [14].
A. Notations
A d -dimensional discrete signal [ ]x ∈n � with 1 2( , , , ) ddn n n= ∈n … Z has z -transform
( ) [ ]dX x−
∈=∑ n
nz n z
Z. Here 1 2( , , , ) d
dz z z= ∈z … � and we denote 1
id n
iiz
== ∏n
z . We also define Mz as
1 2( , , , )dM M MM z z z=z … , where [ ]1 2 dM M M M= � is a d d× integer matrix with iM as its i th
column. For two sequences [ ]x n and [ ]y n in 2 ( )d� Z , we denote their inner product by ,x y . For an
operator 2 2:A →� � , we use asterisk to denote the adjoint (or transpose conjugate) of the operator; hence
we have *, ,Ax y x A y= . We use asterisk to denote * 1( ) ( )X X −=z z , which is the conjugate of ( )X z on
the unit circle and is equivalent to the time reverse of the sequence [ ]x −n . We use x to represent the
greatest integer less than or equal to x .
We denote a multivariate polynomial in continuous time as ( ) dp a+∈
=∑ m
mmt t
Z where d∈t � ,
1 2( , , , )dm m m=m … , and { }0,1d dim i d+ = ∈ ≥ ≤ ≤mZ Z . Similarly, we denote a discrete-time polynomial as
[ ]p a=∑ m
mmn n . The degree of a monomial m
t or mn is denoted as
1
d
iim
==∑m and we use NP for the
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
4
set of polynomials with degree less than N .
We represent the d d× sampling matrix by M and define det( )MM � . We show downsampling and
upsampling operators with the sampling matrix M by M↓ and M↑ , respectively.
B. Two-Channel Filter Banks and Wavelets
We consider a multidimensional two-channel critically-sampled filter bank as depicted in Fig. 1. Here
M is the sampling matrix with size d d× , ( )H z and ( )H z� are lowpass analysis and synthesis filter pair
and likewise, ( )G z and ( )G z� are the highpass filter pair. We can express the filter bank in the polyphase
domain using the polyphase matrices [27], [28]
0 1
0 1
( ) ( )( )
( ) ( )p
H H
G G
=
z zA z
z z and 0 1
0 1
( ) ( )( )
( ) ( )p
H H
G G
=
z zS z
z z
� �
� �, (1)
where for perfect reconstruction we have ( ) ( )Tp p I=A z S z . Further, the analysis and synthesis lowpass
filters are expressed as 1
0( ) ( )i M
iiH H
==∑ c
z z z and 1
0( ) ( )i M
iiH H
−=
=∑ cz z z� � , and similarly the highpass
filters as 1
0( ) ( )i M
iiG G
==∑ c
z z z and 1
0( ) ( )i M
iiG G
−=
=∑ cz z z� � where 0 1{ , }c c are cosets of the sampling
matrix M assuming 0 =c 0 .
By iterating the filter bank, we obtain the analysis scaling and wavelet functions obeying [18]:
( ) [ ] ( )d h Mϕ ϕ∈
= − −∑nt n t n
Z, and ( ) [ ] ( )d g Mψ ϕ
∈= − −∑n
t n t nZ
. (2)
[Thus the wavelet analysis filters are [ ]h −n and [ ]g −n , equivalent to *( )H z and *( )G z .] And also we
have / 2, ( ) ( )j j
j Mϕ ϕ− −= −n
t t nM , where j ∈Z is the scale and we define det( )MM � . Similarly, we have
synthesis scaling and wavelet functions as
Fig. 1. A two-channel multidimensional filter bank.
M
x
M
MM
( )H z
H
( )H z�
rx
( )G z ( )G z�
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
5
( ) [ ] ( )d h Mϕ ϕ∈
= −∑nt n t n�� �
Z, and ( ) [ ] ( )d g Mψ ϕ
∈= −∑n
t n t n� ��Z
. (3)
Vanishing moments in wavelets are critical regarding the regularity and also the approximation power
of wavelets [18], [22]. If the wavelet analysis and synthesis stages have N and N� vanishing moments,
respectively, we have
( ) 0dψ =∫m
t t t , for N<m , and ( ) 0dψ =∫m
t t t� , for N<m � .
These conditions are expressed for the highpass filters and polynomial ( )P z as
( ) *( ) ( ) 0M G P↓ =z z , for ( ) NP ∈z P , (4)
and
( ) ( ) ( ) 0M G P↓ =z z� , for ( )N
P ∈z P� . (5)
[Note that wavelet analysis highpass filter is *( )G z .] As a result, ( )G z has N zeros and ( )G z� has N�
zeros at =ω 0 , but ( )H z and ( )H z� have N� and N zeros at =ω π , respectively. Therefore, if N N> � , the
wavelet transform has more approximation power in the decomposition section and is more regular in
reconstruction. This setting is more desirable in compression since more regular filters are required to
compensate for the quantization in signal reconstruction [18].
C. Neville Filters
Neville filters is a notion introduced in honor of Neville who introduced 1-D polynomial interpolation
algorithm [21] and is referred to a class of interpolating filters, as described below.
Definition 1 [14]: A filter ( )L z is a Neville filter of order N and shift d∈τ � if the following
condition is satisfied:
[ ] [ ] [ ]l p p∗ = +n n n τ , for [ ] Np ∈n P . (6)
This condition is equivalent to having
[ ]( )l − =∑ m m
kk k τ , for N<m . (7)
Remark 1: Here we point out some of the important properties of the Neville filters we need later.
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
6
1) The filter [ ]l −n or *( )L z will be a Neville filter of order N and shift −τ if [ ]l n is a Neville filter of
order N and shift τ . 2) The upsampled version of a Neville filter of order N with shift τ [i.e.
( ) ( )MQ L=z z ] is also a Neville filter of order N but with shift M τ . 3) If 1( )L z and 2 ( )L z are Neville
filters of order 1N and 2N , and shift 1τ and 2τ , then 1 2( ) ( )L Lz z is a Neville filter of order 1 2min( , )N N
and shift 1 2+τ τ . 4) The monomial filter mz ( d∈m � ) is a Neville filter of shift m and order infinity.
Hence, one only needs to build Neville filters with shifts within a hypercube. Neville filters with different
shifts can be obtained using this property.
Remark 2: To find a Neville filter of order N and shift τ , one needs to solve a system of equations
given by (7). For a d -dimensional filter, there are 1N d
qd
+ − =
equations and hence the filter length
will be q .
As a result, a 1-D Neville filter of order N has N nonzero coefficients. In this case, the system of
equations has a Vandermonde matrix which is always invertible. We choose the support of the filter in the
vicinity of τ to avoid extrapolation [14]. Consequently, we select the support of 1-D Neville filters as
2 , ( 1) 2N N− − , where x denotes the greatest integer less than or equal to x . Below we show
that Neville filters with different supports can be obtained from the filters with support
2 , ( 1) 2N N− − .
Proposition 1: If ( )L z is a 1-D Neville filter of order N and shift τ having support on
2 , ( 1) 2N k N k− + − + ( k ∈� ), then 1( ) ( )kL z z L z= where 1( )L z is a Neville filters of order N and
shift kτ − which has the support on 2 , ( 1) 2N N− − .
Proof: This proposition is proved using the property 4 of Remark 1. □
An important class of filters are Deslauriers-Dubuc [10] filters. These filters are Neville filters with
shift 1/ 2 [14]; thus, they can interpolate polynomials at half integers. The coefficients of these filters
when the order N is even are obtained through 2 1
( ) 1( ) ( )
N k kN kk
R z a z z− +
== +∑ , where
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
7
2 1
1( 1) ( 2 0.5 )
( 2 )!( 2 1 )!( 0.5)
Nk N
ik
N ia
N k N k k
+ −=
− + −=
− − + −∏
.
Hence, for even N the filter ( ) ( )NR z is linear phase. Table I shows a few Deslauriers-Dubuc filters. In the
next section we show how we can benefit from Neville filters in order to design two-channel filter banks.
D. Kovačević’s Design for Two-Channel Filter Banks
A two-channel filter bank using a lifting framework provides perfect reconstruction for any kind of
lifting steps. Therefore, one can design filters by simply imposing vanishing moments on the filters and
derive the appropriate lifting steps. This has been done by Kovačević and Sweldens in [14] for filter
banks with two lifting steps (see Fig. 2). In this case if we express the filter bank in the polyphase domain
we have (note that the normalization factors 0K and 1K are not included)
1 ( ) ( ) ( )
( ) 1p
Q L Q
L
− = −
z z zA
z and
1 ( )
( ) 1 ( ) ( )p
L
Q Q L = − −
zS
z z z. (8)
The two-channel filter bank designed by Phoong et al. [19] is similar to the 1-D design of [14], where
they used a ladder network in the polyphase domain. This setting as pointed out by Ansari et al. [1] has a
drawback that the values of lowpass (and highpass) filter pair cannot be made the same at / 2ω π= (one is
1 and the other is 0.5 in the normalized 1-D filters of [19]). We will show that the same problem exists
TABLE I
DESLAURIERS-DUBUC FILTERS ( ) ( )NR z
N ( ) ( )NR z
1 1
2 (1 ) 2z+
3 1 3(3 6 ) 2z z−+ −
4 2 1 4( 9 9 ) 2z z z−− + + −
5 2 1 2 7(5 60 90 20 3 ) 2z z z z− −+ + − +
6 3 2 1 2 8(3 25 150 150 25 3 ) 2z z z z z− −− + + − +
Fig. 2. A two-channel multidimensional filter bank using two-step lifting scheme.
M
x
MM
( )Q z r
x
1cz
( )L z ( )Q z ( )L z
1−cz
M 0K
1K
1
0K
−
1
1K
−
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
8
for the general multidimensional design of two-channel filter banks proposed in [14]. Although in this
paper we focus to address this drawback (dissimilarity between analysis and synthesis filters) in our
design, we will show in Section IV that the designed filters not only are more symmetric, but also are
more regular, have lower frame bounds ratio, and provide better frequency selectivity.
Kovačević and Sweldens showed that by imposing N and N� vanishing moments to the wavelet
analysis and synthesis banks shown in Fig. 1 when using the two-step lifting step depicted in Fig. 2
results in the following solution:
Remark 3 (two-channel two-step lifting design) [14]: In order to have N and N� vanishing moments
in the two-channel wavelet analysis and synthesis banks, ( )L z should be a Neville filter of order N with
shift 10 1M
−=τ c , and *2 ( )Q z should be a Neville filter of order N� with shift 0τ , assuming that N N≥ � .
Remark 4: The lengths of the 1-D filters designed using two-step lifting scheme for N N≥ �
(explained in Remark 3) are
2, 1
length( ) length( )2 1, 1
NG H
N N
== = − >� and
2, 1length( ) length( )
2( ) 3, 1
NH G
N N N
== = + − >
��
.
Proof is straightforward noting that the length of 1-D Neville filters of order N is N . □
The constants 0K and 1K in Fig. 2 are for the appropriate normalization of the filters. Two examples
of the above framework are as follows. We will use these examples in order to compare to our design
Fig. 3. The frequency response of the analysis filters in Example 1.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
X: 0.5Y: 0.7906
Normalized Frequency
Magnitu
de
Analysis Filters
X: 0.5Y: 1.414
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
X: 0.5Y: 0.7906
Normalized Frequency
Magnitu
de
Synthesis Filters
X: 0.5Y: 1.414
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
9
approach.
Example 1: Consider a 1-D case where 2M = and thus 1/ 2τ = where we can use time-reversed
version of Deslauriers-Dubuc filters. Suppose that 3N = and 2N =� . Therefore, we can obtain the
analysis filters ( )2 2 20( ) 1 ( ) ( ) ( )H z K Q z L z zQ z= − + and ( )2
1( ) ( )G z K L z z= − + , using (3)( ) ( )L z R z= and
1(2)( ) (1/ 2) ( )Q z R z−= from Table I with 0 2K = and 1 2 2K = as
( ) 2 2 1 2( ) 2 32 ( 1) ( 2 2 14 3 )H z z z z z z− −= + − − + − and ( ) 1 3( ) 2 16 (1 ) ( 3)G z z z z−= − + .
We see that at / 2ω π= or z j= the magnitude of H and G are 2 and 10 4 . Fig. 3 shows the
frequency responses of the filters.
Example 2: In this example we consider the quincunx filter bank (QFB) design example in [14] with
4N = and 2N =� . Here, we can find that the filters evaluated at ( , )j j=z give ( , ) 2H j j = and
( , ) 2 2G j j = (see the frequency responses in Fig. 7).
III. THREE-STEP LIFTING DESIGN OF TWO-CHANNEL FILTER BANKS
In this section we address the drawback of the Kovačević’s Design by adding an additional lifting
step to the framework of Fig. 2. The proposed scheme is depicted in Fig. 4. Ignoring the normalization
factors, we can write the analysis and synthesis polyphase matrices in terms of the lifting steps as
1 ( ) 1 0 1 ( )
0 1 ( ) 1 0 1
1 ( ) ( ) ( ) ( ) ( ) ( ) ( ),
( ) 1 ( ) ( )
p
U L
Q
Q U L U L Q U
Q L Q
= −
− + − = − −
z zA
z
z z z z z z z
z z z
(9)
and
1 1 ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( ) 1 ( ) ( )T
p p
L Q Q
L U L Q U Q U− − = = − − + −
z z zS A
z z z z z z z. (10)
Fig. 4. A two-channel multidimensional filter bank using a three-step lifting scheme.
M
x
MM
( )Q z r
x
1cz
( )L z ( )Q z ( )L z
1−cz
M 0K
1K
1
0K
−
1
1K
−
( )U z ( )U z
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
10
Now we apply N vanishing moments to the wavelet analysis branch. Using (1), and (9) and we have
( )1( ) ( ) 1 ( ) ( )M M MG Q L Q= − + −cz z z z z .
Thus, by employing (4) we obtain
1 1[ ] [ ] [ ] [ ] [ ] [ ] 0q p M p M q l p M− − ∗ + − − − ∗ − ∗ − =n n n c n n n c ( Np ∈P ). (11)
Suppose that *( )Lk L z , where Lk is a constant, is a Neville filter of order N and shift 10 1M −=τ c , then
we have 1[ ] [ ] [ ]Lk l p M p M− ∗ − =n n c n . Hence, we can rewrite (11) as
1(1 1 ) [ ] [ ] [ ]Lk q p M p M+ − ∗ = −n n n c ( Np ∈P ),
which implies that *( )Qk Q z with
1 1Q Lk k= + , (12)
is a Neville filter of order N and shift 0−τ , or ( )Qk Q z is a Neville filter of order N and shift 0τ .
Now we impose N� vanishing moments on the analysis bank. We can obtain analysis highpass filter
by using (1) and (10) as
( )1( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( )M M M M M M MG L Q U L U Q U−= − − + −c
z z z z z z z z z� .
Applying (5) to the derived ( )G z� we have
1 1
[ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] 0 ( ).N
u q l p M l p M
u p M p M u q p M p
∗ ∗ ∗ − ∗ −
∗ + − − ∗ ∗ − = ∈
n n n n n n
n n n c n n n c P� (13)
If we assume N N≥ � , since *( )Lk L z and ( )Qk Q z are Neville filters with shift 0τ , we can reduce (13) to
1
1
1 1[ ] [ ] [ ]
1[ ] [ ] [ ] [ ] [ ] 0 ( ),
L Q L
NQ
u p M p Mk k k
u p M p M u p M pk
∗ − − −
∗ + − − ∗ = ∈
n n n c
n n n c n n P�
or
1[ ] [ ] [ ] ( ),U Nk u p M p M p∗ = − ∈n n n c P�
where by using (12) we have
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
11
2
2
2
1
LU
L
kk
k=
−. (14)
As a result, *( )Uk U z is a Neville filter of order N� and shift 10 1M −=τ c . We summarize the proposed
design in the following theorem.
Theorem 1: A two-channel multidimensional wavelet having three steps of lifting (see Fig. 4) will
have N and N� vanishing moments in the analysis and synthesis banks, respectively if the following
conditions are satisfied:
1) *( )Lk L z and ( )Qk Q z are Neville filters of order N and shift 10 1M −=τ c . Lk is a free parameter and
Qk is given by (12).
2) *( )Uk U z is a Neville filter of order N� and shift 10 1M −=τ c with Uk given in (14). □
As seen, in the proposed design we have a free parameter where we can adjust the filters to have more
symmetry and as a matter of fact, we expect filters with more regularity and other desirable properties
given in Section IV. For the above setting we assumed N N≥ � , which is more useful in compression as
described before. If we need to set N N≥� vanishing moments, we can reverse the directions and also
signs of the lifting steps ( )L z , ( )Q z , and ( )U z in the filter bank shown in Fig. 4.
In the next section we present a few design examples using the proposed scheme.
A. 1-D Filter Banks
In the 1-D case, since 2M = we have 0 1/ 2τ = , and hence, we can use the Deslauriers-Dubuc filters.
Thus, for a filter bank with N vanishing moments in the analysis bank and N� vanishing moments in the
synthesis stage assuming N N≥ � , we have *( )( ) ( )L Nk L z R z= or 1
( )( ) ( )L Nk L z R z−= , and ( )( ) ( )Q Nk Q z R z= ,
and finally 1
( )( ) ( )U N
k U z R z−= � . Using (9), (12), and (14), we derive the highpass analysis filter as
( ) ( )( )2 2 21 ( ) ( ) ( )( ) 1 ( ) 1 1 ( ) ( ) ( )Q N L Q N NG z K k R z z k k R z R z
− = − + −
,
equivalent to
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
12
( )2 2 21( ) ( ) ( )( ) ( ) 1 ( ) ( )
1L N L N N
L
KG z k R z z k R z R z
k
− = − + + − +, (15)
and the lowpass analysis filter is derived as
2 20 1( ) 1 ( ) ( ) ( )H z K U z G z K zL z = + + ,
or
2 2 2 201 ( )( )2
( ) 2 ( 1) ( ) ( ) 2 ( )2
L L L NN
L
KH z k k R z G z K z k R z
k
− − = + − + � . (16)
Note that we included positive constants 0K and 1K for proper normalization (see Fig. 4). From
Table I, since ( ) (1) 1NR = , we have
0(1) ( 1) / 2L LH K k k= + = , and 1( 1) 2 /( 1) 2L LG K k k− = + = . (17)
Now we can find the parameters by evaluating
( ) ( )H j G j= . (18)
In the case when both vanishing moments are of even order, we have ( ) ( )( 1) ( 1) 0N N
R R− = − =� , hence, (18)
yields 0 1K K= and from (17) we have 0 1 1K K= = , and 2 1Lk = + where this solution gives
( ) ( ) 1H j G j= = . If N or N� , or both are odd then we should solve (17) and (18) to find the parameters.
We can also obtain the synthesis filters using the following equations
1( ) ( )H z z G z−= − −� , and 1( ) ( )G z z H z−= −� .
Remark 5: The lengths of the 1-D filters designed using three-step lifting scheme for N N≥ � are
2, 1
length( ) length( )4 3, 1
NG H
N N
== = − >� and
2, 1length( ) length( )
4 2 5, 1
NH G
N N N
== = + − >
��
.
Here we point out that the design method of Ansari et al. [1] for regular filters is similar to the
proposed 1-D three-step lifting design when N N= � .
Below we provide examples for 1-D filter bank design using the proposed method.
Example 3: Consider the design criteria in Example 1, that is, we desire 3N = and 2N =� vanishing
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
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13
moments. By solving (17) and (18) we obtain the parameters as 2.2686Lk = , 0 0.9816K = , and
1 1.0188K = where we have ( ) ( ) 1.005H j G j= = . Fig. 5 shows the frequency responses of the filters
designed in this example. The symmetry of these filter pairs is clear when compared with the design using
two lifting steps and same number of vanishing moments (see Example 1 and Fig. 3).
Note that to avoid finding a solution for the parameters, we can set 0 1 1K K= = and 2.414Lk = where
they satisfy (17), but we obtain ( ) 1.021H j = and ( ) 0.9919G j = , which are very close together.
Example 4: In this example we design the smallest filter of this kind where 2N = and 1N =� . (Note
that the case 1N N= =� leads to the degenerate Haar filters.) Again we find the parameters by solving (17)
and (18) resulting in 2.1479Lk = , 0 0.9650K = , and 1 1.0363K = . The analysis filters are calculated as
2 1( ) 0.03( 1)( 3.2958 21.372 6.4846 )H z z z z z−= − + + − − , and 2 1( ) 0.0823( 1) ( 6.2958 )G z z z z−= − − + + .
In Fig. 6 we demonstrate the frequency responses of the designed filters.
We see that the design of 1-D filters using the proposed method is easier when we choose both the
order of vanishing moments even. In addition, having linear-phase lifting steps is another advantage of
this setting.
B. Quincunx Filter Banks
Quincunx filter banks (QFB) are 2-D two-channel filter banks where they can provide multiresolution
Fig. 5. The frequency responses of the filters in Example 3.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
Normalized Frequency
Magnitude
Synthesis Filters
X: 0.5Y: 1.005
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
X: 0.5Y: 1.005
Normalized Frequecy
Magnitu
de
Analysis Filters
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
14
analysis under certain regularity conditions. Suppose that we choose the sampling matrix as
1 1
1 1M
= − ,
where we have the nonzero coset vector as 1 (1, 0)T=c and hence 10 1 (1/ 2, 1/ 2)TM −= =τ c . As a result, we
should use Neville filters with shift 0τ . Table II shows the Neville filters with shift 0 (1/ 2, 1/ 2)T=τ using
the de Boor and Ron algorithm [7], [8] (also reported in [14]).
Similar to the 1-D case we choose * 1 1( ) ( ) 1 2( ) ( ) ( , )L N Nk L R R z z− −= =z z , ( )( ) ( )Q Nk Q R=z z , and
1
( )( ) ( )U N
k U R−=z z� . Therefore, the analysis filters are derived as
( )1( ) 1 ( ) ( )( ) ( ) 1 ( ) ( )
1
M M ML N L N N
L
KG k R z k R R
k
− = − + + − +
z z z z , (19)
and
2 201 1 ( )( )2
( ) 2 ( 1) ( ) ( ) / 2 ( )2
M ML L L NN
L
KH k k R G K z k R
k
− − = + − + z z z z� . (20)
We can find the parameters through (1,1) ( 1, 1) 2H G= − − = and ( , ) ( , )H j j G j j= . Note that due to the
symmetry of the filters, H and G on four points ( , )j j± ± and on the points connecting these four vertices
( , )j j± ± (which form a diamond) have same values.
In the next example we use these filters to design the quincunx filters.
Fig. 6. The frequency responses of the filters in Example 4.
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
X: 0.5Y: 1.036
Normalized Frequency
Magnitu
de
Analysis Filters
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
Normalized Frequency
Magnitu
de
Synthesis Filters
X: 0.5Y: 1.036
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
15
Example 5: Suppose that we desire a QFB with 4N = and 2N =� vanishing moments. Therefore, we
have the filters with order 2N = and 4N = as
2(2) 1 2 1 2 1 2( , ) (1 ) 2R z z z z z z= + + + ,
and
2 2 2 2 1 1 1 1 5(4) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2( , ) 10(1 ) ( ) 2R z z z z z z z z z z z z z z z z z z− − − − = + + + − + + + + + + + .
To find the proper parameters we first evaluate (1,1) ( 1, 1) 2H G= − − = , which leads to
0 1( 1) 2 ( 1) 2L L L LK k k K k k+ = + = ,
and the next requirement ( , ) ( , )H j j G j j= yields 0 1K K= [note that 11 2 1 2( , )M z z z z−=z ]. Therefore, we
have similar result to the one in the 1-D case when we have even orders of vanishing moments, that is,
0 1 1K K= = , and 2 1Lk = + . In Fig. 7 we depicted the frequency responses of the designed analysis filters
as well as the designed quincunx filters using Kovačević’s method given in Example 2.
IV. PROPERTIES OF THE DESIGNED FILTERS
In this section we investigate and compute some of the properties of the designed filters and compare
them to those of the two-step lifting design. We assume N N≤� and use 1 2Lk = + unless otherwise is
mentioned.
TABLE II
2-D NEVILLE FILTERS WITH SHIFT (1/ 2,1/ 2)T , ( ) ( )NR z
N ( ) ( )NR z
2 2
(1)( ) 2r z
4 5
(1) (2)10 ( ) ( ) 2r r − z z
6 9
(1) (2) (3) (4)174 ( ) 27 ( ) 2 ( ) 3 ( ) 2r r r r − + + z z z z
n ( ) ( )nr z
1 1 2 1 21 z z z z+ + +
2 2 2 2 2 1 1 1 11 2 1 2 1 2 1 2 1 2 1 2z z z z z z z z z z z z
− − − −+ + + + + + +
3 2 2 1 2 1 1 2 11 2 1 2 1 2 1 2z z z z z z z z
− − − −+ + +
4 3 3 3 3 2 2 2 21 2 1 2 1 2 1 2 1 2 1 2z z z z z z z z z z z z
− − − −+ + + + + + +
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
16
A. Symmetry
Symmetry between analysis and synthesis filters is a desirable property. It could be used as a measure
of near-orthogonality, although such measures fail at some cases [9]. In the last section we showed that
the proposed filters have same magnitude at z j= for 1-D case (and ( , )j j=z in the case of quincunx
filters). Here we use 2|| ||h h− � to measure the dissimilarity of the designed filters. Table III shows this
measure for normalized filters of both two-step and three-step lifting designs. As seen, the symmetry of
the proposed filters is higher (dissimilarity is lower) in all cases. Moreover, the symmetry is very high
when both N and N� are even.
We also found the optimal Lk for the proposed filters that minimizes 2|| ||h h− � . Since the calculations
are very cumbersome, we used symbolic toolbox of Matlab to find optimal Lk . The resulting values are
shown in Table IV. Except for a few cases, the symmetry measure is close to those of Table III for the
Fig. 7. The frequency responses of the quincunx analysis filters. Top: Two-step lifting with 4N = and 2N =� in
Example 2. Bottom: The proposed three-step lifting with 4N = and 2N =� in Example 5.
-1
0
1
-1
0
10
0.5
1
1.5X: 0.5Y: -0.5Z: 1
Normalized Frequency
Magnitude
-1
0
1
-1
0
10
0.5
1
1.5
X: 0.5Y: 0.5Z: 1
Normalized Frequency
Magnitu
de
-1
0
1
-1
0
10
0.5
1
1.5
2X: 0.5Y: -0.5Z: 1.414
Normalized Frequency
Magnitu
de
-1
0
1
-1
0
10
0.5
1
1.5X: 0.5Y: 0.5Z: 0.7071
Normalized Frequency
Magnitu
de
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
17
proposed filters. Interestingly, when N N= � the optimal Lk is 1 2+ which is used as the default value.
B. Regularity
As we observed previously, the proposed filter design yields more symmetric wavelet filters and
hence, we expect they have more regularity. To determine regularity, we measure Sobolev index as
proposed in [11] and [29].
Table V shows the measured Sobolev regularity for both two-step and three-step lifting design
methods for 1-D analysis lowpass filters. Negative values indicate that the wavelet filter is not stable. For
three-step lifting design we always used 0 1 1K K= = and 2 1Lk = + . The results for the three-step lifting
design, shown in boldface, outperform those of the designed filters using two-step lifting for all cases. To
compare the filters with same lengths, consider two-step lifting design with 4N N= =� (to have a fair
comparison we consider same number of vanishing moments of analysis and synthesis filters) which has a
filter length of 13 and the three-step design with 3N N= =� having same filter length (see Remarks 4 and
5). We observe that the proposed design has better regularity (1.61 vs. 1.18 ). Another example showing
better regularity of the three-step design is when 6N N= =� leading to length 21= for the two-step design
as compared to the proposed design with 4N N= =� having a smaller length of 19 (1.77 vs. 1.94 ).
We also compared the regularities for a few designed quincunx filters shown in Table VI using the
TABLE III
DISSIMILARITY MEASURE (3 2
10 || ||h h− � ) FOR BOTH
1-D TWO-STEP AND THREE-STEP
(SHOWN IN BOLDFACE) LIFTING DESIGNS
NN
� 1 2 3 4 5 6
500 1
500
250 93.75 2
42.89 0.46
296.88 105.47 116.58 3
161.63 51.20 86.47
257.81 83.50 89.05 65.05 4
43.00 0.60 6.97 0.34
284.36 91.51 97.04 69.86 74.68 5
110.74 24.18 48.75 22.35 38.96
267.06 82.13 85.26 59.81 63.22 52.63 6
44.27 1.84 6.74 0.11 3.22 0.28
TABLE IV
OPTIMAL Lk FOR MAXIMUM SYMMETRY AND THE DISSIMILARITY
MEASURE (IN BOLDFACE) FOR 1-D THREE-STEP LIFTING DESIGN
NN
� 1 2 3 4 5 6
- 1
500
1.7267 2.4142 2
30.207 0.46
1.0011 2.7357 2.4142 3
116.58 50.61 86.47
1.5914 2.2838 2.1745 2.4142 4
24.697 0.44 6.41 0.34
1.1587 2.3348 2.0365 2.5795 2.4142 5
71.992 24.13 47.93 22.21 38.96
1.5175 2.179 2.0912 2.3489 2.2732 2.4142 6
22.12 1.34 5.77 0.08 3.08 0.28
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
18
method proposed in [20]. At some instances the regularity algorithm fails to find a solution mask in 2L
which we indicated by “nr” in the table. Again we observe that the proposed approach results in more
regular filters for quincunx filters. A major application of quincunx filters is in the construction of
directional filter banks (DFB) [2]. In [12] we showed that the HWD (Hybrid Wavelets and Directional
filter banks) transform utilizing DFBs constructed with three-step lifting quincunx filters (in comparison
to HWD with DFBs having two-step lifting) demonstrates less ringing artifacts and greater PSNR values
in nonlinear approximation.
C. Frame Bounds
Orthogonality of a basis is desirable in applications such as compression. Energy is preserved in such
transforms and hence, the design of quantizers is not difficult. Relaxing the orthogonality criterion,
however, makes the design of filter banks more flexible. To measure the degree of non-orthogonality, we
use frame bounds which give energy preserving property for bases and dictionaries [9]. For the two-
channel filter bank shown in Fig. 1, the set of { }[ ] [ (2 )], [ ] [ (2 )]m mh n h m n g n g m n= − − = − − form a frame in
2 ( )� � if we have [3], [18]
2 22 2
, ,m mmA x x h x g B x
∞
=−∞≤ + ≤∑
for 0B A≥ > . (This definition is easily extended to multidimensional multi-channel filter banks.)
There are a few approaches to find the frame bounds [3], [9], where we use the one proposed in [3].
TABLE V
REGULARITY OF 1-D LOWPASS ANALYSIS FILTERS FOR BOTH
TWO-STEP AND THREE-STEP (SHOWN IN BOLDFACE) LIFTING DESIGNS
NN
� 1 2 3 4 5 6
0.5 1
0.5
-2.38 0.44 2
-0.92 1.07
-6.49 -1.15 1.20 3
-5.11 0.46 1.61
-10.79 -5.48 -0.98 1.18 4
-9.31 -3.83 0.66 1.94
-15.12 -9.72 -5.29 -0.40 1.77 5
-13.45 -7.93 -3.38 1.50 2.40
-19.44 -14.02 -9.58 -4.68 -0.14 1.77 6
-17.53 -11.95 -7.37 -2.41 2.10 2.63
TABLE VI
REGULARITY OF QUINCUNX FILTERS FOR BOTH
TWO-STEP AND THREE-STEP LIFTING DESIGNS
NN
� 2 4
nr 2
2.150
nr 1.059 4
2.038 2.324
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
19
Defining the frame operator in the polyphase domain as
* *( ) (1/ ) ( )p pz z z=F A A
where *(.) denotes the conjugate transpose and ( )p zA is the analysis polyphase matrix for the normalized
filters, we can find the frame bounds A and B as the infimum and supremum of the eigen values of F on
the unit circle 2j fz e π= [3]. Hence, we first form 2 * 2 2( ) ( ) ( )j f j f j fp pe e eπ π π=F A A (using symbolic toolbox
of Matlab) and then evaluate 2( )j fe πF for several frequency points uniformly distributed in 0 1f≤ ≤ . At
each frequency we find eigen values of 2( )j fe πF . Then A and B are found as the minimum and
maximum of all the eigen values computed in the last step, respectively.
In Table VII we show the frame bounds as well as their ratio B A for the proposed and two-step filter
bank designs. As seen, B A for the proposed design are significantly less than those for the two-step
lifting filters (even when filters with same length are considered) and are close to 1 at most cases which
indicates that the designed filters are near-orthogonal.
D. Energy of the Error
The last property we investigate for the designed filters is the frequency selectivity. In particular, we
measure the energy of the error between the designed normalized (lowpass) filter and the ideal one. We
define the total energy of the error as
TABLE VII
FRAME BOUNDS AND THEIR RATIO FOR 1-D LOWPASS ANALYSIS FILTERS FOR BOTH
TWO-STEP AND THREE-STEP (SHOWN IN BOLDFACE) LIFTING DESIGNS
NN
�
1
A B B/A
2
A B B/A
3
A B B/A
4
A B B/A
5
A B B/A
6
A B B/A
1 1 1 1
1 1 1
0.38 2.62 6.85 0.5 2 4 2
0.66 1.51 2.28 0.95 1.05 1.10
0.57 1.77 3.12 0.46 2.16 4.68 0.59 1.69 2.85 3
0.78 1.28 1.64 0.85 1.18 1.39 0.95 1.05 1.10
0.38 2.62 6.85 0.5 2 4 0.46 2.16 4.68 0.5 2 4 4
0.66 1.51 2.28 0.95 1.06 1.12 0.81 1.23 1.51 0.95 1.05 1.10
0.48 2.07 4.29 0.48 2.09 4.38 0.56 1.77 3.15 0.48 2.09 4.38 0.55 1.82 3.31 5
0.74 1.35 1.83 0.89 1.13 1.27 0.91 1.09 1.20 0.89 1.12 1.26 0.95 1.05 1.10
0.38 2.62 6.85 0.5 2 4 0.46 2.16 4.68 0.5 2 4 0.48 2.09 4.38 0.5 2 4 6
0.66 1.51 2.28 0.91 1.11 1.22 0.81 1.23 1.51 0.98 1.03 1.05 0.86 1.17 1.36 0.95 1.05 1.10
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
20
2 2/ 2
0 / 21 ( ) ( )
j jE H e d H e d
π πω ω
πω ω= − +∫ ∫ .
From Tables VIII and IX we observe that the proposed designed filters for both 1-D and quincunx
settings provide better frequency selectivity when compared to the two-step lifting design. This is valid
even if we consider filters with same length.
V. MULTI-CHANNEL FILTER BANKS
In this section we extend our analysis to a general perfect reconstruction multi-channel filter bank.
Fig. 8 illustrates a critically-sampled M -channel filter bank where ( )H z and ( )H z� are the lowpass filter
pair and ( )iG z (1 1i≤ ≤ −M ) along with ( )iG z� represent highpass filters in the analysis and synthesis
banks, respectively and det( )M=M . Again, we can express the filters in the polyphase domain as
0 1
1,0 1, 1
1,0 1, 1
( ) ( )
( ) ( )( )
( ) ( )
p
H H
G G
G G
−
−
− − −
=
z z
z zA z
z z
M
M
M M M
�
�
�
and
0 1
1,0 1, 1
1,0 1, 1
( ) ( )
( ) ( )( )
( ) ( )
p
H H
G G
G G
−
−
− − −
=
z z
z zS z
z z
M
M
M M M
� ��
� ��
� ��
, (21)
The analysis and synthesis lowpass filters are expressed as 1
0( ) ( )j M
jjH H
−
==∑
cz z z
M and
1
0( ) ( )j M
jjH H
− −
==∑
cz z z
M� � , whereas 1
0( ) ( )j M
i ijjG G
−
==∑
cz z z
M and
1
0( ) ( )j M
i ijjG G
− −
==∑
cz z z
M� � (1 1i≤ ≤ −M )
show how we can derive the highpass filters from the polyphase components. Here 0 1{ }j j≤ ≤ −cM
are cosets
TABLE VIII
ERROR OF 1-D LOWPASS ANALYSIS FILTERS FOR BOTH
TWO-STEP AND THREE-STEP (SHOWN IN BOLDFACE) LIFTING DESIGNS
NN
� 1 2 3 4 5 6
0.100 1
0.100
0.284 0.150 2
0.097 0.066
0.208 0.138 0.087 3
0.090 0.057 0.051
0.294 0.136 0.124 0.104 4
0.089 0.055 0.050 0.045
0.246 0.131 0.095 0.099 0.076 5
0.086 0.050 0.046 0.042 0.039
0.301 0.130 0.120 0.098 0.093 0.084 6
0.085 0.049 0.046 0.041 0.038 0.036
TABLE IX
ERROR OF QUINCUNX ANALYSIS FILTERS FOR BOTH
TWO-STEP AND THREE-STEP LIFTING DESIGNS
NN
� 2 4
0.2372 2
0.18034
0.22723 0.23164 4
0.18632 0.19332
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
21
of the sampling matrix M assuming 0 =c 0 .
Similar to what we observed for designing a two-channel filter bank in the last section, we can design
multi-channel filter banks by imposing the desired number of vanishing moments to the highpass
channels. Applying N and N� vanishing moments to the filter bank depicted in Fig. 8 is equivalent to
conditioning the highpass filters as
( ) *( ) ( ) 0iM G P↓ =z z (1 1i≤ ≤ −M ), for ( ) NP ∈z P , (22)
and
( ) ( ) ( ) 0iM G P↓ =z z� (1 1i≤ ≤ −M ), for ( )N
P ∈z P� . (23)
Next we briefly review the multi-channel two-step lifting scheme design by Kovačević and Sweldens
[14] and propose our three-step lifting design subsequently.
A. Multi-Channel Two-Step Lifting Design by Kovačević
Fig. 9 shows a multi-channel filter bank using two-step lifting scheme which is used as a framework
to build filter banks with any number of vanishing moments in [14]. Ignoring the normalization factors,
we can form the analysis polyphase matrix as
1
1 1 1 11
1 1
1 1
1 0 01 1
1 00 1 0 1 0( )
0 10 0 1 0 1
i ii
p
Q Q Q L Q Q
L L
L L
−− −=
− −
− − − = = − −
∑A z
M
M M
MM
�� �
�� �
, (24)
and the synthesis matrix is found to be
Fig. 8. A single-level multi-channel filter bank.
M
x
M
1( )G z
H
M M
M M 1( )G − zM
H
( )H z
H
1( )G − zM�
1( )G z�
( )H z�
rx
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
(Draft) September 23, 2009
22
1 2 1
1 1 1 1 2 1 1
2 2 1 2 2 2 1
1 1 1 1 1
1
1
( ) 1
1
p
L L L
Q Q L Q L Q L
Q Q L Q L Q L
Q Q L Q L
−
−
−
− − − −
− − − − = − − − − − − −
S z
M
M
M
M M M M
�
�
�
. (25)
Remark 6 (multi-channel two-step lifting design) [14]: The authors in [14] proved that if ( )iL z
(1 1i≤ ≤ −M ) are Neville filters with shift 1i iM −=τ c and order N , and *( )iQ zM (1 1i≤ ≤ −M ) are
Neville filters with shift 1i iM −=τ c and order N� , then the filter bank will have N and N� vanishing
moments in the analysis and synthesis banks.
Using a similar strategy as in the case of two-cannel filter banks, we extend the above design to have
a more flexibility as explained next.
B. Multi-Channel Three-Step Lifting Design
Having three steps of lifting in a multi-channel filter bank as shown in Fig. 10 provides more degrees
of freedom in developing the filters.
In this setting we can find the analysis polyphase matrix as
1 1 1 1
1
1
1 1 1
1 1 1 1 1 11 1 1
1 1 1 1 2 1 1
2 2 1 2 2 2
1 0 01 1
1 00 1 0 0 1 0( )
0 10 0 1 0 0 1
1
1
1
p
i i i i i ii i i
U U L L
Q
Q
U Q L U L U Q L U L U Q
Q Q L Q L Q L
Q Q L Q L Q L
− −
−
− − −− − −= = =
−
−
−
= −
− + − + −− − − −
= − − − −
∑ ∑ ∑
A z
M M
M
M M M
M M M
M
M
�� �
�� �
�
�
� 1
1 1 1 1 2 1 11Q Q L Q L Q L− − − − −
− − − − M M M M M
�
, (26)
and similarly the synthesis matrix is obtained as
Fig. 9. A multi-channel multidimensional filter bank using a two-step lifting scheme.
M
x MM
1( )Q z
rx1c
z
1( )L z
1−cz
M 0K
1K
1
0K
−
1
1K
−
MM
1( )Q − z
M
1−cz M
1( )L − z
M
1−−c
z M 1K −M
1
1K
−−M
1( )Q z
1( )L z
1( )L − z
M
1( )Q − z
M
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
23
1
1 2 111
1 1 1 1 1 1 2 1 1 11
1
1 1 1 1 1 2 1 1 11
1
1( ) ( ( ))
1
i ii
T i iip p
i ii
L Q Q Q Q
L U U L Q Q U Q U Q U
L U U L Q Q U Q U Q U
−−=
−− −=
−− − − − − − −=
− − − + − − −= = − − + − − −
∑∑
∑
S z A z
M
M
M
M
M
M M M M M M M
�
�
�
. (27)
As a result, for the three-step lifting framework of Fig. 10 we can express the analysis highpass filters
using (26) as
1
0
1
0
( ) ( ) (1 1)
( ) ( ) ( ) ,
j
j i
Mi ijj
M M Mi i jj
G G i
Q Q L
−
=
−
=
= ≤ ≤ −
= − − +
∑
∑
c
c c
z z z
z z z z z
M
M
M
(28)
and using (27) the synthesis filters are obtained as
1
0
1
0
1
0
( ) ( ) (1 1)
( ) ( ) ( ) ( ) ( )
( ) ( ).
j
ji
Mi ijj
M M M M Mi i i j jj
M Mi jj
G G i
L U U L Q
U Q
− −
=
−
=
− −−=
= ≤ ≤ −
= − − +
+ −
∑
∑
∑
c
cc
z z z
z z z z z
z z z z
M
M
M
M� �
(29)
Now to design the filters we would like to have N vanishing moments in the analysis bank and N�
vanishing moments in the synthesis bank. For the former one we employ (22) to (28)
1
1[ ] [ ] [ ] [ ] [ ] [ ] 0i i j j ij
q p M q l p M p M−
=− − ∗ − − ∗ − ∗ − + − =∑n n n n n c n c
M (1 1i≤ ≤ −M , Np ∈P ). (30)
Assuming that [ ]L ik l −n (1 1i≤ ≤ −M ) are Neville filters of order N and shift 1i iMτ −= c , or
[ ] [ ] [ ]L i ik l p M p M− ∗ = +n n n c (1 1i≤ ≤ −M , Np ∈P ), (31)
we can write (30) as
( )1 [ ] [ ] [ ]L i ik q p M p M+ − ∗ = −n n n cM ,
Fig. 10. A multi-channel multidimensional filter bank using a three-step lifting scheme.
M
x MM
1( )U z
rx1c
z
1( )Q z
1−cz
M 0K
1K
1
0K
−
1
1K
−
MM
1( )U − z
M
1−c
z M
1( )Q − z
M
1−−cz M 1
K −M 1
1K
−−M
1( )U z
1( )Q z
1( )Q − z
M
1( )U − z
M
1( )L z
1( )L − z
M
1( )L z
1( )L − z
M
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
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or
[ ] [ ] [ ]Q i ik q p M p M∗ = +n n n c (1 1i≤ ≤ −M , Np ∈P ). (32)
Consequently, [ ]Q ik q n (1 1i≤ ≤ −M ) are Neville filters of order N and shift 1i iMτ −= c , where
1Q Lk k= +M . (33)
Now we need to find the filters [ ]iu n (1 1i≤ ≤ −M ) to conclude our multi-channel filter design. This
is accomplished through imposing N� vanishing moments as stated in (23) to the synthesis highpass filters
given in (29):
1
1
1
1
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] 0
i i i j jj
i i j jj
l p M u p M u l q p M
p M u q p M
−
=
−
=
− ∗ − ∗ − ∗ ∗ ∗
+ − − ∗ ∗ − =
∑
∑
n n n n n n n n
n c n n n c
M
M (1 1i≤ ≤ −M ,
Np ∈P� ). (34)
Using (31) and (32) and the fact that N N≥ � , we can reduce (34) to
1 1[ ] [ ] [ ] [ ] [ ]
1[ ] [ ] [ ] 0,
i i i
L Q L
i i
Q
p M u p M u p Mk k k
p M u p Mk
−− − − ∗ − ∗
−+ − − ∗ =
n c n n n n
n c n n
M
M
and after applying (33) it turns out that
[ ] [ ] [ ]U i ik u p M p M− ∗ = +n n n c (1 1i≤ ≤ −M , N
p ∈P� ), (35)
where
2
21 (2 )
LU
L L
kk
k k=
− + − −
M
M M. (36)
We summarize our multi-channel design in the next theorem.
Theorem 2: A multi-channel multidimensional filter bank with three steps of lifting (see Fig. 10)
will have N and N� vanishing moments in the analysis and synthesis banks, respectively if the following
conditions are satisfied:
1) *( )L ik L z (1 1i≤ ≤ −M ) and ( )Q ik Q z are Neville filters of order N and shift 1i iM −=τ c . Similar to
the two-channel case, Lk is a free parameter, and Qk is given by (33).
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
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2) *( )U ik U z are Neville filters of order N� and shift 1i iM −=τ c with Uk given in (36). □
Therefore, we have a free parameter Lk that can be employed in designing the wavelet filters. In
contrast to the two-channel wavelets, setting this parameter does not always simply lead to more
symmetric filters in the multi-channel case. We can, however, change the filters responses using Lk . In
the following we present an example demonstrating this fact.
C. Design Example
Example 6: In this example we design filters of a four-channel filter bank with sampling matrix
2 0
0 2M
= . The nonzero coset vectors are 1 (1,0)T=c , 2 (0,1)T=c , and 3 (1,1)T=c . Hence, the
corresponding shifts are 1 (1/ 2,0)=τ , 2 (0,1/ 2)=τ , and 3 (1/ 2,1/ 2)=τ . For 1τ and 2τ we can use 1-D
Deslauriers-Dubuc filters (see Table I) in 1z and 2z , and for the shift 3τ the Neville filters are given in
Table II. Here we design wavelet filters for both two-step and three-step lifting schemes with 4N = and
2N =� vanishing moments in the analysis and synthesis banks.
For the two-step lifting scheme as mentioned in Section V-A (Remark 6), we have 1 (4) 1( ) ( )L R z=z ,
2 (4) 2( ) ( )L R z=z , and 3 (4)( ) ( )L R=z z , and likewise the update filters are expressed as 11 (2) 1( ) ( ) / 4Q R z−=z ,
12 (2) 2( ) ( ) / 4Q R z−=z , and 1
3 (2)( ) ( ) / 4Q R −=z z . Now we can form the analysis filters ( )H z and ( )iG z
(1 3i≤ ≤ ) using the polyphase matrix given in (24) and the constants iK . To find these constants, we set
(1,1) 2H = =M and ( 1, 1) 2iG − − = . Since ( )iL z ( 0 3i≤ ≤ ) and 4 ( )iQ z are one at (1,1)=z , we have
Fig. 11. The frequency response of the highpass analysis filters of Example 6 using a two-step lifting scheme.
-1
0
1
-1
0
10
1
2
ω1
G1
ω2
Magnitude
-1
0
1
-1
0
10
1
2
ω1
G2
ω2
Magnitude
-1
0
1
-1
0
10
1
2
ω1
G3
ω2
Magnitude
ESLAMI AND RADHA: DESIGN OF REGULAR WAVELETS USING A THREE-STEP LIFTING SCHEME
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0(1,1)H K= and ( 1, 1) 2i iG K− − = resulting in 0 2K = and 1iK = (1 3i≤ ≤ ). Fig. 11 shows the frequency
responses of the resultant analysis highpass filters.
Now we provide a possible wavelet design using our proposed three-step lifting for this example.
Using Theorem 2, we choose the first set of lifting filters as 11 (4) 1( ) ( )Lk L R z−=z , 1
2 (4) 2( ) ( )Lk L R z−=z , and
13 (4)( ) ( )Lk L R −=z z . And the second-step filters are expressed as 1 (4) 1( ) ( )Qk Q R z=z , 2 (4) 2( ) ( )Qk Q R z=z , and
3 (4)( ) ( )Qk Q R=z z . The third set of filters are Neville filters of order 2N =� defined as 11 (2) 1( ) ( )Uk U R z−=z ,
12 (2) 2( ) ( )Uk U R z−=z , and 1
3 (2)( ) ( )Uk U R −=z z .
We can derive the analysis filters using the corresponding polyphase matrix given in (26) and
incorporating the constants iK ( 0 3i≤ ≤ ). In addition to these constants we have a free parameter that we
can set to obtain our desirable filter set. One possible constraint is to equate the analysis (or synthesis)
filter magnitudes at ( , )j j=z , which yields 0iK K= (1 3i≤ ≤ ). By setting (1,1) 2H = and 3 ( 1, 1) 2G − − = ,
and using the relations between constants Lk , Qk , and Uk given in (33) and (36), we obtain 1iK =
( 0 3i≤ ≤ ) and 1Lk = − which accordingly we can find the filters.
In Fig. 12 we illustrate the magnitude responses of the resulting highpass analysis filters. The filters
in this design provide a frequency partitioning similar to the separable wavelets and since they form
disjoint passbands, the proposed design would be advantageous to the two-step design.
Fig. 12. The frequency response of the highpass analysis filters of Example 6 using the proposed three-step lifting
scheme.
-1
0
1
-1
0
10
1
2
ω1
G1
ω2
Magnitude
-1
0
1
-1
0
10
1
2
ω1
G2
ω2
Magnitude
-1
0
1
-1
0
10
1
2
ω1
G3
ω2
Magnitude
Submitted to IEEE TRANSACTIONS ON SIGNAL PROCESSING
(Draft) September 23, 2009
27
VI. CONCLUSION
In this work we proposed a design method of multidimensional wavelet filters with any order of
vanishing moments based on a triplet of lifting steps. Our design approach provides one more degree of
freedom when compared to the wavelet filter design using two steps of lifting scheme. One can use this
degree of freedom in better shaping the frequency responses of the wavelet filters. For two-channel
wavelets we employed this parameter in order to obtain more symmetry between analysis and synthesis
filters. The proposed filter design in comparison to the lifting design with two steps provides higher
regularity, lower frame bounds ratio, and lower energy of the error leading to better frequency selectivity.
Further, we extended our design for multidimensional multi-channel filter banks of any order of vanishing
moments. Again, using three steps of lifting provides more flexibility in filter design.
ACKNOWLEDGMENT
The authors would like to thank Professors A. Ron, Z. Shen, and K. Toh for providing us with the
Sobolev regularity analysis algorithm code of [20]. We also thank H. Ojanen for the software to measure
Sobolev regularity of 1-D filters.
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